Introduction to energy

WEBVTT
Kind: captions
Language: en

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- [Tutor] Let's talk about one
of the most fundamental ideas

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in science and that is
the notion of energy

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and energy definitely has some
meaning in our everyday life,

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if we kind of imagine
things that are moving

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or hot or bright as being energetic,

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but what I wanna talk about in this video

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is a more formal definition of it,

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a more scientific definition
and the most typical one,

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that's often given is
the ability to do work,

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ability to do and I'm
gonna put work in quotes,

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because the notion of work here

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isn't the everyday notion of
work, where you go to your job

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and you work nine to
five and you get paid,

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work in a physics context is a little bit,

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it's not completely unrelated

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to our everyday notion of work,

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but I'll give you an example

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just to get a better idea of it,

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so let's say that you
have some object here

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and you were to apply a force,

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you were to apply a
force in that direction

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and the magnitude of that
force, let's say it's 10 newtons

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and if the units, newtons in force

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isn't too familiar to
you, don't worry too much,

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but you can also review it on
those videos on Khan Academy,

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but you apply a force to
the right on this object

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and by doing that you're
able to move the object,

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you're able to displace the object

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in the same direction as that force,

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so you're able to displace
it, let's say 10 meters,

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so after you've done it, the object,

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the object is right over here,

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so when you do this, you apply a force

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and it's causing the
object to be displaced

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in the direction of that force,

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you would say that work has been performed

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and the amount of work
that has been performed

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would be 10 newtons times 10 meters

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and so 10 times 10, it
would be equal to 100

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and then the units are
newton meters of work,

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100 newton meters,

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'cause you're multiplying
newtons times meters

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of newton meters of
work and newton meters,

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that has been defined as the joule,

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which is the unit of work
and also the unit of energy,

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so this is the same thing as 100,

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I could write it out as joules, 100 joules

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or we could just abbreviate it with a J,

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so 100 joules of work
has been performed here

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by moving this, so we've
done something here

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and this is considered to
be work, 100 joules of work,

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if we move this twice as far,

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then it would be 200 joules of work

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and so energy is the ability
to do this type of work.

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Now let's look at these pictures here,

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which are depicting
different forms of energy

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and let's see if we can
identify the forms of energy

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and then think about how they can relate

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to actually doing work.

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So if we look at the fire here,

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there's some maybe
obvious forms of energy,

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we have some thermal energy,

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fires are definitely
hot, so thermal energy,

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but we should think about

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what is thermal energy fundamentally?

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A system's temperature is really about

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the average kinetic
energy of its molecules,

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so thermal energy is really
about the energy of movement,

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it's really about all these
little molecules here,

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because of the combustion
reaction going on,

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they're getting excited and
they have higher kinetic energy

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and so the temperature goes up,

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their average kinetic energy goes up,

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so thermal energy is really a
form of energy due to movement

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and the general term for
energy due to movement

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is kinetic energy, so thermal
energy is really a form

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of kinetic, kinetic,

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kinetic energy.

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You also have light being emitted,

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that has energy as well, we
call that radiant energy,

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so that light being emitted,
that's the reason why

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we can see this fire,
radiant, radiant energy.

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Now you might say,

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"Okay, maybe that's all of
the energy in the system,"

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but I'll say no, there's
another form of energy

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and actually even in this picture,

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that's probably where
most of the energy is

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and that's potential energy,

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so where is the potential energy?

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Well, it's sitting in the
bonds of the fuel over here,

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so these are either chips of
wood or charcoal of some kind,

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but these are formed
by carbon-carbon bonds,

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so you have these carbon-carbon bonds

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and they could be bonded to other carbons

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or other things and they're
also going to be bound

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to some hydrogens here and there,

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so you're gonna have, you're
gonna have bonds like this,

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that actually store energy in them,

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they have the potential to be released,

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if you're able to break these bonds,

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those electrons are gonna get
into a lower energy state,

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or they might bond with other
things and in the process,

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they're going to release energy,

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that's gonna be radiant energy
and thermal/kinetic energy,

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so how does this happen?

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How do these bonds actually get broken?

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Well, that's our good old
friend, the combustion reaction,

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that's our good old friend,
the combustion reaction,

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where you take some oxygen,

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you take some heat or we
could say some energy,

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so it takes a little
energy to get started,

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that's why you might have to light this

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with a match to begin with,

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so oxygen plus energy,

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plus energy

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and then you could say plus
these carbon-carbon bonds

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and you could say plus
you know, whatever it is,

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these fuels, which are made out of carbon,

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either charcoal or wood, so
plus, I'll do it like this,

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I'll draw some carbon-carbon
bond right over here,

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that's going to combust and
I'll do this in a color,

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so that is going to,

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I'm really having trouble changing colors,

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this is going to combust,

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combust

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and it's going to release,
it's going to release

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water, because the fuel
has hydrogens in it,

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it's going to release carbon dioxide

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and it's going to release
a lot more energy,

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I'll do that in caps,

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it's going to release a lot more energy

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and that energy we see in the form

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of the kinetic energy of the molecules

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and the radiant energy being emitted.

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Now you might say, "Okay, I can buy that,

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"I have this potential energy here,"

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and this potential energy

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that's in the bonds between these atoms,

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we call that chemical potential,

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so we have chemical,

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chemical potential energy,

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potential, potential energy

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is right over there, but you
might say, "Okay, I buy that,

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"the chemical energy is being converted

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"into the thermal energy
and the radiant energy,"

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and this is actually an interesting point,

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energy, this is the law
of conservation of energy,

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energy cannot be created or destroyed,

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it can only be converted
from one form or another,

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but you might be saying,

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"Okay, I can convert
from one form to another,

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"but how can this actually do work

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"in the way that I've even depicted here?"

