US History Overview 2 - Reconstruction to the Great Depression

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Language: en

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- [Instructor] Where we
left off in the last video,

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the North had just won the Civil War.

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Unfortunately for Abraham Lincoln,

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it was two months after
he was assassinated.

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But now the North was dominant

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and essentially occupied the South.

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And we enter a period
called Reconstruction,

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and Reconstruction can
refer to one of two things

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and they're somewhat related.

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One is just the
Reconstruction from the war.

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Obviously there was a lot of
damage done on both sides.

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But it's usually referred
to the actual Reconstruction

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of the South and to some degree,

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kind of the reform of the South.

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I'm going to glaze over a lot of details,

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like I did in the last video.

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I might ignore some major events

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that you might find important.

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And I'll get back to them, don't worry.

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But the three big things

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that happened during Reconstruction,

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other than the fact of the
North occupying the South

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and essentially, to a large degree,

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suspending democracy in the South

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and installing its own
politicians, its own lawmakers,

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is that the United States passed

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the 13th, 14th, and 15th Amendments.

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These are known as the
Reconstruction Amendments.

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In 1865, you have the 13th Amendment

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and this abolished slavery.

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Let me write this here.

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This ended slavery.

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We talked about the
Emancipation Proclamation

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and that was essentially Abraham
Lincoln's executive order,

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this was a speech he made.

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But now it became official law in 1865.

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Then in 1868, you have the 14th Amendment

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which made everyone,

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every person born in the
United States a citizen,

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and this includes the freed slaves,

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so it's kind of like,
the slaves are now free

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and they are also citizens.

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And then in 1870, you
have the 15th Amendment

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which gave all free men the right to vote.

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And obviously, now all men were free.

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There were no non-free men.

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So, the right to vote.

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And I emphasize the men
because even at this point,

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women did not have the right to vote.

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The right to vote.

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And the 14th Amendment also
introduced a due process which,

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I won't go into the details
here, but it essentially said,

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"Look, the government has
to go under a due process

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"where it's subject to its
own laws when determining

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"whether it can take away property from

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"or in some way,

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"infringe on rights of other people."

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But we'll probably do a whole
video on that in the future.

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But these were the real takeaways.

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So it really brought the
former slaves, at least by law,

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by these amendments, on equal standing.

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But we know that in
practice, that didn't happen.

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And you fast forward to
1877, and you essentially

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have the Reconstruction
period formally ending.

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The occupation of the South formally ends.

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And as soon as the occupation
of the South formally ends,

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and democracy comes
about, you have a bunch

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of people coming to power.

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And at this point of time,

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the Republicans were
essentially the North,

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and these were the people
who were anti-slavery.

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Abraham Lincoln was a Republican.

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And the Democrats come
to power in the South.

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And we can talk about how
the different perceptions

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of the different parties change over time.

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But at this point, as soon
as the occupation ended,

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and I put democracy in quotes
because even in this period,

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the North has essentially
not occupied anymore,

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but the elections, these were things

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that were heavily contested.

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You have both sides of
them exerting force.

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And in particular, you
have the Jim Crow Laws

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being passed in the South.

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And they're called the Jim Crow based

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on this parody in the early 1800s.

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It was a practice for

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white men in the South at this time,

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or even well before this,
in order to parody blacks

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they would paint their face black

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and they would act silly and all of this.

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And Jim Crow was the name
of one of these characters

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that was portrayed in the early 1800s.

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I think it was Jumping
Jim Crow, was the name.

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That's where the laws come from.

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But the Jim Crow Laws essentially

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segregated blacks and whites in the South.

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Even though the idea might
have been that they were equal,

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the reality were that the
conditions for blacks,

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the places that they were
separated to, were far inferior.

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They had to use separate
drinking fountains,

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they had to use separate bathrooms.

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They couldn't sit in the
same parts of theaters

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or in the same parts of buses.

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And these lasted all the way until

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the Civil Rights Movement,
all the way to the 1960s.

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Now, at the same time that
all of this was happening,

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you kinda had this post Civil
War boom in the economy,

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where you had this massive building

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of the railroads and steam engines.

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To some degree, it was the first,

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well, I don't wanna say the first.

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There was many ages of mass innovation.

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But all of these things
tend to always lead

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to a little bit of a bubble.

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And then in 1873, what
you have happening is

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a lot of the governments of
the world start going off

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of the gold and silver standard,

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and they go to the gold standard.

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And what then happens is,
is that anyone who's left

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on the silver standard or partially both,

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the gold and silver standard,

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their currency would devalue.

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And back then, it was viewed
as an unambiguous negative

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for your currency to devalue.

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We can later talk about that,
there's more nuance there.

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So the United States
decide to follow suit.

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And actually, the big
actor here was Germany

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that decided to go off
of the silver standard

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and going on pure gold standard.

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And so the United States
decide to follow suit

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with the Coinage Act in 1873.

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But this leads to a huge, they
call it the Panic of 1873.

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There's a couple of things here.

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One, it completely demolishes
the price of silver,

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although this was already
happening on a global basis.

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It hurts the silver miners
and the industries associated

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with the silver miners.

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But I guess more importantly,

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now this restricts the money supply.

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And I won't go into all
of the economics of it.

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When you restrict the money supply,

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you essentially increase interest rates,

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and it essentially popped
the bubble that was forming

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due to the railroads and
all of the booming business.

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And then you essentially have

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the United States entering a depression.

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And that depression lasts from 1873,

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when the Coinage Act and
kind of this bubble burst,

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all the way to 1879.

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But lucky for the United
States, after that time period,

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after it recovered from the depression,

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it actually recovered
at this super fast rate,

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and this was one of the
fastest economic growths

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in US history.

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You had this huge influx of immigrants,

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tens of millions, from Europe.

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And by 1890, the United States
was now the richest country

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in the world on a per capita basis,

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which is amazing because
only a hundred years ago,

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it was kind of this
colony of Great Britain,

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or part of the British Empire.

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It was kind of this thing that

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the European powers didn't
view as that relative.

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But now it was the richest
country in the world.

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And then you fast forward to 1898.

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And it starts to, essentially,
become a bit of an empire.

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Until this time, United
States kind of kept to itself.

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It wasn't really interested

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in controlling other
nations or other people.

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But in 1898,

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until 1898,

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Cuba was a Spanish colony.

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And there had been many revolts try

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against the Spanish by the Cubans.

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And the United States, or the Americans,

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were fairly sympathetic to the Cubans.

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After all, here's another
country in the New World,

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again, trying to revolt
against a European power.

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And the Spanish were pretty infamous

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for cracking down pretty hard.

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So in 1898, while there was
a revolt against the Spanish,

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the United States set some

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ships over to Havana Harbor,

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essentially to protect American interests.

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This might resonate a little bit relative

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to maybe the Mexican-American War,

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that we kind of send things
close to another country

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to protect our interests

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and make sure nothing crazy happens.

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Then while in Savannah
Harbor, not Savannah,

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while in Havana Harbor,
you have a battleship,

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a US battleship called
the Maine, the Maine,

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that explodes and sinks.

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And this is an actual picture of it.

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This is fun because we're
entering the point in history

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where pictures start to become relevant.

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Although, even in the mid
1860s, you had pictures.

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That's a picture of Abraham Lincoln.

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The Maine gets sunk.

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The people who wanna
declare war on Spain say,

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"Hey, Spain must've blown up the Maine."

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Although it's still a complete mystery

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on what was the actual cause.

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Some people say it was
just a random explosion.

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There's even conspiracy
theorists who believe

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that the United States did
it to itself intentionally

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to justify entering the
war, while some say,

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"Hey, no. Spain did it
for whatever reason.

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"It didn't like this American
fleet in Havana Harbor."

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But regardless to say,
after this happened,

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it allowed, it made the
American public angry,

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the American government angry,

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and they declared war on Spain.

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And it was actually a
very short-lived war.

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They won pretty handedly.

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The big takeaway from
the Spanish-American War

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is that the United States
essentially became an empire.

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It started to have control
of other countries,

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and in particular, it had
temporary control of Cuba.

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But it also, because it
won, it got control of Guam,

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which is right over there, and
it still has control of Guam.

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It also got control of the
Philippines from Spain,

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and it maintained control
of the Philippines

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until the end of World War II.

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And it got control of Puerto Rico,

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which is still part of the United States.

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It's not an official state

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but it is United States territory.

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So at this point, the United
States becomes an empire.

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And then you fast forward to 1914.

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War breaks out in Europe.

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I need to do a whole series
of videos on World War I.

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But war breaks out in Europe.

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Particularly, the two strongest powers

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that are really at each
other at this time period,

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are the British Empire and Germany.

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You have this situation
where United States

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is trying its hardest to stay neutral.

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Obviously, the American
people were predominantly

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of English descent, it's an
English speaking country.

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So there were some sympathies
for the British Empire,

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for Great Britain.

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But they wanted to stay neutral.

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But what you had happening
is that the British

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had a blockade of the Germans.

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They really had a strangle hold.

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And the Germans wanted to
have a blockade of the British

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because Great Britain was an island.

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It was an island, it could
really maybe win the war

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if it could somehow strangle the island,

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if it could blockade the island.

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But unfortunately for Germany,

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it did not have as strong of a navy.

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So you get close to 1917,
actually 1915, 1916, 1917,

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Germany starts to get desperate.

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So it sends its submarines
into the Atlantic.

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They say, "Well, if we can't
blockade Great Britain,

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"at least maybe we can
start harassing ships

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"or even blowing up ships that are trying

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"to trade with Great Britain
and that'll make people

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"afraid to, it'll
essentially be the equivalent

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"of a blockade."

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And at first, Germany
does some minor things.

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But as the war goes on, it
gets more and more desperate.

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It gets more and more desperate,
and it starts attacking

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civilian ships, cruise liners,

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Americans start dying
because German u-boats

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are just willy-nilly, just
essentially torpedoing ships.

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So the US doesn't tolerate it anymore,

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enters the war in 1917.

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Germany didn't take the
United States that seriously

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up to that point.

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But it learned, and we'll
do a whole series of videos

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on this, that it should have.

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And then you fast forward to 1918,

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and the United States
was definitely one of,

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the British were doing alright,

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but the United States
really turned the tides.

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No one really expected
how large of a power

00:11:57.850 --> 00:12:00.376
they had essentially
gotten involved in the war.

00:12:00.376 --> 00:12:04.694
Then you fast forward to
1918 and the war ends.

00:12:04.694 --> 00:12:07.022
The real takeaway of this, I
mean there's a bunch of these

00:12:07.022 --> 00:12:09.607
and we'll talk more about this
in depth in future videos,

00:12:09.607 --> 00:12:12.514
is that it ended some of the nations

00:12:12.514 --> 00:12:14.391
that were on the losing end.

00:12:14.391 --> 00:12:17.142
Austria-Hungary no longer was a nation,

00:12:17.142 --> 00:12:18.030
at least in this form.

00:12:18.030 --> 00:12:21.333
The Ottoman Empire no longer
was a nation in this form.

00:12:21.333 --> 00:12:25.964
And as we will learn later,
there were huge reparations

00:12:25.964 --> 00:12:29.054
by the victors on Germany,
and that to a large degree,

00:12:29.054 --> 00:12:31.323
may have led to World War II.

00:12:31.323 --> 00:12:33.866
But we won't talk in depth
about that right now.

00:12:33.866 --> 00:12:36.385
The other things that started
to happen at this point,

00:12:36.385 --> 00:12:38.452
in 1920 you have the 18th

00:12:38.452 --> 00:12:41.157
and the 19th Amendments being passed.

00:12:41.157 --> 00:12:43.474
The 18th enacted Prohibition,

00:12:43.474 --> 00:12:47.095
where all of a sudden,
you made alcohol illegal

00:12:47.095 --> 00:12:48.791
in the United States.

00:12:48.791 --> 00:12:50.527
And you know the irony of it is,

00:12:50.527 --> 00:12:52.408
that's when you have all of these movies

00:12:52.408 --> 00:12:55.233
about these bootleggers and
you have this whole crime scene

00:12:55.233 --> 00:12:58.932
that develops around illegal alcohol.

00:12:58.932 --> 00:13:00.631
But at the same time
the 19th Amendment was

00:13:00.631 --> 00:13:03.639
maybe a little less controversial.

00:13:03.639 --> 00:13:07.714
And the 19th Amendment, it finally gave

00:13:07.714 --> 00:13:09.714
women the right to vote.

00:13:11.470 --> 00:13:13.275
Right to vote.

00:13:13.275 --> 00:13:15.739
And one of the arguments

00:13:15.739 --> 00:13:19.549
against having women the right
to vote before this time was,

00:13:19.549 --> 00:13:22.062
"Hey, you know, only men are
fighting for the country.

00:13:22.062 --> 00:13:23.514
"Only they have the right to vote.

00:13:23.514 --> 00:13:26.166
"Only they can be soldiers."

00:13:26.166 --> 00:13:27.585
But during World War I, and this happened

00:13:27.585 --> 00:13:30.981
not just in the US, this happened
worldwide in World War I,

00:13:30.981 --> 00:13:33.335
that because so many men were fighting,

00:13:33.335 --> 00:13:37.000
women really had to take
up the slack domestically.

00:13:37.000 --> 00:13:39.235
And they, essentially, were
a big part of the war effort

00:13:39.235 --> 00:13:40.789
in terms of just working at the factories

00:13:40.789 --> 00:13:42.678
and producing things.

00:13:42.678 --> 00:13:45.005
So that was probably one
of the big things that,

00:13:45.005 --> 00:13:47.390
on kind of a global
basis, all of a sudden,

00:13:47.390 --> 00:13:50.640
women started to get the right to vote.

00:13:51.815 --> 00:13:54.299
Also, at this period,
you have in the 1920s,

00:13:54.299 --> 00:13:57.862
you have another post-war economic boom

00:13:57.862 --> 00:14:00.719
that really develops into
a post-war economic bubble

00:14:00.719 --> 00:14:02.636
all the way until 1929.

00:14:03.529 --> 00:14:06.070
And then you have the stock market crash,

00:14:06.070 --> 00:14:07.673
and then I think some of us know that,

00:14:07.673 --> 00:14:12.092
after that period, the
Great Depression ensues.

00:14:12.092 --> 00:14:14.011
And the Great Depression continues.

00:14:14.011 --> 00:14:16.613
And this was a global Great Depression,

00:14:16.613 --> 00:14:18.065
and it continues all the way

00:14:18.065 --> 00:14:20.561
to the US entry in World War II.

00:14:20.561 --> 00:14:22.644
And I'll leave you there.


US History Overview 1: Jamestown to the Civil War

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:02.247
What I thought I
would attempt to do

00:00:02.247 --> 00:00:04.080
in this and the next
few videos is just give

00:00:04.080 --> 00:00:05.720
a scaffold of American history.

00:00:05.720 --> 00:00:09.181
I'm clearly going to glaze
over a lot of the details,

00:00:09.181 --> 00:00:10.680
but hopefully it'll
give you a sense

00:00:10.680 --> 00:00:12.490
of how everything at
least fits together,

00:00:12.490 --> 00:00:14.849
at least the major events
in American history.

00:00:14.849 --> 00:00:17.015
So you can kind of, and
when I say American history,

00:00:17.015 --> 00:00:18.960
I'm talking about
United States history.

00:00:18.960 --> 00:00:21.610
And so the first real
successful settlement

00:00:21.610 --> 00:00:24.220
in what's now the United
States was at Jamestown.

00:00:24.220 --> 00:00:27.210
That's Jamestown,
Virginia right over here.

00:00:27.210 --> 00:00:28.380
And it was 1607.

00:00:28.380 --> 00:00:30.380
It was set up as kind of
a commercial settlement

00:00:30.380 --> 00:00:32.619
and then shortly after
that, and we always

00:00:32.619 --> 00:00:37.060
learned this in school, you know
the pilgrims on the Mayflower,

00:00:37.060 --> 00:00:39.190
sailing the oceans
blue and all the rest.

00:00:39.190 --> 00:00:43.414
They were kind of the next major
settlement in the New World.

00:00:43.414 --> 00:00:45.830
Or I guess we should say the
next major successful English

00:00:45.830 --> 00:00:46.300
settlement.

00:00:46.300 --> 00:00:48.966
There were obviously the Spanish
and the Portuguese were already

00:00:48.966 --> 00:00:52.240
settling the New World
with a good bit of success

00:00:52.240 --> 00:00:54.940
at this point, but we're talking
about the English settlements.

00:00:54.940 --> 00:00:58.420
And so the pilgrims
settled what's

00:00:58.420 --> 00:01:02.390
now Plymouth,
Massachusetts in 1620.

00:01:02.390 --> 00:01:06.340
And obviously from 1620
until the mid-1700s,

00:01:06.340 --> 00:01:09.640
you just had a huge influx of
people migrating and cities

00:01:09.640 --> 00:01:10.560
developing.

00:01:10.560 --> 00:01:14.072
But I'm going to fast forward
all the way to the mid-1700s.

00:01:14.072 --> 00:01:15.780
So this is actually
a huge amount of time

00:01:15.780 --> 00:01:19.110
that I'm just not
providing any details over.

00:01:19.110 --> 00:01:21.620
Because I'm really just quite
focused on the major events

00:01:21.620 --> 00:01:23.910
in American history.

00:01:23.910 --> 00:01:26.560
And so this is a 130-year
period where things were just

00:01:26.560 --> 00:01:29.430
getting built out more, they
were getting more developed.

00:01:29.430 --> 00:01:32.320
And I'm going to fast
forward to 1754, because

00:01:32.320 --> 00:01:34.390
at this period you
had essentially

00:01:34.390 --> 00:01:36.870
the entire east coast
of what's now the US.

00:01:36.870 --> 00:01:39.654
These were the 13
colonies of the-- well,

00:01:39.654 --> 00:01:41.070
they're not the
United States yet,

00:01:41.070 --> 00:01:42.790
they're the 13 British colonies.

00:01:42.790 --> 00:01:44.350
But these are
English settlements,

00:01:44.350 --> 00:01:47.580
and then if you go a little bit
to the northwest from there,

00:01:47.580 --> 00:01:49.790
you have all the
French settlements.

00:01:49.790 --> 00:01:53.120
And obviously still in these
parts of Quebec and Canada,

00:01:53.120 --> 00:01:54.170
people speak French.

00:01:54.170 --> 00:01:55.830
But you had the
French settlements

00:01:55.830 --> 00:01:59.579
up in this area over here.

00:01:59.579 --> 00:02:01.120
I'm not going to go
into the details.

00:02:01.120 --> 00:02:03.244
Each of these can be a
whole series of videos,

00:02:03.244 --> 00:02:04.660
and hopefully in
the future I will

00:02:04.660 --> 00:02:06.500
make them whole
series of videos.

00:02:06.500 --> 00:02:10.740
But you fast forward
to 1754, and you

00:02:10.740 --> 00:02:13.900
start having the French
and the British start

00:02:13.900 --> 00:02:16.670
getting into squabbles
over territory

00:02:16.670 --> 00:02:19.460
where Pittsburgh,
Pennsylvania is right now.

00:02:19.460 --> 00:02:21.020
In 1754.

00:02:21.020 --> 00:02:23.970
And that starts the
French and Indian War.

00:02:23.970 --> 00:02:27.037
And I want to be very clear
here because this is maybe

00:02:27.037 --> 00:02:28.620
one of the biggest
points of confusion

00:02:28.620 --> 00:02:30.130
when people first
learn American history.

00:02:30.130 --> 00:02:31.590
Since it's called the
French and Indian War,

00:02:31.590 --> 00:02:33.750
they think it's between
the French and the Indians.

00:02:33.750 --> 00:02:34.540
But it's not.

00:02:34.540 --> 00:02:36.570
It was the French
and the Indians

00:02:36.570 --> 00:02:43.250
against the British
and the colonists.

00:02:43.250 --> 00:02:45.170
So in this war, the
British and the colonists

00:02:45.170 --> 00:02:48.229
were on the same side against
the French and the Indians.

00:02:48.229 --> 00:02:49.770
And obviously there
were some Indians

00:02:49.770 --> 00:02:52.250
that were also on the
side of the British,

00:02:52.250 --> 00:02:55.860
but it's called the French
and Indian War because these

00:02:55.860 --> 00:03:00.110
were the people that the
British were fighting against.

00:03:00.110 --> 00:03:02.907
Now if anyone outside
of the United States

00:03:02.907 --> 00:03:04.490
talks about the
French and Indian War,

00:03:04.490 --> 00:03:06.550
they will not call it the
French and Indian War.

00:03:06.550 --> 00:03:08.950
They'll really just call
that the American theater

00:03:08.950 --> 00:03:11.430
of the Seven Years' War
because it eventually

00:03:11.430 --> 00:03:14.290
evolves into a much bigger
conflict between Great Britain

00:03:14.290 --> 00:03:16.084
and France that's
going on in Europe,

00:03:16.084 --> 00:03:17.750
and the French and
Indian War was really

00:03:17.750 --> 00:03:20.100
just the American theater of it.

00:03:20.100 --> 00:03:22.945
So between-- the
French and Indian

00:03:22.945 --> 00:03:26.049
War starts in 1754 based on
these disputes over Pittsburgh.

00:03:26.049 --> 00:03:27.340
But that wasn't the only thing.

00:03:27.340 --> 00:03:29.280
You had all of these
other things, all

00:03:29.280 --> 00:03:31.690
of these other tensions
that were developing.

00:03:31.690 --> 00:03:34.820
The thing that starts the
war is never the only factor.

00:03:34.820 --> 00:03:36.960
It's always just
the tipping point.

00:03:36.960 --> 00:03:40.240
But that leads to a
bigger war in Europe.

00:03:40.240 --> 00:03:43.830
And that's the Seven Years'
War that starts in 1756.

00:03:43.830 --> 00:03:48.720
And they both end because
they're really the same war.

00:03:48.720 --> 00:03:53.590
They both end in 1763
with the Treaty of Paris.

00:03:53.590 --> 00:03:55.800
Treaty of Paris, 1763.

00:03:55.800 --> 00:04:00.620
And the big takeaway of that is
that really most of what France

00:04:00.620 --> 00:04:05.510
had in the New World
now becomes essentially

00:04:05.510 --> 00:04:07.630
a part of the
British Empire, now

00:04:07.630 --> 00:04:10.170
becomes British colonies
or British territories.

00:04:10.170 --> 00:04:13.481
And even Louisiana goes
over to Spain at this point.

00:04:13.481 --> 00:04:14.980
And we'll see it
goes back to France

00:04:14.980 --> 00:04:17.160
for a little bit
in 1800, and then

00:04:17.160 --> 00:04:18.910
it goes back to the
United States in 1803,

00:04:18.910 --> 00:04:20.720
but we'll see that in a second.

00:04:20.720 --> 00:04:25.560
So 1763, the British-- it
was this huge costly war--

00:04:25.560 --> 00:04:26.896
but they were able to win.

00:04:26.896 --> 00:04:29.020
And at least from the point
of view of the British,

00:04:29.020 --> 00:04:32.590
they felt that the main
beneficiaries of this war

00:04:32.590 --> 00:04:33.870
were the Americans.

00:04:33.870 --> 00:04:35.940
They were able to get all
this new territory, all

00:04:35.940 --> 00:04:38.070
this new area that they
can now trade with,

00:04:38.070 --> 00:04:40.600
or they could now
potentially settle.

00:04:40.600 --> 00:04:44.700
And so the British
decide to start

00:04:44.700 --> 00:04:49.220
taxing the Americans
to recoup some portion

00:04:49.220 --> 00:04:50.500
of the cost of the war.

00:04:50.500 --> 00:04:54.500
So in 1765 they
pass the Stamp Act.

00:04:54.500 --> 00:04:58.160
And this wasn't a tax on stamps.

00:04:58.160 --> 00:05:00.040
What this was is
that they essentially

00:05:00.040 --> 00:05:02.390
declared that a
whole set of paper

00:05:02.390 --> 00:05:04.240
that had to be used
in the New World.