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Well, the entire Industrial Revolution

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is all about trying to convert
from one form of energy

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to another and also to do work,

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so a steam engine is fundamentally
based upon combustion,

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heating up some steam and
then that steam can expand

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and then it can push a piston
to do all sorts of things,

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including move a train,

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combustion is what's going
on in your car engines,

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where the pistons are expanding
due to the thermal energy

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and then that helps drive the drive train

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of the actual car, so
it can clearly do work.

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So here we have some other
examples, this is lightning

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and so when you see the lightning,

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there's something clearly
very kinetic is going on,

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you have electrons, you have electrons

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moving from the cloud, from
the cloud to the ground

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and you might say, so
this right over here,

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that is, you could say
that's kinetic energy,

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kinetic, kinetic energy and you might say,

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"Well, how can I do work with that?"

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Well, that's what the
whole electronics industry

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is all about, that's what
power lines are all about,

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movement of electrons, that's current

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and current can be used to do
all sorts of amazing things,

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you can actually have an electric motor

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is one way to actually do it,

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so that's kinetic energy there,

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there's clearly radiant energy going on,

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we can see the lightning and
that radiant energies do,

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because the air gets
ionized and gets heated

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and so there's also thermal energy,

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as the electrons go down,

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there's heat that is
actually being generated.

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Now, where did this energy come from?

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It just doesn't come from anywhere,

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well, you have all of
this potential energy,

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that starts building up in these clouds

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as the water vapor rises and the mechanism

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isn't fully understood
of how this happens,

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but because of energy from the sun,

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you have water vapor rising,

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as the water vapor rises
through the clouds,

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the bottom part of the
cloud becomes more negative,

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it becomes more electron rich

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and the top part of the
cloud become more positive

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and so you have these electrons,

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that really wanna get
down here onto the ground,

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because the air above the
ground becomes more negative,

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the ground starts becoming more positive

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and so you can imagine these electrons,

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more and more wanna get down here,

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but this air isn't a natural conductor,

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but once the electric
potential gets high enough,

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these electrons find a way, the
air essentially gets ionized

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and the electrons are able to find a path,

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so while this is all building up,

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you have this electrostatic
potential building,

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so this is electrostatic,

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static, you can't see it that well,

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electrostatic potential
and how this forms,

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once again, it's an area
that people are still,

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there's some good theories out
there about how this forms,

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but it's not 100% well established

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and over here in this third old drawing

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of this person doing a handstand dive,

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this is probably the most typical example

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of potential energy being
converted into kinetic energy

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that you might find in a physics textbook.

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Over here at the top of the diving board,

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this gentleman has potential energy

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by virtue of his position and over here,

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it's very clear he has
the potential to fall

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and it has the potential to
turn it into kinetic energy

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and so once he falls over, at this point,

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most of his potential energy

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has been converted into kinetic energy,

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so here it's potential,
here it's potential

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and here it is kinetic,

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kinetic energy.

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So the big takeaway is energy
has the ability to do work,

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it cannot be created or destroyed,

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but it can be converted
from one form to another

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and all of the forms at their essence,

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you can really think about
them in two big buckets,

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you can think about
them as potential energy

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or kinetic energy,

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or kinetic energy

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and as a last example, you might say,

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"Well, how can this guy do work?"

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Well, you can imagine if there
was some type of system here,

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new system here, where, you know,

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we'll create some machinery,

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maybe there's a pulley right over here

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and then it's lifting a
weight right over here,

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well, if he jumped on this,
he won't fall down as fast,

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but then if, as long as he's
heavier than this weight,

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it's gonna pull this down

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and then this weight is gonna go up,

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so he has the potential to do work

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by virtue of his position,

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there just isn't this pulley system there

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to get that actual work done.


Example of subtracting fractions with unlike denominators

WEBVTT
Kind: captions
Language: en

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Let's figure out
7/12 minus 6/13.

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And in order to be
able to do this,

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we need to find a
common denominator.

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We notice that they're not
common right over here.

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Here, the denominator is 12.

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Here, it is 13.

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And the common
denominator is going

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to be the least common
multiple of 12 and 13.

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And 12 and 13 share no
common factors other than 1,

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so their least common
multiple is actually

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just going to be 12 times 13.

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Actually, I'm just
going to leave

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it written as 12 times 13.

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So let's first try to
rewrite this right over here.

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And I'll do the
denominator first.

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So 7/12.

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Instead of the
denominator being 12,

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I'm going to write
it as 12 times 13.

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Well, if we multiply the
denominator by 13 in order

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to not change the
value of the fraction,

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we need to multiply
the numerator times 13.

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So notice, all I
did is I multiplied

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the numerator and
the denominator

00:00:56.740 --> 00:00:59.520
by the same number that's
equivalent to just multiplying

00:00:59.520 --> 00:01:01.930
it by 13/13, or 1.

00:01:01.930 --> 00:01:05.190
And so this is still
equivalent to 7/12.

00:01:05.190 --> 00:01:08.740
Similarly-- let me do this
one in blue right over here--

00:01:08.740 --> 00:01:12.760
the common denominator,
we want is 12 times 13.

00:01:12.760 --> 00:01:15.800
So here in the
denominator, we multiply--

00:01:15.800 --> 00:01:18.510
let me now write
it as 13 times 12.

00:01:18.510 --> 00:01:21.140
13 times 12.

00:01:21.140 --> 00:01:23.490
Well, if we multiply
the denominator by 12,

00:01:23.490 --> 00:01:26.500
we have to multiply the
numerator by 12 as well.

00:01:26.500 --> 00:01:29.060
So I'll write the
numerator as 6 times 12.

00:01:29.060 --> 00:01:31.870
Notice, here we multiplied the
numerator and the denominator

00:01:31.870 --> 00:01:32.912
by 13.

00:01:32.912 --> 00:01:35.120
Here, we multiplied the
numerator and the denominator

00:01:35.120 --> 00:01:36.100
by 12.

00:01:36.100 --> 00:01:38.700
How did I know to
multiply this by 12?

00:01:38.700 --> 00:01:42.150
Well, the common
denominator is 13 times 12.