00:05:04.240 --> 00:05:06.740
So the stuff for
legal documents,

00:05:06.740 --> 00:05:08.400
stuff that maybe even newspaper.

00:05:08.400 --> 00:05:12.650
That that paper would have to
be produced in Great Britain,

00:05:12.650 --> 00:05:15.040
and it had to have a
special stamp on it in order

00:05:15.040 --> 00:05:17.460
for the contracts or
whatever was on top of it,

00:05:17.460 --> 00:05:19.380
in order for them
to be legitimate.

00:05:19.380 --> 00:05:23.000
So essentially it was a huge
tax on paper and on documents.

00:05:23.000 --> 00:05:25.910
And essentially, this is
what societies ran on.

00:05:25.910 --> 00:05:29.580
So it was just a way to extract
money from the colonists

00:05:29.580 --> 00:05:33.000
in order to, I guess, help
pay back some of the costs

00:05:33.000 --> 00:05:36.340
that the empire felt
that they had incurred

00:05:36.340 --> 00:05:37.650
on behalf of the colonists.

00:05:37.650 --> 00:05:41.410
You could debate whether who
was the main beneficiary, but

00:05:41.410 --> 00:05:44.154
regardless you could
imagine this didn't make--

00:05:44.154 --> 00:05:46.570
this whole period over here--
the colonists weren't happy.

00:05:46.570 --> 00:05:48.160
Especially because
they didn't have

00:05:48.160 --> 00:05:50.030
any representation
in Parliament.

00:05:50.030 --> 00:05:53.127
This was done without
anybody from the colony

00:05:53.127 --> 00:05:54.960
saying, hey, wait I
don't think that's fair.

00:05:54.960 --> 00:05:56.890
Or this is fair or whatever.

00:05:56.890 --> 00:05:58.280
And so you fast forward.

00:05:58.280 --> 00:06:01.710
1773, you have the
Boston Tea Party

00:06:01.710 --> 00:06:05.090
where you have a bunch of
people who, for whatever reason,

00:06:05.090 --> 00:06:06.900
and there's multiple
interests here.

00:06:06.900 --> 00:06:10.830
But there was three ships
in Boston Harbor full of tea

00:06:10.830 --> 00:06:14.450
and the tea was owned by
the East India Tea company.

00:06:14.450 --> 00:06:16.660
And they decide, in
protest, and there

00:06:16.660 --> 00:06:19.982
was a whole series of
acts and other taxes

00:06:19.982 --> 00:06:21.940
that went back and forth,
but once again, we're

00:06:21.940 --> 00:06:23.980
not going to go into
the details here.

00:06:23.980 --> 00:06:27.020
But in revolt they
dumped the tea.

00:06:27.020 --> 00:06:29.840
They dressed up as Indians,
as American Indians,

00:06:29.840 --> 00:06:31.930
and they dumped the
tea into Boston Harbor,

00:06:31.930 --> 00:06:34.144
and then you could
imagine well that was

00:06:34.144 --> 00:06:36.060
kind of a very exciting
act for the colonists,

00:06:36.060 --> 00:06:38.497
but it didn't make the
British very happy.

00:06:38.497 --> 00:06:40.580
And then after that, they
passed the Coercive Act.

00:06:40.580 --> 00:06:42.980
They essentially did
a blockade of Boston.

00:06:42.980 --> 00:06:44.680
So things started to
get really, really,

00:06:44.680 --> 00:06:48.270
really tense in the early 1770s.

00:06:48.270 --> 00:06:50.730
And then you fast
forward to 1775,

00:06:50.730 --> 00:06:54.084
you have essentially the first
conflicts of the American

00:06:54.084 --> 00:06:55.500
Revolutionary War,
and we're going

00:06:55.500 --> 00:06:58.520
to do a whole series
of videos on really

00:06:58.520 --> 00:07:00.160
the whole Revolutionary War.

00:07:00.160 --> 00:07:04.380
1776, you have the
Declaration of Independence.

00:07:04.380 --> 00:07:07.320
This is them right here
drafting the Declaration

00:07:07.320 --> 00:07:08.120
of Independence.

00:07:08.120 --> 00:07:10.530
And that's really just
saying, hey, we've

00:07:10.530 --> 00:07:13.400
had enough of you Great Britain!

00:07:13.400 --> 00:07:16.540
We are now declaring ourselves
as an independent country.

00:07:16.540 --> 00:07:19.860
No more of this
colonies business.

00:07:19.860 --> 00:07:22.920
And so all the
way until 1783 you

00:07:22.920 --> 00:07:24.880
have the American
Revolutionary War.

00:07:24.880 --> 00:07:27.090
And once again, you can do
a lot of videos on this,

00:07:27.090 --> 00:07:28.690
but I'm just going
to go over it just

00:07:28.690 --> 00:07:30.700
so you have a sense of
when everything happened

00:07:30.700 --> 00:07:31.783
and when everything ended.

00:07:31.783 --> 00:07:33.790
And we can later dig
deeper into the scaffold.

00:07:33.790 --> 00:07:36.010
And it ends with
the Treaty of Paris.

00:07:36.010 --> 00:07:41.500
The US becomes a free
independent state.

00:07:41.500 --> 00:07:42.750
And then you fast forward.

00:07:42.750 --> 00:07:45.490
Until this point,
the US has been

00:07:45.490 --> 00:07:48.400
governed by Congress, and the
Articles of Confederation.

00:07:48.400 --> 00:07:50.290
But the Constitution
that we have now,

00:07:50.290 --> 00:07:52.482
it was drafted in 1787.

00:07:52.482 --> 00:07:54.940
It was ratified-- it had to
get at least nine of the states

00:07:54.940 --> 00:07:57.130
to ratify it-- that
happened in 1788.

00:07:57.130 --> 00:07:59.960
And then it went
into effect in 1789.

00:07:59.960 --> 00:08:02.630
So it depends what you consider
the birth of the country.

00:08:02.630 --> 00:08:05.590
Well, it would definitely be
the Declaration of Independence,

00:08:05.590 --> 00:08:07.290
but the country in
its current form,

00:08:07.290 --> 00:08:10.020
with its current institutions,
with this current constitution,

00:08:10.020 --> 00:08:12.130
started in 1789.

00:08:12.130 --> 00:08:15.620
And that was also the
beginning of Washington's first

00:08:15.620 --> 00:08:21.240
of two terms as president,
and those ended in 1797.

00:08:21.240 --> 00:08:23.510
And then John Adams
comes into the picture.

00:08:23.510 --> 00:08:25.320
And the reason why I
put this-- obviously

00:08:25.320 --> 00:08:27.130
this is actually the only
president that I showed--

00:08:27.130 --> 00:08:28.713
is that it was
actually very important

00:08:28.713 --> 00:08:31.590
that he decided to step
down after two terms.

00:08:31.590 --> 00:08:33.210
He was hugely popular.

00:08:33.210 --> 00:08:35.614
If he wanted to,
he probably could

00:08:35.614 --> 00:08:37.280
have become one of
these characters that

00:08:37.280 --> 00:08:39.850
stick around maybe
a little bit longer

00:08:39.850 --> 00:08:41.600
than some people would want.

00:08:41.600 --> 00:08:44.376
So it was really good that he
set this example of stepping

00:08:44.376 --> 00:08:45.750
down after two
terms, and that he

00:08:45.750 --> 00:08:48.480
wasn't this kind of
power hungry dude.

00:08:48.480 --> 00:08:50.300
You fast forward
a little bit more.

00:08:50.300 --> 00:08:55.144
1803, I mentioned that after
the French and Indian War what's

00:08:55.144 --> 00:08:57.310
Louisiana-- I want to be
clear when I say Louisiana.

00:08:57.310 --> 00:09:00.220
Louisiana isn't just what's
the current state of Louisiana.

00:09:00.220 --> 00:09:01.720
It's this whole
region that includes

00:09:01.720 --> 00:09:03.530
the state of Louisiana,
but all the way up

00:09:03.530 --> 00:09:06.850
to roughly what
the United States'

00:09:06.850 --> 00:09:09.490
current border with Canada.

00:09:09.490 --> 00:09:11.060
And after the French
and Indian War,

00:09:11.060 --> 00:09:13.850
all of this business
over here went to Spain.

00:09:13.850 --> 00:09:17.070
And then in 1800, it
went back to France.

00:09:17.070 --> 00:09:20.080
But then in 1803, Napoleon
had a bunch of stuff

00:09:20.080 --> 00:09:22.600
that he-- his Naval
fleet was destroyed,

00:09:22.600 --> 00:09:28.330
he had a suffered some
defeats in the West Indies,

00:09:28.330 --> 00:09:31.250
I guess we could call it.

00:09:31.250 --> 00:09:35.444
In particular in
Haiti, and he said,

00:09:35.444 --> 00:09:36.860
well, you know I
probably won't be

00:09:36.860 --> 00:09:38.530
able to control this
territory anyway,

00:09:38.530 --> 00:09:40.500
so he sold it to
the United States

00:09:40.500 --> 00:09:44.240
for what turned out to be a
very, very, very cheap price.

00:09:44.240 --> 00:09:45.625
But it was kind
of like, it's not

00:09:45.625 --> 00:09:47.250
like he could have
protected it anyway.

00:09:47.250 --> 00:09:47.860
The United States
might have been

00:09:47.860 --> 00:09:50.980
able to take it from him without
him being able to do anything.

00:09:50.980 --> 00:09:54.010
So he might as well
get some money for it

00:09:54.010 --> 00:09:56.630
so that he could fund
his battles in Europe.

00:09:56.630 --> 00:10:00.340
So in 1803, the United States
almost doubled in size.

00:10:00.340 --> 00:10:03.960
It went from these
territories that it

00:10:03.960 --> 00:10:09.230
had after the American
Revolution for Independence,

00:10:09.230 --> 00:10:14.070
and now it got all of this
region over here in 1803.

00:10:14.070 --> 00:10:17.750
Then you fast forward a bit.

00:10:17.750 --> 00:10:19.840
And the War of 1812,
it's an interesting one,

00:10:19.840 --> 00:10:24.040
because there weren't any
really serious outcomes from it.

00:10:24.040 --> 00:10:26.920
But what was interesting about
it, this whole time period,

00:10:26.920 --> 00:10:29.250
even after independence,
the British

00:10:29.250 --> 00:10:30.820
continued to harass America.

00:10:30.820 --> 00:10:37.980
They continued to arm Native
Americans who would maybe

00:10:37.980 --> 00:10:40.590
revolt or cause
trouble for settlers.

00:10:40.590 --> 00:10:42.495
They would impress
American seamen--

00:10:42.495 --> 00:10:43.870
and when I say
impress, it didn't

00:10:43.870 --> 00:10:45.703
mean that they were
doing something special.

00:10:45.703 --> 00:10:47.390
It meant that they
were-- impressment

00:10:47.390 --> 00:10:51.560
of seamen meant that they
were taking over these boats,

00:10:51.560 --> 00:10:53.390
taking the sailors,
and forcing them

00:10:53.390 --> 00:10:56.440
to become part of
the British military.

00:10:56.440 --> 00:10:57.966
So they were doing
a whole series

00:10:57.966 --> 00:11:00.340
of things that was really kind
of antagonizing the United

00:11:00.340 --> 00:11:01.050
States.

00:11:01.050 --> 00:11:04.550
In 1812, the United States
declares war on Great Britain.

00:11:04.550 --> 00:11:06.240
You have the War of 1812.

00:11:06.240 --> 00:11:08.850
It ends in 1815 with the
Battle of New Orleans.

00:11:12.250 --> 00:11:14.935
But there wasn't
any real transfer

00:11:14.935 --> 00:11:17.660
of a territory or anything
like that over here.

00:11:17.660 --> 00:11:20.470
What was good, some people call
it the Second War for American

00:11:20.470 --> 00:11:25.217
Independence, is it really
asserted that America was here

00:11:25.217 --> 00:11:27.550
to stay, or I should say that
the United States was here

00:11:27.550 --> 00:11:28.580
to stay.