00:01:42.150 --> 00:01:44.330
So here, I multiplied
the denominator by 12

00:01:44.330 --> 00:01:45.990
so I have do the
numerator by 12.

00:01:45.990 --> 00:01:49.210
The common denominator
here is 13 times 12.

00:01:49.210 --> 00:01:51.900
But here, I had to multiply
it by 13 to get it,

00:01:51.900 --> 00:01:55.250
so I have to multiply the
numerator by 13 as well.

00:01:55.250 --> 00:01:57.430
But now we are
ready to subtract.

00:01:57.430 --> 00:01:58.440
You might say, hey, Sal.

00:01:58.440 --> 00:01:58.939
Hold on.

00:01:58.939 --> 00:02:00.680
Wait, what do these
actually evaluate to?

00:02:00.680 --> 00:02:01.971
Well, maybe that's a good idea.

00:02:01.971 --> 00:02:03.470
Let's figure that out first .

00:02:03.470 --> 00:02:07.310
So this is equal to 7 times 13.

00:02:07.310 --> 00:02:10.660
Let's see, 7 times
13 is 70 plus 21.

00:02:10.660 --> 00:02:14.960
It's 91 over 12 times 13.

00:02:14.960 --> 00:02:18.200
Let's see, I haven't
memorized my 13 times tables.

00:02:18.200 --> 00:02:21.360
So we know that 12
times 12 is 144.

00:02:21.360 --> 00:02:24.950
You put on one more
12, you get to 156.

00:02:24.950 --> 00:02:27.240
156.

00:02:27.240 --> 00:02:32.270
So 91/156 is the exact
same thing as 7/12.

00:02:32.270 --> 00:02:36.230
I just multiplied the numerator
and denominator by 13.

00:02:36.230 --> 00:02:42.510
And from that we are going
to subtract 6 times 12.

00:02:42.510 --> 00:02:45.310
6 times 12 is 72.

00:02:45.310 --> 00:02:46.940
72 over.

00:02:46.940 --> 00:02:49.690
Well, we already figured
out what 12 times 13 is

00:02:49.690 --> 00:02:52.080
or what 13 times 12 is.

00:02:52.080 --> 00:02:55.800
It is 156.

00:02:55.800 --> 00:02:57.550
And now that we have
a common denominator,

00:02:57.550 --> 00:03:13.440
we can rewrite this as being
equal to 91 minus 72 over 156.

00:03:13.440 --> 00:03:17.980
I'll just write that in a
neutral color, over 156.

00:03:17.980 --> 00:03:20.910
Let's see, if it was 92
minus 72, it would be 20.

00:03:20.910 --> 00:03:25.260
That's 1 less, so it's
going to be 19/156.

00:03:25.260 --> 00:03:31.200
So this is going to
be equal to 19/156.

00:03:31.200 --> 00:03:34.230
And as far as I can
tell, let's see.

00:03:34.230 --> 00:03:37.150
19 does not go into 156.

00:03:37.150 --> 00:03:40.640
Let me just make sure of that,
that somehow, magically, 156

00:03:40.640 --> 00:03:44.540
isn't a multiple of 19
so I could simplify this.

00:03:44.540 --> 00:03:47.570
So let's see, 19 is almost 20.

00:03:47.570 --> 00:03:50.130
So maybe, let's see
if it will go 7 times.

00:03:50.130 --> 00:03:52.670
7 times 9 is 63.

00:03:52.670 --> 00:03:54.570
7 times 1 is 7.

00:03:54.570 --> 00:03:56.270
Plus 6 is 13.

00:03:56.270 --> 00:03:57.520
Let's see the difference here.

00:03:57.520 --> 00:04:00.120
Actually, I could have
gone in one more time.

00:04:00.120 --> 00:04:02.810
So let's do it eight times.

00:04:02.810 --> 00:04:06.550
8 times 9 is 72.

00:04:06.550 --> 00:04:07.650
8 times 1 is 8.

00:04:07.650 --> 00:04:09.800
Plus 7 is 15.

00:04:09.800 --> 00:04:11.210
Yeah, you have a remainder here.

00:04:11.210 --> 00:04:12.690
So this doesn't go evenly.

00:04:12.690 --> 00:04:15.014
This isn't divisible by 19.

00:04:15.014 --> 00:04:16.930
And they don't share any
other common factors,

00:04:16.930 --> 00:04:19.029
so we've simplified it
about as much as we can.

00:04:19.029 --> 00:04:22.210
This is equal to 19/156.


Adding fractions with unlike denominators: units of 10

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:04.590
Let's see if we can
add 3/10 to 7/100.

00:00:04.590 --> 00:00:07.080
And I encourage you to
try adding these two

00:00:07.080 --> 00:00:11.050
fractions on your own first
before I work through it,

00:00:11.050 --> 00:00:12.660
and I'll give you one hint.

00:00:12.660 --> 00:00:15.490
Right now, it's very hard
to add these fractions.

00:00:15.490 --> 00:00:18.120
You're adding 3/10 to 7/100.

00:00:18.120 --> 00:00:20.130
You're adding two
different fractions

00:00:20.130 --> 00:00:22.130
with two different denominators.

00:00:22.130 --> 00:00:23.760
So what I would
encourage you to do

00:00:23.760 --> 00:00:27.465
is try to rewrite
3/10 in a way that it

00:00:27.465 --> 00:00:30.020
has 100 as a
denominator, or so it's

00:00:30.020 --> 00:00:31.940
expressed in terms
of hundredths,

00:00:31.940 --> 00:00:34.280
and then see if
you can add them.

00:00:34.280 --> 00:00:36.530
Well, I'm assuming
you've given a go at it.

00:00:36.530 --> 00:00:38.780
Let's see how we
can rewrite 3/10,

00:00:38.780 --> 00:00:40.730
and I'll try to visualize it.

00:00:40.730 --> 00:00:45.570
So what I've done here, so this
you could consider a whole,

00:00:45.570 --> 00:00:47.310
and this is a whole as well.

00:00:47.310 --> 00:00:52.891
And this whole is divided into
tenths-- 1,2, 3, 4, 5, 6, 7, 8,

00:00:52.891 --> 00:00:53.840
9, 10.

00:00:53.840 --> 00:00:55.970
So what would 3/10 look like?

00:00:55.970 --> 00:01:03.600
Well, it would be one, two,
and three of the tenths.

00:01:06.750 --> 00:01:10.170
Now, what would happen if
you took each of those tenths

00:01:10.170 --> 00:01:16.230
and you divided them
into 10 more sections?

00:01:16.230 --> 00:01:18.630
So you're essentially
taking each of those tenths

00:01:18.630 --> 00:01:21.610
and you're dividing
them into 10 sections.

00:01:21.610 --> 00:01:24.090
Well, you have 10 sections
and then each of them

00:01:24.090 --> 00:01:26.400
will have 10 subsections in it.

00:01:26.400 --> 00:01:30.030
So you're going to
have hundredths.

00:01:30.030 --> 00:01:32.720
Then the sections are going
to describe hundredths.

00:01:32.720 --> 00:01:34.880
So 10-- let me
write it this way.

00:01:34.880 --> 00:01:38.210
10 times 10.

00:01:40.950 --> 00:01:44.134
You are then going
to have hundredths.

00:01:44.134 --> 00:01:48.690
And 3/10 would be equivalent to
how many of these hundredths?

00:01:48.690 --> 00:01:52.280
Well, each of these
tenths will now become 10.

00:01:52.280 --> 00:02:00.791
So you're going to
have 10, 20, and-- let

00:02:00.791 --> 00:02:01.790
me color them in better.

00:02:01.790 --> 00:02:07.710
So that's 10, 20, and
then 30 hundredths.

00:02:07.710 --> 00:02:10.250
So this part right over here.

00:02:10.250 --> 00:02:13.450
This-- let me get my tool
right-- this part right

00:02:13.450 --> 00:02:21.550
over here, this is going
to be 3 times 10, which

00:02:21.550 --> 00:02:23.960
is going to be equal to 30.

00:02:23.960 --> 00:02:27.122
10 times 10 is equal to 100.

00:02:27.122 --> 00:02:29.330
So this is how we're going
to change the denominator.

00:02:29.330 --> 00:02:30.820
Instead of thinking
in terms of tenths,

00:02:30.820 --> 00:02:32.653
we're going to think
in terms of hundredths.

00:02:32.653 --> 00:02:38.120
And now our numerator, 3/10,
is equivalent to 30 hundredths

00:02:38.120 --> 00:02:40.470
so we can rewrite this fraction.

00:02:40.470 --> 00:02:46.640
We essentially multiplied
the numerator by 10,

00:02:46.640 --> 00:02:48.470
and we multiplied
the denominator

00:02:48.470 --> 00:02:52.300
by 10, which didn't change
the value of the fraction.

00:02:52.300 --> 00:02:55.160
It still represents
the same-- it still

00:02:55.160 --> 00:02:57.350
represents 3/10 of the whole.

00:02:57.350 --> 00:02:59.270
So when you do
that, you end up--

00:02:59.270 --> 00:03:05.136
we can rewrite this
thing as 30 over 100.

00:03:05.136 --> 00:03:08.640
Three tenths is
equivalent to 30 over 100.

00:03:08.640 --> 00:03:12.389
We add that to 7 over 100.

00:03:12.389 --> 00:03:14.430
Or another way of thinking
about it, three tenths

00:03:14.430 --> 00:03:17.360
is the same thing
as 30 hundredths.

00:03:17.360 --> 00:03:19.820
And we add that to 7 hundredths.

00:03:19.820 --> 00:03:23.300
Well, now we're going to
have 30 plus 7 hundredths.

00:03:23.300 --> 00:03:30.564
So now, we're going to
have 30 plus 7 hundredths.

00:03:33.710 --> 00:03:39.680
Or we could say that this is
now going to be 37 hundredths.

00:03:39.680 --> 00:03:42.960
So this extra 7 that we're
adding here-- maybe I'll

00:03:42.960 --> 00:03:45.780
do this in-- that's
the same color.

00:03:45.780 --> 00:03:49.500
This extra sevenths-- let
me do this in a new color.

00:03:49.500 --> 00:03:54.190
This 7 hundredths that we're
adding, that's going to be one,

00:03:54.190 --> 00:03:57.410
two-- let me paint that
in a little bit clearer.

00:03:57.410 --> 00:04:02.530
So that's 1, 2, so it
gets us all the way to 7.

00:04:02.530 --> 00:04:09.980
Let's see, 1, 2, 3, so 7
hundredths is right over there.

00:04:09.980 --> 00:04:11.920
So let me get back to my pen.

00:04:11.920 --> 00:04:17.740
So that, whoops, this right
over here is 7 out of the 100

00:04:17.740 --> 00:04:23.060
so 30/100 plus 7/100 is
going to get us to 37/100.


Comparing fractions with like numerators and denominators

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:07.544
Let's compare the fraction
4/7 to the fraction 5/7.

00:00:07.544 --> 00:00:09.960
And what I want you to do right
now is to pause this video

00:00:09.960 --> 00:00:11.890
and think about which
of these fractions

00:00:11.890 --> 00:00:14.552
represents a larger quantity.

00:00:14.552 --> 00:00:16.010
I'm assuming you've
had a go at it,

00:00:16.010 --> 00:00:17.330
and the one thing that
might jump out at you

00:00:17.330 --> 00:00:19.980
is that they both have the
same bottom number, which

00:00:19.980 --> 00:00:21.330
we call the denominator.

00:00:21.330 --> 00:00:24.280
They both have a
denominator of 7.

00:00:24.280 --> 00:00:27.330
So one way to think about it is
this is literally 4 sevenths.

00:00:27.330 --> 00:00:28.970
This is literally 5 sevenths.