00:11:28.580 --> 00:11:32.270
That the Revolution wasn't
just some fluke that isn't some

00:11:32.270 --> 00:11:33.450
just fly by night country.

00:11:33.450 --> 00:11:36.530
It was able to defeat one of the
greatest empires in the world

00:11:36.530 --> 00:11:37.320
again.

00:11:37.320 --> 00:11:39.080
So it's kind of here to stay.

00:11:39.080 --> 00:11:42.150
Now you fast forward
a little bit more.

00:11:42.150 --> 00:11:45.500
This part of what
we would call Texas,

00:11:45.500 --> 00:11:47.120
this area right over here.

00:11:47.120 --> 00:11:52.170
It was, before 1836,
it was part of Mexico.

00:11:52.170 --> 00:11:56.020
But the Mexicans
actually encouraged

00:11:56.020 --> 00:11:57.420
English-speaking
settlers-- these

00:11:57.420 --> 00:11:59.240
would be American
English-speaking settlers

00:11:59.240 --> 00:12:02.540
into the area-- just because
it was very sparsely settled.

00:12:02.540 --> 00:12:04.810
But these English-speaking
settlers, a lot of them

00:12:04.810 --> 00:12:09.870
were slave owners, and then
as we kind of go up to 1836,

00:12:09.870 --> 00:12:12.690
the state of Mexico that
this was all governed by,

00:12:12.690 --> 00:12:16.020
they were thinking about
abolishing slavery.

00:12:16.020 --> 00:12:18.660
So you can imagine that
the settlers there, they

00:12:18.660 --> 00:12:19.805
didn't like this idea.

00:12:19.805 --> 00:12:23.870
So in 1836, you had the
War for Texas Independence,

00:12:23.870 --> 00:12:27.600
and that's where you remember
the Alamo, and all of that.

00:12:27.600 --> 00:12:30.470
And then the first president
of Texas is Sam Houston.

00:12:30.470 --> 00:12:32.990
That's why Houston
is named Houston.

00:12:32.990 --> 00:12:36.350
And then you fast forward
all the way to 1845.

00:12:36.350 --> 00:12:38.513
And in this time period,
you have this whole talk

00:12:38.513 --> 00:12:41.350
in the United States
of Manifest Destiny,

00:12:41.350 --> 00:12:45.710
that it's part of our
God-given destiny as Americans,

00:12:45.710 --> 00:12:50.310
to one day extend our territory
all the way to the Pacific

00:12:50.310 --> 00:12:50.810
Ocean.

00:12:50.810 --> 00:12:53.150
So people were already eyeing
a lot of the territory.

00:12:53.150 --> 00:12:56.760
Remember all of this
territory, this was Texas.

00:12:56.760 --> 00:12:59.420
And Mexico still viewed
it as their territory,

00:12:59.420 --> 00:13:01.379
even though it was being
governed independently

00:13:01.379 --> 00:13:02.836
by the people who
called themselves

00:13:02.836 --> 00:13:03.990
the Republic of Texas.

00:13:03.990 --> 00:13:06.040
And you had all
of this territory

00:13:06.040 --> 00:13:07.657
that was Mexican territory.

00:13:07.657 --> 00:13:09.490
So people were starting
to eye this and say,

00:13:09.490 --> 00:13:12.550
wouldn't it be nice to
get a little bit of that?

00:13:12.550 --> 00:13:16.200
So in 1845, and this
was in agreement

00:13:16.200 --> 00:13:19.830
with the settlers in Texas,
with the Republic of Texas,

00:13:19.830 --> 00:13:21.242
the United States annexed Texas.

00:13:21.242 --> 00:13:22.950
The settlers there
wanted this to happen,

00:13:22.950 --> 00:13:25.490
so wasn't a forced
annexation of Texas.

00:13:25.490 --> 00:13:27.570
But Mexico was not
so happy about this

00:13:27.570 --> 00:13:31.360
because Mexico
still viewed Texas

00:13:31.360 --> 00:13:33.740
as part of their territory.

00:13:33.740 --> 00:13:37.040
And America, to some degree,
depends on how you view it,

00:13:37.040 --> 00:13:40.370
it seems like they kind of
wanted to goad Mexico into war,

00:13:40.370 --> 00:13:44.000
so they sent military really
close to the border of Mexico,

00:13:44.000 --> 00:13:47.240
even into some territory where
Mexico might have had better

00:13:47.240 --> 00:13:49.780
claims to it or-- I'm not
going to take sides on this,

00:13:49.780 --> 00:13:53.010
but it seemed like there was
some instigation going on.

00:13:53.010 --> 00:13:55.520
And there's some debate about
the actual course of events.

00:13:55.520 --> 00:13:59.240
But in 1846 you have war
actually breaking out

00:13:59.240 --> 00:14:01.540
between Mexico and
the United States.

00:14:01.540 --> 00:14:06.030
And by 1848, the United States
essentially trounces Mexico,

00:14:06.030 --> 00:14:10.670
and most of the war actually
does go on on Mexican land.

00:14:10.670 --> 00:14:16.270
And because of that, Mexico
cedes over all of this area.

00:14:16.270 --> 00:14:20.270
So, California and all of
the rest of Nevada, Arizona,

00:14:20.270 --> 00:14:22.770
what the part of New Mexico
the didn't come along,

00:14:22.770 --> 00:14:25.300
that the United States
didn't already have.

00:14:25.300 --> 00:14:28.150
And along that same
amount of time,

00:14:28.150 --> 00:14:31.700
you both had the British
and the Americans that

00:14:31.700 --> 00:14:34.510
were eyeing this territory,
the Oregon Territory up here

00:14:34.510 --> 00:14:36.220
and it even included
part of Canada.

00:14:36.220 --> 00:14:39.000
And eventually they were
able to resolve it relatively

00:14:39.000 --> 00:14:40.791
peacefully, and
what they agreed is,

00:14:40.791 --> 00:14:43.040
is that the Americans would
get all of this territory.

00:14:43.040 --> 00:14:44.920
And the British would
get everything north

00:14:44.920 --> 00:14:46.520
of this line right over here.

00:14:46.520 --> 00:14:49.210
And that's why Vancouver
and British Columbia and all

00:14:49.210 --> 00:14:51.340
of that, is Canada now.

00:14:51.340 --> 00:14:56.370
It stayed as part of the British
Empire for a little bit longer.

00:14:56.370 --> 00:15:00.130
So by 1848, the Manifest Destiny
essentially had happened.

00:15:00.130 --> 00:15:02.840
The United States had gotten
everything from California

00:15:02.840 --> 00:15:06.820
all the way from the Pacific
coast to the Atlantic coast.

00:15:06.820 --> 00:15:12.270
And clearly I'm really just
covering the high levels,

00:15:12.270 --> 00:15:16.250
30,000 level foot view
of American history here.

00:15:16.250 --> 00:15:18.530
This whole time you had
this tension developing.

00:15:18.530 --> 00:15:22.930
From the birth of the country
through the election of Abraham

00:15:22.930 --> 00:15:25.030
Lincoln, you have this
tension over slavery.

00:15:25.030 --> 00:15:27.280
A lot of people in the
North didn't like it

00:15:27.280 --> 00:15:28.910
on moral grounds.

00:15:28.910 --> 00:15:31.820
A lot of people in the
South didn't like it-- well,

00:15:31.820 --> 00:15:34.750
they wanted slavery
regardless of what

00:15:34.750 --> 00:15:37.080
they thought of it morally--
the South's economy,

00:15:37.080 --> 00:15:40.830
to a large degree,
was based on slavery.

00:15:40.830 --> 00:15:44.352
And so all of this, the
tipping point happened in 1860.

00:15:44.352 --> 00:15:49.920
Where Abraham Lincoln, who was
pretty vocal about the fact

00:15:49.920 --> 00:15:52.460
that he did not like
slavery, that he wanted

00:15:52.460 --> 00:15:54.770
to curb the spread
of slave states.

00:15:54.770 --> 00:15:57.470
And up to this point, you
had all of these compromises

00:15:57.470 --> 00:15:59.260
every time a state
came into the Union.

00:15:59.260 --> 00:16:01.535
The slave states wanted it
to be another slave state.

00:16:01.535 --> 00:16:03.660
The free states wanted it
to be another free state.

00:16:03.660 --> 00:16:05.530
So you always had
this people kind

00:16:05.530 --> 00:16:08.510
of jocking for whoever
could have the most

00:16:08.510 --> 00:16:10.800
states on their
side of the camp.

00:16:10.800 --> 00:16:14.690
But all of this pro-slavery
and anti-slavery

00:16:14.690 --> 00:16:17.400
hit a tipping point in 1860
when Abraham Lincoln, who

00:16:17.400 --> 00:16:21.030
was fairly vocal about
not extending slavery,

00:16:21.030 --> 00:16:22.360
he was elected.

00:16:22.360 --> 00:16:26.050
Then a bunch of what we now
consider southern states

00:16:26.050 --> 00:16:27.890
seceded from the Union.

00:16:27.890 --> 00:16:32.287
And then in 1861
in South Carolina.

00:16:32.287 --> 00:16:34.870
South Carolina said, hey, we are
not part of the United States

00:16:34.870 --> 00:16:37.610
anymore, but there was still a
United States military garrison

00:16:37.610 --> 00:16:39.070
there, so they attacked it.

00:16:39.070 --> 00:16:40.990
That started the Civil War.

00:16:40.990 --> 00:16:45.140
And so during the Civil War--
it lasts until 1865-- Abraham

00:16:45.140 --> 00:16:47.570
Lincoln makes the
Emancipation Proclamation

00:16:47.570 --> 00:16:50.430
in 1863, which
essentially proclaims

00:16:50.430 --> 00:16:52.900
all the slaves should be free.

00:16:52.900 --> 00:16:56.160
This lays the groundwork
for the 13th Amendment

00:16:56.160 --> 00:16:57.620
to the Constitution.

00:16:57.620 --> 00:17:00.690
And then unfortunately,
he dies two months

00:17:00.690 --> 00:17:02.400
before the end of the Civil War.

00:17:02.400 --> 00:17:05.510
But in 1865, the South
surrenders and so

00:17:05.510 --> 00:17:07.099
they're not able to secede.

00:17:07.099 --> 00:17:09.920
And essentially,
we no longer have

00:17:09.920 --> 00:17:11.560
slavery in the United States.

00:17:11.560 --> 00:17:14.130
So I'm going to leave--
and it's fascinating,

00:17:14.130 --> 00:17:17.000
and just to give you a sense
of things, here's the map.

00:17:17.000 --> 00:17:20.890
The navy blue are the Union
states, the northern states,

00:17:20.890 --> 00:17:23.280
the light blue are the
territory controlled

00:17:23.280 --> 00:17:24.790
by the northern states.

00:17:24.790 --> 00:17:27.339
This orange color
are the states that

00:17:27.339 --> 00:17:30.970
seceded from the
Union, the Confederacy.

00:17:30.970 --> 00:17:34.370
And this light orange,
these are territories

00:17:34.370 --> 00:17:37.950
that they controlled,
but they were disputed.

00:17:37.950 --> 00:17:41.270
These yellow states right here
were members of the Union.

00:17:41.270 --> 00:17:42.700
They didn't secede
from the Union.

00:17:42.700 --> 00:17:44.390
They didn't join
the Confederacy,

00:17:44.390 --> 00:17:45.855
but they were slave states.

00:17:45.855 --> 00:17:47.480
But probably the most
fascinating thing

00:17:47.480 --> 00:17:50.480
about the Civil War,
other than the fact

00:17:50.480 --> 00:17:53.080
that it ended slavery
in the United States,

00:17:53.080 --> 00:17:54.580
that was probably
its biggest thing,

00:17:54.580 --> 00:17:55.955
but it was also
the bloodiest war

00:17:55.955 --> 00:17:59.160
that ever happened in the
United States history.

00:17:59.160 --> 00:18:03.330
During the Civil War-- and
these are unbelievable numbers--

00:18:03.330 --> 00:18:12.070
18% of white males
in the South died.

00:18:12.070 --> 00:18:15.590
18%, almost one out of every
five white males in the South,

00:18:15.590 --> 00:18:17.530
died during the Civil War.