00:00:28.970 --> 00:00:35.550
So we could rewrite 4/7
as literally 4 times 1/7.

00:00:35.550 --> 00:00:40.970
And we can rewrite 5/7
as literally 5 times 1/7.

00:00:40.970 --> 00:00:41.657
It's 5/7.

00:00:44.910 --> 00:00:47.880
So now if I have 4
of something versus 5

00:00:47.880 --> 00:00:51.320
of something, which is going
to be a larger quantity?

00:00:51.320 --> 00:00:55.200
Well, clearly 5 of this
1/7 is going to be more.

00:00:55.200 --> 00:00:58.730
So 4/7 is smaller,
5/7 is larger.

00:00:58.730 --> 00:01:01.370
And so what we can do is we
can write a less than symbol.

00:01:01.370 --> 00:01:03.640
The way that I remember
less than and greater than,

00:01:03.640 --> 00:01:06.260
is that the point, the
small side of the symbol,

00:01:06.260 --> 00:01:09.370
always is on the same side
as the smaller number.

00:01:09.370 --> 00:01:13.670
So this could be read
as 4/7 is less than 5/7,

00:01:13.670 --> 00:01:19.819
or that 4 times 1/7 is
less than 5 times 1/7.

00:01:19.819 --> 00:01:21.610
Now, let's do another
scenario, but instead

00:01:21.610 --> 00:01:26.220
of having the same denominator,
let's have the same numerator.

00:01:26.220 --> 00:01:34.380
So let's say we
want to compare 3/4

00:01:34.380 --> 00:01:45.430
versus-- let's say versus 3/9.

00:01:45.430 --> 00:01:48.325
Which of these two fractions
is a larger number?

00:01:48.325 --> 00:01:49.950
And once again, pause
the video and try

00:01:49.950 --> 00:01:52.850
to think about it on your own.

00:01:52.850 --> 00:01:57.220
Well, as we mentioned, we don't
have the same denominator here.

00:01:57.220 --> 00:01:58.990
We have the same
top number instead.

00:01:58.990 --> 00:02:01.120
Here is the same bottom
number, same denominator.

00:02:01.120 --> 00:02:03.510
Here we're going to
have the same numerator.

00:02:03.510 --> 00:02:05.900
We have the 3 right over there.

00:02:05.900 --> 00:02:12.450
And we could view 3/4 as
literally 3 times 1/4.

00:02:12.450 --> 00:02:16.660
And we could view 3/9 as
literally 3 times 1/9.

00:02:23.030 --> 00:02:25.487
So we have 3/4 and we have 3/9.

00:02:25.487 --> 00:02:27.070
So we really just
have to about what's

00:02:27.070 --> 00:02:30.010
larger, a fourth or a ninth?

00:02:30.010 --> 00:02:32.840
Well, think about if you
start with a whole-- think

00:02:32.840 --> 00:02:35.865
about starting with
a whole like this.

00:02:40.000 --> 00:02:44.910
And let me make a
whole right over here,

00:02:44.910 --> 00:02:46.482
so the same sized whole.

00:02:46.482 --> 00:02:48.065
A fourth is literally
taking the whole

00:02:48.065 --> 00:02:49.366
and dividing it into 4 pieces.

00:02:49.366 --> 00:02:51.740
While a ninth is taking the
whole and dividing it into 9.

00:02:51.740 --> 00:02:54.490
9 equal sections, I could say.

00:02:54.490 --> 00:02:56.920
So let's divide this
into 4 equal sections.

00:02:56.920 --> 00:03:01.700
So my best attempt to
hand draw equal sections.

00:03:01.700 --> 00:03:04.940
So that's 2 equal
sections and then

00:03:04.940 --> 00:03:07.960
that looks pretty close
to 4 equal sections.

00:03:07.960 --> 00:03:10.990
So that right over there is 1/4.

00:03:10.990 --> 00:03:12.790
And let me draw ninths here.

00:03:12.790 --> 00:03:16.120
So let me first split this
into 3 equal sections.

00:03:16.120 --> 00:03:17.120
So those would be third.

00:03:17.120 --> 00:03:21.500
And then split each of
those into 3 equal sections.

00:03:21.500 --> 00:03:25.430
So this is my best
attempt at that,

00:03:25.430 --> 00:03:27.560
at hand drawing
9 equal sections,

00:03:27.560 --> 00:03:29.834
splitting the whole
into 9 equal sections.

00:03:29.834 --> 00:03:32.250
So when you see here-- and you
might have already realized

00:03:32.250 --> 00:03:35.074
this-- if you divide something
into 4 equal sections,

00:03:35.074 --> 00:03:36.490
each section is
going to be bigger

00:03:36.490 --> 00:03:39.180
than if you divide it
into 9 equal sections.

00:03:39.180 --> 00:03:41.660
A ninth is smaller
than a fourth.

00:03:41.660 --> 00:03:43.890
A ninth is smaller than 1/4.

00:03:43.890 --> 00:03:47.450
So 3/9 is going to
be smaller than 3/4.

00:03:47.450 --> 00:03:50.100
So once again, when you want to
do less than or greater than,

00:03:50.100 --> 00:03:53.330
you want to put the point,
the small side of the symbol,

00:03:53.330 --> 00:03:55.450
on the same side as
the smaller number.

00:03:55.450 --> 00:03:57.490
So it would look like this.

00:03:57.490 --> 00:03:59.230
And this is the
greater than symbol,

00:03:59.230 --> 00:04:02.790
because what you have on the
left is the larger number.

00:04:02.790 --> 00:04:05.330
3/4 is greater than 3/9.

00:04:05.330 --> 00:04:08.520
And if you actually wanted to
represent not just 1/4 and 1/9,

00:04:08.520 --> 00:04:09.850
we could actually color it in.

00:04:09.850 --> 00:04:15.980
3 times one fourth,
well that's 1, 2, 3.

00:04:15.980 --> 00:04:22.550
While 3 times 1/9,
or 3/9, is 1, 2, 3.

00:04:22.550 --> 00:04:24.880
And when you look at it
that way, it's pretty clear.

00:04:24.880 --> 00:04:27.020
But the important
thing to realize

00:04:27.020 --> 00:04:31.400
is, is that when the
denominator is larger,

00:04:31.400 --> 00:04:34.920
you're dividing the
whole into more pieces

00:04:34.920 --> 00:04:38.430
so each piece will be smaller.

00:04:38.430 --> 00:04:42.470
So making the denominator larger
makes the fraction smaller.

00:04:42.470 --> 00:04:47.009
Making the numerator larger
makes the fraction larger.


Rewriting fraction as a decimal to the hundredths place

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:05.300
Let's see if we can
write 21/60 as a decimal.

00:00:05.300 --> 00:00:07.630
And I'll give you a little hint.

00:00:07.630 --> 00:00:10.830
See if you can rewrite
this as a fraction with 100

00:00:10.830 --> 00:00:11.780
as the denominator.

00:00:11.780 --> 00:00:14.690
Or another way to say it is
see if you can rewrite this

00:00:14.690 --> 00:00:17.250
so it's a certain
number of hundredths,

00:00:17.250 --> 00:00:22.010
and then you can represent
that as a decimal.

00:00:22.010 --> 00:00:23.340
So we've done this before.

00:00:23.340 --> 00:00:24.730
We can rewrite a fraction.

00:00:24.730 --> 00:00:26.320
We can get an
equivalent fraction

00:00:26.320 --> 00:00:29.440
if we either multiply the
numerator and the denominator

00:00:29.440 --> 00:00:31.206
by the same quantity
or we divide

00:00:31.206 --> 00:00:32.580
the numerator and
the denominator

00:00:32.580 --> 00:00:34.140
by the same quantity.

00:00:34.140 --> 00:00:36.089
And this numerator and
this denominator, it

00:00:36.089 --> 00:00:36.880
looks pretty clear.

00:00:36.880 --> 00:00:40.760
21 is divisible by 3
and 7 and 1 and 21.

00:00:40.760 --> 00:00:43.510
And 60 is clearly
divisible by 3 as well.

00:00:43.510 --> 00:00:45.589
It's not divisible by 7 or 21.

00:00:45.589 --> 00:00:47.130
Well, of course,
it's divisible by 1,

00:00:47.130 --> 00:00:48.750
but that doesn't
really help you much.

00:00:48.750 --> 00:00:50.499
So let's see if we can
rewrite this, maybe

00:00:50.499 --> 00:00:52.285
with lower numbers,
where we divide both

00:00:52.285 --> 00:00:53.660
the numerator and
the denominator

00:00:53.660 --> 00:00:57.380
by that common factor of 3.

00:00:57.380 --> 00:00:58.615
So we're dividing by 3.

00:00:58.615 --> 00:01:00.990
So I'm just rewriting this as
an equivalent fraction that

00:01:00.990 --> 00:01:03.030
might make it a little
bit easier for our heads

00:01:03.030 --> 00:01:04.330
to get around it.

00:01:04.330 --> 00:01:08.990
So 21 divided by
3 is equal to 7.

00:01:08.990 --> 00:01:13.570
And 60 divided by
3 is equal to 20.

00:01:13.570 --> 00:01:17.690
So we've rewritten
21/60 as 7/20.

00:01:17.690 --> 00:01:20.130
So you might be saying, Sal,
why did you even do this?

00:01:20.130 --> 00:01:22.330
Aren't we trying to get
it in terms of hundredths?

00:01:22.330 --> 00:01:26.570
Well, this one helps simplify
it in my brain a little bit.

00:01:26.570 --> 00:01:29.040
And what's extra good
about writing it as 7/20

00:01:29.040 --> 00:01:32.720
it is that it's easier
to go from 20 to 100.

00:01:32.720 --> 00:01:36.970
To go from 20 to 100 we
just have to multiply by 5.

00:01:36.970 --> 00:01:39.490
Well, if each section is going
to be five times as many then

00:01:39.490 --> 00:01:42.570
these seven sections are going
to be five times as many.

00:01:42.570 --> 00:01:44.560
So, once again, we're
multiplying the numerator

00:01:44.560 --> 00:01:46.690
and the denominator
by the same thing.

00:01:46.690 --> 00:01:56.330
And so this is going to be
equal to 35 over 100, or 35/100.

00:01:56.330 --> 00:02:00.591
35-- let me write it
a little bit-- 35/100,

00:02:00.591 --> 00:02:01.840
which is what we wanted to do.

00:02:01.840 --> 00:02:04.890
We wanted to rewrite this
in terms of hundredths.

00:02:04.890 --> 00:02:06.760
And what is 35/100?

00:02:06.760 --> 00:02:08.600
Well, let's just
remind ourselves

00:02:08.600 --> 00:02:13.300
when we're writing a decimal,
that's the ones place.

00:02:13.300 --> 00:02:18.900
This right over here, this next
place, is the tenths place.

00:02:18.900 --> 00:02:21.560
And the next place is
the hundredths place.

00:02:24.980 --> 00:02:29.740
And so 35/100, well, you
could write that like this.

00:02:29.740 --> 00:02:33.000
You could write that
as 35 hundredths.

00:02:33.000 --> 00:02:38.290
So you could literally
write this as 0.35.

00:02:38.290 --> 00:02:41.020
And you might say, wait, you
put a 3 in the tenths place.

00:02:41.020 --> 00:02:42.420
Why is this 35 hundredths?

00:02:42.420 --> 00:02:46.690
I get that this is 5 hundredths,
but why is this 35 hundredths?

00:02:46.690 --> 00:02:50.730
Well, 3/10 is 30/100.

00:02:50.730 --> 00:02:52.180
So this is 35/100.

00:02:52.180 --> 00:02:53.680
Or another way of
thinking about it,

00:02:53.680 --> 00:02:56.140
you could rewrite
this right over here.