00:18:17.530 --> 00:18:19.450
And for the North, it
was slightly better.

00:18:19.450 --> 00:18:20.150
It was 6%.

00:18:20.150 --> 00:18:25.270
But still, a huge percentage
of the men in the United States

00:18:25.270 --> 00:18:28.330
died fighting the Civil War.


Addition and subtraction with significant figures

WEBVTT
Kind: captions
Language: en

00:00:00.882 --> 00:00:04.592
We saw in the last video that when you multiply or you divide

00:00:04.592 --> 00:00:07.187
numbers, or (I guess I should say when you multiply or

00:00:07.187 --> 00:00:12.569
divide measurements) your result can only have as many

00:00:12.569 --> 00:00:17.067
significant digits as the thing with the smallest significant

00:00:17.067 --> 00:00:20.333
digits you ended up multiplying and dividing.

00:00:20.333 --> 00:00:28.789
So just as a quick example, if I have 2.00 times (I don't know)

00:00:28.789 --> 00:00:36.058
3.5 my answer over here can only have 2 significant digits

00:00:36.058 --> 00:00:41.559
This has 2 significant digits, this has 3. 2 times 3.5

00:00:41.559 --> 00:00:47.311
is 7, and we can get to 1 zero to the right of the decimal.

00:00:47.311 --> 00:00:48.912
Because we can have 2 significant digits.

00:00:48.912 --> 00:00:50.328
This was 3, this is 2.

00:00:50.328 --> 00:00:53.575
We only limited it to 2, because that was the smallest

00:00:53.575 --> 00:00:55.605
number of significant digits we had in all of the things

00:00:55.605 --> 00:00:57.712
that we were taking the product of.

00:00:57.712 --> 00:01:00.152
When we do addition and subtraction, it's a little bit different.

00:01:00.152 --> 00:01:01.627
And I'll do an example first.

00:01:01.627 --> 00:01:06.227
I just do a kind of a numerical example first, and then I'll think of a little bit more of a real world example.

00:01:06.227 --> 00:01:09.027
And obviously even my real world examples aren't really real world.

00:01:09.027 --> 00:01:11.026
In my last video, I talked about laying down carpet

00:01:11.026 --> 00:01:13.776
and someone rightfully pointed out,"Hey, if you are laying down

00:01:13.776 --> 00:01:17.267
carpet, you always want to round up. Just because you don't wanna

00:01:17.267 --> 00:01:20.875
it's easier to cut carpet away, then somehow glue carpet there.

00:01:20.875 --> 00:01:24.467
But that's particular to carpet. I was just saying a general way

00:01:24.467 --> 00:01:26.420
to think about precision in significant figures.

00:01:26.420 --> 00:01:29.286
That was only particular to carpets or tiles.

00:01:29.286 --> 00:01:34.679
But when you add, when you add, or subtract,

00:01:34.679 --> 00:01:39.409
now these significant digits or these significant figures don't matter as much

00:01:39.409 --> 00:01:42.867
as the actual precision of the things that you are adding.

00:01:42.867 --> 00:01:45.333
How many decimal places do you go? For example,

00:01:45.333 --> 00:01:53.996
if I were to add 1.26, and I were to add it to - let's say - to 2.3.

00:01:53.996 --> 00:01:57.200
If you just add these two numbers up, and let's say these are measurements,

00:01:57.200 --> 00:01:58.956
so when you make it (these are clearly 3

00:01:58.956 --> 00:02:02.886
significant digits) we're able to measure to the nearest hundreth.

00:02:02.919 --> 00:02:06.967
Here this is two significant digits so three significant digits

00:02:06.967 --> 00:02:10.303
this is two significant digits, we are able to measure to the nearest tenth.

00:02:10.303 --> 00:02:14.008
Let me label this. This is the hundredth

00:02:14.008 --> 00:02:16.998
and this is the tenth. When you add or subtract numbers,

00:02:16.998 --> 00:02:20.229
your answer, so if you just do this, if we just add these

00:02:20.229 --> 00:02:24.499
two numbers, I get - what? - 3.56.

00:02:24.499 --> 00:02:30.799
The sum, or the difference whatever you take, you don't count significant figures

00:02:30.799 --> 00:02:33.099
You don't say,"Hey, this can only have two significant figures."

00:02:33.099 --> 00:02:38.233
What you can say is, "This can only be as precise as the least precise thing that I had over here.

00:02:38.233 --> 00:02:40.965
The least precise thing I had over here is 2.3.

00:02:40.965 --> 00:02:44.129
It only went to the tenths place, so in our answer

00:02:44.129 --> 00:02:49.233
we can only go to the tenths place. So we need to round

00:02:49.233 --> 00:02:54.166
this guy up. Cause we have a six right here, so we round up

00:02:54.166 --> 00:03:00.383
so if you care about significant figures, this is going to become a 3.7.

00:03:00.383 --> 00:03:02.466
And I want to be clear. This time it worked out,

00:03:02.466 --> 00:03:04.048
cause this also has 2 significant figures,

00:03:04.048 --> 00:03:06.899
this also has two significant figures. But this could have been...

00:03:06.899 --> 00:03:12.129
(let me do another situation) you could have 1.26 plus

00:03:12.129 --> 00:03:19.768
102.3, and you would get obviously 103.56.

00:03:19.768 --> 00:03:23.396
Then, in this situation - this obviously over here has 4 significant figures,

00:03:23.396 --> 00:03:26.768
this over here has 3 significant figures. But in our answer

00:03:26.768 --> 00:03:29.501
we don't want to have 3 significant figures. We wanna have the...

00:03:29.501 --> 00:03:32.768
only as precise as the least precise thing that we added up.

00:03:32.768 --> 00:03:36.400
The least precise thing we only go one digit behind the decimal over here,

00:03:36.400 --> 00:03:40.035
so we can only go to the tenth, only one digit over the decimal there.

00:03:40.035 --> 00:03:45.234
So once again, we round it up to 103.6.

00:03:45.234 --> 00:03:49.317
And to see why that makes sense, let's do a little bit of an example here

00:03:49.317 --> 00:03:50.769
with actually measuring something.

00:03:50.769 --> 00:03:52.628
So let's say we have a block here,

00:03:52.628 --> 00:03:56.896
let's say that I have a block, we draw that block a little bit neater,

00:03:56.896 --> 00:03:59.069
and let's say we have a pretty good meter stick,

00:03:59.069 --> 00:04:01.501
and we're able to measure to the nearest centimeter,

00:04:01.501 --> 00:04:06.901
we get, it is 2.09 meters.

00:04:06.901 --> 00:04:14.368
Let's say we have another block, and this is the other block right over there.

00:04:14.368 --> 00:04:18.638
We have a, let's say we have an even more precise meter stick,

00:04:18.638 --> 00:04:20.501
which can measure to the nearest millimeter.

00:04:20.501 --> 00:04:26.319
And we get this to be 1.901 meters.

00:04:26.319 --> 00:04:28.402
So measuring to the nearest millimeter.

00:04:28.402 --> 00:04:31.166
And let's say those measurements were done a long time ago,

00:04:31.166 --> 00:04:33.237
and we don't have access to measure them any more,

00:04:33.237 --> 00:04:36.649
but someone says 'How tall is it if I were stack the blue block

00:04:36.649 --> 00:04:40.228
on the top of the red block - or the orange block, or whatever that color that is?"

00:04:40.228 --> 00:04:42.736
So how high would this height be?

00:04:42.736 --> 00:04:45.814
Well, if you didn't care about significant figures or precision,

00:04:45.814 --> 00:04:47.236
you would just add them up.

00:04:47.236 --> 00:04:52.982
You'd add the 1.901 plus the 2.09.

00:04:52.982 --> 00:04:54.168
So let me add those up:

00:04:54.168 --> 00:05:01.643
so if you take 1.901 and add that to 2.09,

00:05:01.643 --> 00:05:05.986
you get 1 plus nothing is 1,

00:05:05.986 --> 00:05:07.968
0 plus 9 is 9,

00:05:07.968 --> 00:05:09.501
9 plus 0 is 9,

00:05:09.501 --> 00:05:10.835
you get the decimal point,

00:05:10.835 --> 00:05:15.168
1 plus 2 is 3. So you get 3.991.

00:05:15.168 --> 00:05:19.508
And the problem with this, the reason why this is a little bit...

00:05:19.508 --> 00:05:21.832
it's kind of misrepresenting how precise you measurement is.

00:05:21.832 --> 00:05:26.811
You don't know, if I told you that the tower is 3.991 meters tall,

00:05:26.811 --> 00:05:29.768
I'm implying that I somehow was able to measure

00:05:29.768 --> 00:05:32.234
the entire tower to the nearest millimeter.

00:05:32.234 --> 00:05:34.442
The reality is that I was only be able to

00:05:34.442 --> 00:05:36.508
measure the part of the tower to the millimeter.

00:05:36.508 --> 00:05:39.835
This part of the tower I was able to measure to the nearest centimeter.

00:05:39.835 --> 00:05:43.101
So to make it clear the our measurement is only good

00:05:43.101 --> 00:05:44.307
to the nearest centimeter,

00:05:44.307 --> 00:05:46.067
because there is more error here, then...

00:05:46.067 --> 00:05:50.101
it might overwhelm or whatever the precision we had on the millimeters there.

00:05:50.101 --> 00:05:53.168
To make that clear, we have to make this only as precise

00:05:53.168 --> 00:05:55.708
as the least precise thing that we are adding up.

00:05:55.708 --> 00:05:58.818
So over here, the least precise thing was, we went to the hundredths,

00:05:58.818 --> 00:06:01.501
so over here we have to round to the hundredths.

00:06:01.501 --> 00:06:04.832
So, since 1 is less then 5, we are going to round down,

00:06:04.832 --> 00:06:06.968
and so we can only legitimately say,

00:06:06.968 --> 00:06:08.646
if we want to represent what we did properly

00:06:08.646 --> 00:06:12.311
that the tower is 3.99 meters.

00:06:12.311 --> 00:06:16.174
And I also want to make it clear that this doesn't just apply

00:06:16.174 --> 00:06:17.811
to when there is a decimal point.

00:06:17.811 --> 00:06:23.105
If I were tell you that... Let's say that I were to measure...

00:06:23.105 --> 00:06:25.239
I want to measure a building.

00:06:25.239 --> 00:06:28.035
I was only able to measure the building to the nearest 10 feet.

00:06:28.035 --> 00:06:32.155
So I tell you that that building is 350 feet tall.

00:06:32.155 --> 00:06:34.615
So this is the building.

00:06:34.615 --> 00:06:37.830
This is a building.

00:06:37.830 --> 00:06:42.708
And let's say there is a manufacturer of radio antennas, so...

00:06:42.708 --> 00:06:44.366
or radio towers.

00:06:44.366 --> 00:06:49.378
And the manufacturers has measured their tower to the nearest foot.

00:06:49.378 --> 00:06:53.715
And they say, their tower is 8 feet tall.

00:06:53.715 --> 00:06:56.174
So notice: here they measure to the nearest 10 feet,

00:06:56.174 --> 00:06:58.100
here they measure to the nearest foot.

00:06:58.100 --> 00:07:00.234
And actually to make it clear, because once again,

00:07:00.234 --> 00:07:01.900
as I said, this is ambiguous,

00:07:01.900 --> 00:07:04.167
it's not 100% clear how many significant figures there are.

00:07:04.167 --> 00:07:09.241
Maybe it was exactly 350 feet or maybe they just rounded it to the nearest 10 feet.

00:07:09.241 --> 00:07:10.878
So a better way to represent this,

00:07:10.878 --> 00:07:14.234
they... would be to say instead of writing it 350,

00:07:14.234 --> 00:07:21.234
a better way to write it would be 3.5 times 10 to the second feet tall.

00:07:21.234 --> 00:07:22.835
And when you are writing in scientific notation,

00:07:22.835 --> 00:07:26.635
that makes it very clear that there is only 2 significant digits here,

00:07:26.635 --> 00:07:28.635
you are only measuring to the nearest 10 feet.