00:02:56.140 --> 00:03:14.210
You could rewrite this as being
equal to 30/100 plus 5/100.

00:03:14.210 --> 00:03:16.150
And what is 30 over
100 if you wanted

00:03:16.150 --> 00:03:18.009
to rewrite it in
terms of tenths?

00:03:18.009 --> 00:03:20.550
Well, you could just divide the
numerator and the denominator

00:03:20.550 --> 00:03:35.150
by 10, and you would
get 3/10 plus 5/100.

00:03:35.150 --> 00:03:37.900
And we see that right
over here, 3/10,

00:03:37.900 --> 00:03:42.110
that's the tenths place, plus
5/100, that's hundredths place.

00:03:42.110 --> 00:03:45.540
Or this is sometimes
referred to as 35/100.


Basic example of rewriting a fraction as a decimal

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.930
Let's see if we can
write 3/5 as a decimal.

00:00:04.930 --> 00:00:06.640
And I encourage you
to pause this video

00:00:06.640 --> 00:00:08.940
and think about if you
can do it on your own.

00:00:08.940 --> 00:00:10.470
And I'll give you a hint here.

00:00:10.470 --> 00:00:12.690
Can we rewrite this
fraction so, instead

00:00:12.690 --> 00:00:17.370
of it being in terms of fifths,
it can be in terms of tenths?

00:00:17.370 --> 00:00:19.570
So I'm assuming you've
given a go at it.

00:00:19.570 --> 00:00:23.930
Let's try to rewrite this
as a fraction with 10

00:00:23.930 --> 00:00:25.150
as the denominator.

00:00:25.150 --> 00:00:27.340
But let's just first
visualize this.

00:00:27.340 --> 00:00:28.405
So we have fifths.

00:00:31.750 --> 00:00:33.995
So let's say that's 1/5.

00:00:33.995 --> 00:00:35.830
Actually, let me just
copy and paste this.

00:00:39.170 --> 00:00:41.720
That is 2/5.

00:00:41.720 --> 00:00:47.850
That is 3/5, and that is 4/5.

00:00:47.850 --> 00:00:53.610
And that is 5/5, or this
would be a whole now.

00:00:53.610 --> 00:00:54.970
So that is our whole.

00:00:54.970 --> 00:00:57.980
And we want to color
in 3 of those 5,

00:00:57.980 --> 00:01:00.920
so we want to think
about what 3/5 are.

00:01:00.920 --> 00:01:03.070
So let me get my magenta out.

00:01:03.070 --> 00:01:06.110
So that's 1/5.

00:01:06.110 --> 00:01:10.570
I can actually make this
bigger even-- 2/5 and 3/5.

00:01:10.570 --> 00:01:11.190
There you go.

00:01:11.190 --> 00:01:12.530
Color that in.

00:01:12.530 --> 00:01:16.000
That is 3/5.

00:01:16.000 --> 00:01:18.170
Now, how could I
write this in terms

00:01:18.170 --> 00:01:22.470
of tenths-- instead of 3/5,
a certain number of tenths?

00:01:22.470 --> 00:01:24.900
Well, let's split this
whole into tenths.

00:01:24.900 --> 00:01:29.010
And the easiest way to
split this whole into tenths

00:01:29.010 --> 00:01:34.010
is to take each of those
fifths and turn them into 2/10.

00:01:34.010 --> 00:01:35.360
So let's do that.

00:01:35.360 --> 00:01:39.160
So If we were to do
this right over here,

00:01:39.160 --> 00:01:42.470
we now have twice
as many sections.

00:01:42.470 --> 00:01:45.600
So another way of
thinking about it,

00:01:45.600 --> 00:01:49.260
we are multiplying the
number of sections by 2.

00:01:49.260 --> 00:01:51.340
We now have 10 sections.

00:01:51.340 --> 00:01:53.460
Each of these is a tenth.

00:01:53.460 --> 00:01:57.970
And the 3 of those sections are
now going to be twice as many.

00:01:57.970 --> 00:02:00.460
What we have in magenta, we
now have twice as many sections

00:02:00.460 --> 00:02:01.000
in magenta.

00:02:01.000 --> 00:02:03.800
So we're going to multiply
that by 2 as well.

00:02:03.800 --> 00:02:06.390
Notice we just multiplied the
numerator and the denominator

00:02:06.390 --> 00:02:07.190
by 2.

00:02:07.190 --> 00:02:09.150
But hopefully it makes
conceptual sense.

00:02:09.150 --> 00:02:11.500
Every piece, when we're
talking about fifths,

00:02:11.500 --> 00:02:15.720
we've now doubled so
that instead of every 1/5

00:02:15.720 --> 00:02:17.690
is now 2/10.

00:02:17.690 --> 00:02:21.990
You have a 1/10
now and a 1/10 now.

00:02:21.990 --> 00:02:24.960
And we could just keep
writing 1/10 if we like.

00:02:24.960 --> 00:02:31.130
Each of these things right
over here are a tenth.

00:02:31.130 --> 00:02:35.150
And then each of the 3 are
now twice as many tenths.

00:02:35.150 --> 00:02:39.400
So the 3/5 is now 6/10.

00:02:39.400 --> 00:02:41.200
So let's write that down.

00:02:41.200 --> 00:02:44.256
So this is going to
be equal to 6/10.

00:02:49.950 --> 00:02:51.320
Now why is this interesting?

00:02:51.320 --> 00:02:59.400
You can literally
view this as 6/10--

00:02:59.400 --> 00:03:04.055
let me write it this
way-- 6 times 1/10.

00:03:07.440 --> 00:03:09.340
I'm going to do that in blue.

00:03:09.340 --> 00:03:12.130
6 times 1/10.

00:03:12.130 --> 00:03:16.360
Well what's another way to
represent 6/10 or 6 times 1/10?

00:03:16.360 --> 00:03:18.440
Well you can express
that as a decimal,

00:03:18.440 --> 00:03:21.220
where we go to the tenths place.