00:07:28.635 --> 00:07:30.167
Other way to represent it:

00:07:30.167 --> 00:07:33.242
you could write 350 this notation has done less,

00:07:33.242 --> 00:07:36.768
but sometimes the last significant digit has a line on the top of it,

00:07:36.768 --> 00:07:39.635
or the last significant digit has a line below it.

00:07:39.635 --> 00:07:42.460
Either of those are ways to specify it,

00:07:42.460 --> 00:07:43.968
this is probably the least ambiguous,

00:07:43.968 --> 00:07:47.794
but assuming that they only make measure to the nearest 10 feet,

00:07:47.794 --> 00:07:52.100
If someone were ask you: "How tall is the building plus the tower?"

00:07:52.100 --> 00:07:54.301
Well, your first reaction were, let's just add

00:07:54.301 --> 00:07:58.213
the 350 plus 8, you get 358.

00:07:58.213 --> 00:08:04.715
You'd get 358 feet. So this is the building plus the tower. 358 feet.

00:08:04.715 --> 00:08:06.900
For once again, we are misrepresenting it.

00:08:06.900 --> 00:08:08.794
We are making it look like we were able

00:08:08.794 --> 00:08:10.842
to measure the combination to the nearest foot.

00:08:10.842 --> 00:08:13.642
But we were able to measure only the tower to the nearest foot.

00:08:13.642 --> 00:08:19.568
So in order to represent our measurement at the level of precision at we really did,

00:08:19.568 --> 00:08:22.042
we really have to round this to the nearest 10 feet.

00:08:22.042 --> 00:08:24.635
Because that was our least precise measurement.

00:08:24.635 --> 00:08:26.840
So we would really have to round this up to,

00:08:26.840 --> 00:08:29.234
8 is greater-than-or-equal to 5,

00:08:29.234 --> 00:08:33.710
so we round this up to 360 feet.

00:08:33.710 --> 00:08:35.768
So once again, whatever is...

00:08:35.768 --> 00:08:39.044
Just to make it clear, even this ambiguous,

00:08:39.044 --> 00:08:42.301
maybe we put a line over to show, that is our level of precision,

00:08:42.301 --> 00:08:43.968
that we have 2 significant digits.

00:08:43.968 --> 00:08:49.301
Or we could write this as 3.6 times 10 to the second.

00:08:49.301 --> 00:08:50.301
Which is times 100.

00:08:50.301 --> 00:08:53.382
3.6 times 10 to the second feet in scientific notation.

00:08:53.382 --> 00:08:57.567
And this makes it very clear that we only have 2 significant digits here.


Multiplying and dividing with significant figures

WEBVTT
Kind: captions
Language: en

00:00:00.250 --> 00:00:01.930
Now that we have a
decent understanding

00:00:01.930 --> 00:00:04.429
of how to figure out how many
significant figures we're even

00:00:04.429 --> 00:00:06.780
dealing with, let's
think about a situation

00:00:06.780 --> 00:00:12.150
where significant figures
will or might become relevant.

00:00:12.150 --> 00:00:14.870
So let's say that I
have a carpet here.

00:00:14.870 --> 00:00:17.860
And using maybe a
meter stick, I'm

00:00:17.860 --> 00:00:21.130
able to measure the carpet
to the nearest centimeter.

00:00:21.130 --> 00:00:25.160
And so I get the carpet as--
to the nearest centimeter,

00:00:25.160 --> 00:00:28.760
I get it to being 1.69 meters.

00:00:28.760 --> 00:00:32.390
So this 9, obviously, this
is to the nearest centimeter.

00:00:32.390 --> 00:00:37.840
This 9/100 of a meter is the
same thing as 9 centimeters.

00:00:37.840 --> 00:00:41.180
And let's say I'm able
to measure the width here

00:00:41.180 --> 00:00:44.754
as 2.09 meters.

00:00:44.754 --> 00:00:46.485
I use the same meter stick.

00:00:46.485 --> 00:00:47.860
And you were to
ask me, Sal, what

00:00:47.860 --> 00:00:50.990
is the area of your carpet?

00:00:50.990 --> 00:00:53.320
And so I'd just do the
straight-up calculation.

00:00:53.320 --> 00:00:55.980
The area is just going to be
the length times the width,

00:00:55.980 --> 00:01:03.530
so it would be 1.69
meters times 2.09 meters.

00:01:03.530 --> 00:01:05.670
And we could do this by
hand, but let me just get

00:01:05.670 --> 00:01:10.210
the calculator out just to make
things move along a little bit

00:01:10.210 --> 00:01:11.060
faster.

00:01:11.060 --> 00:01:17.620
And so we have 1.69 times 2.09.

00:01:17.620 --> 00:01:21.330
And that gives us 3.5321.

00:01:21.330 --> 00:01:26.370
Let me write that down-- 3.5321.

00:01:26.370 --> 00:01:28.880
So let me write
this in a new color.

00:01:28.880 --> 00:01:33.180
So this gives us
3.5321, and we have

00:01:33.180 --> 00:01:35.230
a meters times a
meters, which gives us

00:01:35.230 --> 00:01:37.710
meters squared or square meters.

00:01:37.710 --> 00:01:39.620
And so I might very
proudly tell you,

00:01:39.620 --> 00:01:44.880
hey, the area here is
3.5321 square meters.

00:01:44.880 --> 00:01:48.200
And the problem here
is that when I give you

00:01:48.200 --> 00:01:49.987
this thing that has
all of these numbers

00:01:49.987 --> 00:01:52.195
behind the decimal point
and all of these what we now

00:01:52.195 --> 00:01:53.700
know to be significant
figures, it

00:01:53.700 --> 00:01:58.190
implies that I had a really
precise way of measuring

00:01:58.190 --> 00:01:58.690
the area.

00:01:58.690 --> 00:02:02.060
Well, in reality, I only
was able to measure the area

00:02:02.060 --> 00:02:04.090
to the nearest centimeter.

00:02:04.090 --> 00:02:06.860
So the way we would
do this so that I

00:02:06.860 --> 00:02:08.550
don't make it look
like my measurement

00:02:08.550 --> 00:02:10.110
is more precise
than it really is.

00:02:10.110 --> 00:02:13.240
Or this calculation that's
derived from my measurements--

00:02:13.240 --> 00:02:18.110
I make sure that it has no more
significant figures than either

00:02:18.110 --> 00:02:20.010
of the numbers
that I multiplied.

00:02:20.010 --> 00:02:25.650
So in this situation, I have
three significant figures here.

00:02:25.650 --> 00:02:30.750
And over here, I have
three significant figures.

00:02:30.750 --> 00:02:33.740
And so in general, when
you multiply or divide,

00:02:33.740 --> 00:02:41.890
the significant figures in your
product or-- I always forget.

00:02:41.890 --> 00:02:45.608
There's the divisor,
there's the dividend,

00:02:45.608 --> 00:02:46.985
and there's the quotient.

00:02:50.840 --> 00:02:54.680
The significant figures in your
product or your quotient cannot

00:02:54.680 --> 00:02:58.331
be any more than the least
number of significant digits

00:02:58.331 --> 00:03:00.580
in whatever you are using
to come up with that product

00:03:00.580 --> 00:03:01.440
or quotient.

00:03:01.440 --> 00:03:04.250
So over here, both of these
have three significant figures.

00:03:04.250 --> 00:03:06.700
So I can only have three
significant figures

00:03:06.700 --> 00:03:09.189
in my product.

00:03:09.189 --> 00:03:10.730
If this had three
significant figures

00:03:10.730 --> 00:03:12.410
and this had two
significant figures,

00:03:12.410 --> 00:03:14.580
I could have only have
two significant figures

00:03:14.580 --> 00:03:15.680
in my product.

00:03:15.680 --> 00:03:18.520
So in order to be
kind of legit here,

00:03:18.520 --> 00:03:21.090
I have to round this to
three significant figures.

00:03:24.390 --> 00:03:27.370
So I'm going to round to
the nearest hundredth here.

00:03:27.370 --> 00:03:30.200
And so this 2 we'll round down.

00:03:30.200 --> 00:03:38.330
So this gets us to
3.53 meters squared.

00:03:38.330 --> 00:03:41.770
And now we're cool with
the significant figures.

00:03:41.770 --> 00:03:44.360
Let's do another
situation with division.

00:03:44.360 --> 00:03:48.680
Let's say that I'm laying
tiles down in my bathroom.

00:03:48.680 --> 00:03:52.170
And so the diagram
will look very similar.

00:03:52.170 --> 00:04:03.229
And I measure the width of
my bathroom to be-- I'll

00:04:03.229 --> 00:04:04.020
do it in feet, now.

00:04:04.020 --> 00:04:09.012
So let's say it is 10.1 feet.

00:04:09.012 --> 00:04:11.470
And this is the precision that
I'm able to measure it with.

00:04:11.470 --> 00:04:14.460
So I'm able to measure it to
the nearest tenth of a foot.

00:04:14.460 --> 00:04:22.880
And let's say that the length
of my floor-- I'll just make up

00:04:22.880 --> 00:04:27.420
a number-- is 12 point--
and for whatever reason,

00:04:27.420 --> 00:04:30.090
I was able to measure this with
slightly higher precision, so

00:04:30.090 --> 00:04:33.400
12.07 feet.

00:04:33.400 --> 00:04:39.140
And let's say that I have tiles.

00:04:39.140 --> 00:04:43.930
And the tile has an area--
so someone else measured it

00:04:43.930 --> 00:04:46.920
for me.

00:04:46.920 --> 00:04:54.756
Let's say that the area of
this tile is 1.07 feet squared.

00:04:54.756 --> 00:04:56.462
And this was just
measured for me.

00:04:56.462 --> 00:04:59.770
And I want to do is I want
to figure out how many tiles

00:04:59.770 --> 00:05:03.280
can fit on this bathroom floor.

00:05:03.280 --> 00:05:05.710
So what I would do
is I would figure out

00:05:05.710 --> 00:05:08.230
the area of this bathroom
floor and then divide

00:05:08.230 --> 00:05:10.330
by the area of the tiles.

00:05:10.330 --> 00:05:17.910
And so the area of the
bathroom floor, so floor area

00:05:17.910 --> 00:05:25.570
is going to be equal to
10.1 feet times 12.07 feet.

00:05:25.570 --> 00:05:31.690
And so that will give
us-- let's calculate it.

00:05:31.690 --> 00:05:37.065
It is 10.1 times 12.07 feet.

00:05:37.065 --> 00:05:42.370
So it gives us 121.907.

00:05:42.370 --> 00:05:45.150
So this is equal to-- let
me scroll over a little bit

00:05:45.150 --> 00:05:47.790
to the right.

00:05:47.790 --> 00:05:49.740
I could go a little
bit more to the right.

00:05:49.740 --> 00:05:56.580
This is equal to 121.907
feet squared, or square feet.

00:05:56.580 --> 00:05:58.660
Now, we're not done
with our calculation.

00:05:58.660 --> 00:06:00.410
But there might be a
temptation right here

00:06:00.410 --> 00:06:03.490
to say, look, I had four
significant figures over here.

00:06:03.490 --> 00:06:05.900
I have three significant
figures over here.

00:06:05.900 --> 00:06:08.890
There would be a temptation to
say, look, my area should not

00:06:08.890 --> 00:06:10.830
have more than three
significant figures.

00:06:10.830 --> 00:06:13.770
And that temptation
would be OK if this

00:06:13.770 --> 00:06:15.940
is all you were looking
for, if the final answer

00:06:15.940 --> 00:06:18.650
you were looking for was
the area of the floor.

00:06:18.650 --> 00:06:20.720
But we're not done
with our calculation.

00:06:20.720 --> 00:06:23.680
We want to figure out
how many of these tiles

00:06:23.680 --> 00:06:25.614
will fit into this area.

00:06:25.614 --> 00:06:27.030
And so the general
rule of thumb--

00:06:27.030 --> 00:06:29.530
because you don't want
to lose information.

00:06:29.530 --> 00:06:31.360
The general rule
of thumb is don't

00:06:31.360 --> 00:06:33.430
round to significant
figures until you

00:06:33.430 --> 00:06:37.120
are done with your calculation,
especially if you're just

00:06:37.120 --> 00:06:39.490
doing a bunch of
multiplying and dividing.

00:06:39.490 --> 00:06:41.680
Because otherwise,
if you round here,

00:06:41.680 --> 00:06:44.620
you actually will introduce
more error into your calculation

00:06:44.620 --> 00:06:45.500
than you'd want to.

00:06:45.500 --> 00:06:48.250
So what you do is you keep it
as kind of the full number.

00:06:48.250 --> 00:06:49.509
Now you do the division.

00:06:49.509 --> 00:06:50.550
So let's do the division.

00:06:50.550 --> 00:06:57.920
So the tiles per floor-- I
guess we could say my bathroom,

00:06:57.920 --> 00:07:04.540
or tiles in the bathroom,
tiles fitting in bathroom,

00:07:04.540 --> 00:07:06.180
in the floor of this bathroom.

00:07:06.180 --> 00:07:11.640
It would be the area of the
bathroom, so 121.907 feet

00:07:11.640 --> 00:07:15.120
squared, divided by the
area of the tile, divided

00:07:15.120 --> 00:07:19.180
by 1.07 feet squared.

00:07:19.180 --> 00:07:22.460
And once again, let me
get the calculator out.

00:07:22.460 --> 00:07:29.680
And so we have 121.907
divided by 1.07.

00:07:29.680 --> 00:07:32.500
And you get this crazy thing
with all of these digits.

00:07:32.500 --> 00:07:34.320
But this is going to
be our final answer,

00:07:34.320 --> 00:07:36.970
so here we do care about
significant figures.

00:07:36.970 --> 00:07:38.790
So tiles fitting
in the bathroom,

00:07:38.790 --> 00:07:41.160
we get something that
actually just keeps going.

00:07:41.160 --> 00:07:42.620
Let me write this
in a new color.

00:07:42.620 --> 00:07:49.760
We get 113.931775701--
and it actually

00:07:49.760 --> 00:07:51.941
just keeps going-- feet squared.

00:07:51.941 --> 00:07:53.440
And since this is
the final answer--

00:07:53.440 --> 00:07:56.700
we care about how many tiles
will fit onto this bathroom

00:07:56.700 --> 00:07:59.924
floor-- now the significant
figures come into play.

00:07:59.924 --> 00:08:01.340
And the way to
think about this is

00:08:01.340 --> 00:08:04.090
I have four significant
figures over here.

00:08:04.090 --> 00:08:06.720
I have two significant
figures over here.

00:08:06.720 --> 00:08:09.400
I have three significant
figures over here.

00:08:09.400 --> 00:08:15.720
And since we did just a bunch
of multiplying and dividing,

00:08:15.720 --> 00:08:17.250
we have to have the minimum.

00:08:17.250 --> 00:08:19.280
Whatever is the minimum
significant figures

00:08:19.280 --> 00:08:20.780
of the things that
we computed with,

00:08:20.780 --> 00:08:22.520
that's how many
significant figures

00:08:22.520 --> 00:08:23.950
we can have in our final answer.

00:08:23.950 --> 00:08:24.950
Oh, and let me make this clear.

00:08:24.950 --> 00:08:27.366
This isn't two significant
figures, this is three-- the 1,

00:08:27.366 --> 00:08:28.350
the 0, and the 1.

00:08:28.350 --> 00:08:33.980
So our final answer can only
have three significant figures.

00:08:33.980 --> 00:08:36.919
So we need to round
to the nearest foot.

00:08:36.919 --> 00:08:42.120
The next digit over is a 9,
so we're going to round up.

00:08:42.120 --> 00:08:45.480
So this will get us to 114.

00:08:45.480 --> 00:08:47.450
Actually, the units here
aren't in square feet.

00:08:47.450 --> 00:08:48.910
This is in tiles.

00:08:48.910 --> 00:08:51.660
This is feet divided
by feet, and so this

00:08:51.660 --> 00:08:56.490
is going to be 114 tiles.

00:08:56.490 --> 00:08:59.290
Obviously, it's not going
to be exactly 114 tiles.

00:08:59.290 --> 00:09:03.040
But based on the precision of
the measurements we've done,

00:09:03.040 --> 00:09:04.499
we can say 114 tiles.

00:09:04.499 --> 00:09:06.290
Now, what I have just
showed you right here

00:09:06.290 --> 00:09:09.187
is when we multiply
and divide measurements

00:09:09.187 --> 00:09:11.270
that have a certain number
of significant figures.

00:09:11.270 --> 00:09:13.055
The general rule of
thumb is whatever

00:09:13.055 --> 00:09:15.370
is the minimum number of
significant figures in any

00:09:15.370 --> 00:09:16.744
of the numbers
you've calculated,

00:09:16.744 --> 00:09:19.390
that's how many significant
figures-- or the least number

00:09:19.390 --> 00:09:22.050
is the number of significant
figures in your final quotient

00:09:22.050 --> 00:09:24.060
or product or answer.

00:09:24.060 --> 00:09:26.184
When you do addition
and subtraction,

00:09:26.184 --> 00:09:27.350
it's a little bit different.

00:09:27.350 --> 00:09:30.770
And we'll cover that
in the next video.


More on significant figures

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:02.500
Based on the examples
in the last video,

00:00:02.500 --> 00:00:05.180
let's see if we can come
up with some rules of thumb

00:00:05.180 --> 00:00:07.760
for figuring out how
many significant figures

00:00:07.760 --> 00:00:09.260
or how many significant
digits there

00:00:09.260 --> 00:00:12.350
are in a number
or a measurement.

00:00:12.350 --> 00:00:14.140
So the first thing
that is pretty obvious

00:00:14.140 --> 00:00:17.140
is that any non-zero
digit and any

00:00:17.140 --> 00:00:19.720
of the zero digits in
between are significant.

00:00:19.720 --> 00:00:21.930
Clearly, the 7 and the
5 here are significant.

00:00:21.930 --> 00:00:24.550
And the 0 in between them, it's
also going to be significant.

00:00:24.550 --> 00:00:25.900
So let's write this over here.

00:00:25.900 --> 00:00:37.940
So any non-zero digits
and zeros in between

00:00:37.940 --> 00:00:42.020
are going to be significant.

00:00:42.020 --> 00:00:44.170
That's pretty straightforward.

00:00:44.170 --> 00:00:47.680
Now, the zeros that are not in
between nonzero digits, these

00:00:47.680 --> 00:00:49.610
become a little
bit more confusing.

00:00:49.610 --> 00:00:52.740
So let's just make sure we
can rule out some of them.

00:00:52.740 --> 00:00:55.730
So you can always rule out
when you're thinking about

00:00:55.730 --> 00:00:57.272
significant figures,
the leading 0's.

00:00:57.272 --> 00:00:58.980
And when I'm talking
about leading zeros,

00:00:58.980 --> 00:01:00.910
I'm talking about
the zeros that come

00:01:00.910 --> 00:01:03.720
before your non-zero digit.

00:01:03.720 --> 00:01:05.670
So these are
leading zeroes here.

00:01:05.670 --> 00:01:07.220
These are leading zeroes.

00:01:07.220 --> 00:01:09.061
There is no leading zeroes here.

00:01:09.061 --> 00:01:11.310
No leading zeroes in this
one, this one, and this one.

00:01:11.310 --> 00:01:14.200
But in any situation,
the leading zeros

00:01:14.200 --> 00:01:15.960
are not significant.

00:01:15.960 --> 00:01:17.830
So leading zeros
not significant,

00:01:17.830 --> 00:01:18.830
I'll write it over here.

00:01:18.830 --> 00:01:26.180
Leading zeros not significant.

00:01:26.180 --> 00:01:28.320
And so the last question,
all you have left,

00:01:28.320 --> 00:01:31.389
I mean you only have non-zero
digits and zeros in between.

00:01:31.389 --> 00:01:33.680
You could have some leading
zeros, which you've already

00:01:33.680 --> 00:01:35.020
said are not significant.

00:01:35.020 --> 00:01:37.269
And so the only thing left
that you have to figure out

00:01:37.269 --> 00:01:41.440
is what do you do with the
trailing zeroes, the zeroes

00:01:41.440 --> 00:01:44.450
behind the last
non-zero, or to the right

00:01:44.450 --> 00:01:46.820
of the last non-zero digit.

00:01:46.820 --> 00:01:49.680
So these trailing zeroes here.

00:01:49.680 --> 00:01:52.090
There's actually two
trailing zeroes over here.

00:01:52.090 --> 00:01:54.490
And then there's three
trailing zeroes over here.

00:01:54.490 --> 00:01:57.365
So let me make a little--
so trailing zeroes.

00:02:00.790 --> 00:02:03.066
Trailing zeroes, what
do we do with them?

00:02:03.066 --> 00:02:04.440
So the easy way
to think about is

00:02:04.440 --> 00:02:08.490
if you have a decimal, if
there's a decimal anywhere

00:02:08.490 --> 00:02:10.550
in your number, count them.

00:02:10.550 --> 00:02:13.550
If you have a
decimal, count them.

00:02:13.550 --> 00:02:15.560
Count them as significant.

00:02:15.560 --> 00:02:20.790
They are significant,
count them as significant.

00:02:20.790 --> 00:02:25.790
If there's no decimal
anywhere in the number,

00:02:25.790 --> 00:02:27.810
then it's kind of ambiguous.

00:02:27.810 --> 00:02:31.190
You're kind of not sure
and this is a situation.

00:02:31.190 --> 00:02:33.980
So clearly over here, there's
a decimal in the number,

00:02:33.980 --> 00:02:35.560
so you count the
trailing zeroes.

00:02:35.560 --> 00:02:37.540
These are adding
to the precision.

00:02:37.540 --> 00:02:40.950
Over here there's a decimal,
so you count the trailing zero.

00:02:40.950 --> 00:02:43.580
There's a decimal here, so
you count the trailing zeroes.

00:02:43.580 --> 00:02:45.440
There are no
trailing zeroes here.

00:02:45.440 --> 00:02:49.960
And over here-- well, the way
I later put a decimal here.

00:02:49.960 --> 00:02:51.450
Here you would count it.

00:02:51.450 --> 00:02:52.940
So if you have
the decimal there,

00:02:52.940 --> 00:02:54.460
you would count all five.

00:02:54.460 --> 00:02:56.640
If you didn't have the
decimal, if you just

00:02:56.640 --> 00:03:01.650
had 37,000 like
that, it's ambiguous.

00:03:01.650 --> 00:03:04.420
And if someone doesn't
give you more information,

00:03:04.420 --> 00:03:06.800
your best assumption is
probably that they just

00:03:06.800 --> 00:03:08.779
measured to the
nearest thousand.

00:03:08.779 --> 00:03:10.820
That they didn't measure
exactly the one and just

00:03:10.820 --> 00:03:13.610
happened to get
exactly on 37,000.

00:03:13.610 --> 00:03:15.910
So if there's no
decimal, let me write

00:03:15.910 --> 00:03:20.820
it this way-- it's
ambiguous, which

00:03:20.820 --> 00:03:23.590
means that you're not
sure what it means,

00:03:23.590 --> 00:03:25.820
it's not clear what it means.

00:03:25.820 --> 00:03:28.755
And you're probably safer
assuming to not count it.

00:03:35.769 --> 00:03:38.310
If someone really does measure,
if you were to really measure

00:03:38.310 --> 00:03:40.610
something to the
exact one, then you

00:03:40.610 --> 00:03:43.810
should put a decimal
at the end like that.

00:03:43.810 --> 00:03:46.970
And there is a notation
for specifying.

00:03:46.970 --> 00:03:49.970
Let's say you do measure-- and
let me do a different number.

00:03:49.970 --> 00:03:54.270
Let's say you do measure 56,000.

00:03:54.270 --> 00:03:57.630
And there is a notation for
specifying that 6 definitely

00:03:57.630 --> 00:03:59.360
is the last significant digit.

00:03:59.360 --> 00:04:01.990
And sometimes you'll see
a bar put over the 6,

00:04:01.990 --> 00:04:04.740
sometimes you'll see
a bar put under the 6.

00:04:04.740 --> 00:04:07.440
And that could be
useful because maybe

00:04:07.440 --> 00:04:09.980
your last significant digit
is this zero over here.

00:04:09.980 --> 00:04:12.030
Maybe you were able to
measure to the hundreds

00:04:12.030 --> 00:04:13.930
with a reasonable
level of precision.

00:04:13.930 --> 00:04:20.250
And so then you would write
something like 56,000,

00:04:20.250 --> 00:04:22.710
but then you would put
the bar above that zero,

00:04:22.710 --> 00:04:24.540
or the bar below
that zero to say

00:04:24.540 --> 00:04:26.804
that that was the last
significant digit.

00:04:26.804 --> 00:04:28.220
So if you saw
something like this,

00:04:28.220 --> 00:04:30.910
you would say three
significant digits.

00:04:30.910 --> 00:04:33.170
This isn't used so frequently.

00:04:33.170 --> 00:04:35.850
A better way to show
that you've measured

00:04:35.850 --> 00:04:37.800
to three significant
digits would

00:04:37.800 --> 00:04:40.001
be to write it in
scientific notation.

00:04:40.001 --> 00:04:41.250
There's a whole video on that.

00:04:41.250 --> 00:04:43.540
But to write this in
scientific notation,

00:04:43.540 --> 00:04:51.800
you could write this as 5.60
times 10 to the fourth power.

00:04:51.800 --> 00:04:55.397
Because if you multiply
this times to the fourth,

00:04:55.397 --> 00:04:57.230
you would move this
decimal over four spaces

00:04:57.230 --> 00:04:59.630
and get us to 56,000.

00:04:59.630 --> 00:05:02.372
So 5.60 times 10 to the fourth.

00:05:02.372 --> 00:05:04.080
And if this confuses
you, watch the video

00:05:04.080 --> 00:05:05.310
on scientific notation.

00:05:05.310 --> 00:05:07.640
It will hopefully clarify
things a little bit.

00:05:07.640 --> 00:05:10.300
But when you write a number
in scientific notation,

00:05:10.300 --> 00:05:12.440
it makes it very clear
about your precision

00:05:12.440 --> 00:05:14.760
and how many significant
digits you're dealing with.

00:05:14.760 --> 00:05:16.426
So instead of doing
this notation that's

00:05:16.426 --> 00:05:18.700
a little bit outdated-- I
haven't seen it used much

00:05:18.700 --> 00:05:22.170
with these bars below or above
the high significant digit,

00:05:22.170 --> 00:05:24.080
instead you could
represent it with a decimal

00:05:24.080 --> 00:05:25.970
in scientific notation.

00:05:25.970 --> 00:05:28.540
And there it's
very clear that you

00:05:28.540 --> 00:05:30.797
have three significant digits.

00:05:30.797 --> 00:05:32.130
So hopefully that helps you out.

00:05:32.130 --> 00:05:34.380
In the next couple of videos,
we'll explore a little bit more

00:05:34.380 --> 00:05:36.870
why significant digits are
important, especially when you

00:05:36.870 --> 00:05:40.010
do calculations with
multiple measurements.


Significant figures

WEBVTT
Kind: captions
Language: en

00:00:00.094 --> 00:00:01.760
Let's see if we can
learn a thing or two

00:00:01.760 --> 00:00:06.020
about significant
figures, sometimes

00:00:06.020 --> 00:00:07.880
called significant digits.

00:00:07.880 --> 00:00:10.020
And the idea behind
significant figures

00:00:10.020 --> 00:00:13.040
is just to make sure that
when you do a big computation

00:00:13.040 --> 00:00:14.610
and you have a bunch
of digits there,

00:00:14.610 --> 00:00:16.140
that you're not
over-representing

00:00:16.140 --> 00:00:17.790
the amount of
precision that you had,

00:00:17.790 --> 00:00:20.900
that the result isn't more
precise than the things

00:00:20.900 --> 00:00:24.214
that you actually measured, that
you used to get that result.

00:00:24.214 --> 00:00:25.630
Before we go into
the depths of it

00:00:25.630 --> 00:00:27.150
and how you use it
with computation,

00:00:27.150 --> 00:00:28.600
let's just do a
bunch of examples

00:00:28.600 --> 00:00:30.180
of identifying
significant figures.

00:00:30.180 --> 00:00:32.350
Then we'll try to come up
with some rules of thumb.

00:00:32.350 --> 00:00:34.960
But the general way to think
about it is, which digits

00:00:34.960 --> 00:00:38.110
are really giving me
information about how precise

00:00:38.110 --> 00:00:39.720
my measurement is?

00:00:39.720 --> 00:00:41.950
So on this first
thing right over here,

00:00:41.950 --> 00:00:45.700
the significant figures
are this 7, 0, 0.

00:00:45.700 --> 00:00:51.422
So over here, you have
three significant figures.

00:00:51.422 --> 00:00:53.880
And it might make you a little
uncomfortable that we're not

00:00:53.880 --> 00:00:57.010
including these 0's that
are after the decimal point

00:00:57.010 --> 00:00:59.720
and before this 7, that
we're not including those.

00:00:59.720 --> 00:01:02.510
Because you're just like, that
does help define the number.

00:01:02.510 --> 00:01:05.180
And that is true, but
it's not telling us

00:01:05.180 --> 00:01:06.710
how precise our measurement is.

00:01:06.710 --> 00:01:08.840
And to try to understand
this a little bit better,

00:01:08.840 --> 00:01:11.600
imagine if this right over
here was a measurement

00:01:11.600 --> 00:01:19.960
of kilometers, so if we
measured 0.00700 kilometers.

00:01:19.960 --> 00:01:26.300
This would be the exact
same thing as 7.00 meters.

00:01:26.300 --> 00:01:28.630
Maybe, in fact, we just
used a meter stick.

00:01:28.630 --> 00:01:32.010
And we said it's
exactly 7.00 meters.

00:01:32.010 --> 00:01:34.030
So we measured to the
nearest centimeter.

00:01:34.030 --> 00:01:36.880
And we just felt like
writing it in kilometers.

00:01:36.880 --> 00:01:39.252
These two numbers are
the exact same thing.

00:01:39.252 --> 00:01:40.460
They're just different units.

00:01:40.460 --> 00:01:41.960
But I think when
you look over here,

00:01:41.960 --> 00:01:43.720
it makes a lot more
sense why you only

00:01:43.720 --> 00:01:45.590
have three significant figures.

00:01:45.590 --> 00:01:50.140
These 0's are just
shifting it based

00:01:50.140 --> 00:01:53.107
on the units of measurement
that you're using.

00:01:53.107 --> 00:01:55.440
But the numbers that are
really giving you the precision

00:01:55.440 --> 00:01:58.189
are the 7, the 0, and the 0.

00:01:58.189 --> 00:02:00.480
And the reason why we're
counting these trailing 0's is

00:02:00.480 --> 00:02:03.696
that whoever wrote this number
didn't have to write them down.

00:02:03.696 --> 00:02:05.320
They wrote them down
to explicitly say,

00:02:05.320 --> 00:02:06.819
look, I measured this far.

00:02:06.819 --> 00:02:08.360
If they didn't
measure this far, they

00:02:08.360 --> 00:02:10.270
would have just
left these 0's off.

00:02:10.270 --> 00:02:14.540
And they would have just
told you 7 meters, not 7.00.

00:02:14.540 --> 00:02:15.600
Let's do the next one.

00:02:15.600 --> 00:02:18.660
So based on the same idea,
we have the 5 and the 2.

00:02:18.660 --> 00:02:21.330
The non-zero digits are going
to be significant figures.

00:02:21.330 --> 00:02:23.560
You don't include
this leading 0,

00:02:23.560 --> 00:02:28.610
by the same logic that if
this was 0.052 kilometers,

00:02:28.610 --> 00:02:33.290
this would be the same thing as
52 meters, which clearly only

00:02:33.290 --> 00:02:37.410
has two significant figures.

00:02:37.410 --> 00:02:42.050
So you don't want
to count leading

00:02:42.050 --> 00:02:52.770
0's before the first non-zero
digit, I guess we could say.

00:02:52.770 --> 00:02:54.570
You don't want to include those.

00:02:54.570 --> 00:02:57.370
You just want to include all the
non-zero digits and everything

00:02:57.370 --> 00:03:03.750
in between, and trailing 0's
if a decimal point is involved.

00:03:03.750 --> 00:03:06.150
I'll make those ideas a
little bit more formal.

00:03:06.150 --> 00:03:08.720
So over here, the
person did 370.

00:03:08.720 --> 00:03:10.636
And then they wrote
the decimal point.

00:03:10.636 --> 00:03:12.260
If they didn't write
the decimal point,

00:03:12.260 --> 00:03:14.702
it would be a little unclear
on how precise this was.

00:03:14.702 --> 00:03:16.410
But because they wrote
the decimal point,

00:03:16.410 --> 00:03:19.130
it means that they measured
it exactly to be 370.

00:03:19.130 --> 00:03:21.990
They didn't get 372
and then round down.

00:03:21.990 --> 00:03:24.360
Or they didn't have
kind of a roughness only

00:03:24.360 --> 00:03:25.980
to the nearest tens place.

00:03:25.980 --> 00:03:29.120
This decimal tells you that all
three of these are significant.

00:03:29.120 --> 00:03:34.100
So this is three significant
figures over here.

00:03:34.100 --> 00:03:37.090
Then on this next one,
once again, this decimal

00:03:37.090 --> 00:03:40.210
tells us that not only did
we get to the nearest one,

00:03:40.210 --> 00:03:41.960
but then we put another
trailing 0 here,

00:03:41.960 --> 00:03:43.890
which means we got
to the nearest tenth.

00:03:43.890 --> 00:03:45.520
So in this situation,
once again,

00:03:45.520 --> 00:03:48.760
we have three
significant figures.

00:03:48.760 --> 00:03:52.100
Over here, the 7
is in the hundreds.

00:03:52.100 --> 00:03:54.300
But the measurement
went all the way down

00:03:54.300 --> 00:03:55.980
to the thousandths place.

00:03:55.980 --> 00:03:57.760
And even though there
are 0's in between,

00:03:57.760 --> 00:03:59.630
those 0's are part
of our measurement,

00:03:59.630 --> 00:04:02.790
because they are in
between non-zero digits.

00:04:02.790 --> 00:04:06.660
So in this situation,
every digit here,

00:04:06.660 --> 00:04:08.970
the way it's written,
is a significant digit.

00:04:08.970 --> 00:04:12.530
So you have six
significant digits.

00:04:12.530 --> 00:04:14.830
Now, this last one is ambiguous.

00:04:14.830 --> 00:04:17.209
The 37,000-- it's
not clear whether you

00:04:17.209 --> 00:04:19.079
measured exactly 37,000.

00:04:19.079 --> 00:04:21.089
Maybe you measured
to the nearest one,

00:04:21.089 --> 00:04:22.890
and you got an exact number.

00:04:22.890 --> 00:04:25.430
You got exactly 37,000.

00:04:25.430 --> 00:04:29.500
Or maybe you only measured
to the nearest thousand.

00:04:29.500 --> 00:04:33.780
So there's a little
bit of ambiguity here.

00:04:33.780 --> 00:04:36.600
If you just see something
written exactly like this,

00:04:36.600 --> 00:04:39.629
you would probably say, if you
had to guess-- or not guess.

00:04:39.629 --> 00:04:41.170
If there wasn't any
more information,

00:04:41.170 --> 00:04:46.100
you would say that there's
just two significant figures

00:04:46.100 --> 00:04:47.690
or significant digits.

00:04:47.690 --> 00:04:49.730
For this person to
be less ambiguous,

00:04:49.730 --> 00:04:52.620
they would want to put a
decimal point right over there.

00:04:52.620 --> 00:04:55.120
And that lets you know
that this is actually

00:04:55.120 --> 00:04:57.060
five digits of precision,
that we actually

00:04:57.060 --> 00:04:58.590
go to five significant figures.

00:04:58.590 --> 00:05:02.690
So if you don't see that decimal
point, I would go with two.