00:03:21.220 --> 00:03:22.820
So when you write a
decimal-- so let's

00:03:22.820 --> 00:03:27.180
see 0 point-- the place
right to the right

00:03:27.180 --> 00:03:29.130
of the decimal, that
is the tenths place.

00:03:29.130 --> 00:03:31.340
This right over
here is the ones.

00:03:31.340 --> 00:03:34.220
That right over
here is the tenths.

00:03:34.220 --> 00:03:35.730
That's the tenths place.

00:03:35.730 --> 00:03:37.090
So how many tenths do we have?

00:03:37.090 --> 00:03:38.380
We have six tenths.

00:03:38.380 --> 00:03:42.260
So we could write this as 0.6.

00:03:42.260 --> 00:03:43.220
So there you have it.

00:03:43.220 --> 00:03:44.130
Let me write that.

00:03:44.130 --> 00:03:49.940
This is equal to 0.6.

00:03:49.940 --> 00:03:51.150
And we're done.

00:03:51.150 --> 00:03:53.360
We've just expressed
this as a decimal.

00:03:53.360 --> 00:03:56.200
0.6 is the same
thing as 6/10, which

00:03:56.200 --> 00:03:59.770
could be rewritten
as 3/5 or vice versa.


Finding a percentage

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:02.130
Let's give ourselves
a little bit

00:00:02.130 --> 00:00:04.480
of practice with percentages.

00:00:04.480 --> 00:00:13.820
So let's ask ourselves, what
percent of-- I don't know,

00:00:13.820 --> 00:00:18.940
let's say what
percent of 16 is 4?

00:00:18.940 --> 00:00:20.600
And I encourage you
to pause this video

00:00:20.600 --> 00:00:22.830
and to try it out yourself.

00:00:22.830 --> 00:00:25.940
So when you're saying
what percent of 16 is 4,

00:00:25.940 --> 00:00:29.520
percent is another way of
saying, what fraction of 16

00:00:29.520 --> 00:00:30.740
is 4?

00:00:30.740 --> 00:00:35.330
And we just need to write
it as a percent, as per 100.

00:00:35.330 --> 00:00:39.610
So if you said what fraction
of 16 is 4, you would say,

00:00:39.610 --> 00:00:43.150
well, look, this is
the same thing as 4/16,

00:00:43.150 --> 00:00:48.380
which is the same thing as 1/4.

00:00:48.380 --> 00:00:51.660
But this is saying what
fraction 4 is of 16.

00:00:51.660 --> 00:00:53.449
You'd say, well, 4 is 1/4 of 16.

00:00:53.449 --> 00:00:55.240
But that still doesn't
answer our question.

00:00:55.240 --> 00:00:58.116
What percent?

00:00:58.116 --> 00:00:59.740
So in order to write
this as a percent,

00:00:59.740 --> 00:01:03.360
we literally have to write
it as something over 100.

00:01:03.360 --> 00:01:05.915
Percent literally
means "per cent."

00:01:05.915 --> 00:01:08.820
The word "cent" you know
from cents and century.

00:01:08.820 --> 00:01:10.550
It relates to the number 100.

00:01:10.550 --> 00:01:12.419
So it's per 100.

00:01:12.419 --> 00:01:13.960
So you could say,
well, this is going

00:01:13.960 --> 00:01:19.481
to be equal to question mark
over 100, the part of 100.

00:01:19.481 --> 00:01:21.980
And there's a bunch of ways
that you could think about this.

00:01:21.980 --> 00:01:23.980
You could say, well, look,
if in the denominator

00:01:23.980 --> 00:01:27.990
to go from 4 to 100, I
have to multiply by 25.

00:01:27.990 --> 00:01:30.970
In the numerator
to go from-- I need

00:01:30.970 --> 00:01:34.710
to also multiply by 25 in order
to have an equivalent fraction.

00:01:34.710 --> 00:01:37.390
So I'm also going
to multiply by 25.

00:01:37.390 --> 00:01:40.720
So 1/4 is the same
thing as 25/100.

00:01:40.720 --> 00:01:48.040
And another way of saying 25/100
is this is 25 per 100, or 25%.

00:01:48.040 --> 00:01:52.382
So this is equal to 25%.

00:01:52.382 --> 00:01:53.840
Now, there's a
couple of other ways

00:01:53.840 --> 00:01:55.450
you could have thought about it.

00:01:55.450 --> 00:02:00.240
You could have said
well, 4/16, this

00:02:00.240 --> 00:02:02.640
is literally 4 divided by 16.

00:02:02.640 --> 00:02:04.064
Well, let me just
do the division

00:02:04.064 --> 00:02:05.480
and convert to a
decimal, which is

00:02:05.480 --> 00:02:08.229
very easy to convert
to a percentage.

00:02:08.229 --> 00:02:10.780
So let's try to actually do
this division right over here.

00:02:10.780 --> 00:02:15.520
So we're going to
literally divide 4 by 16.

00:02:15.520 --> 00:02:18.225
Now, 16 goes into 4 zero times.

00:02:18.225 --> 00:02:20.880
0 times 16 is 0.

00:02:20.880 --> 00:02:22.835
You subtract, and you get a 4.

00:02:22.835 --> 00:02:24.960
And we're not satisfied
just having this remainder.

00:02:24.960 --> 00:02:30.560
We want to keep adding zeroes to
get a decimal answer right over

00:02:30.560 --> 00:02:31.170
here.

00:02:31.170 --> 00:02:33.250
So let's put a decimal
right over here.

00:02:33.250 --> 00:02:35.870
We're going into
the tenths place.

00:02:35.870 --> 00:02:38.660
And let's throw some
zeroes right over here.

00:02:38.660 --> 00:02:41.190
The decimal makes sure
we keep track of the fact

00:02:41.190 --> 00:02:43.398
that we are now in the
tenths, and in the hundredths,

00:02:43.398 --> 00:02:45.940
and in the thousandths place
if we have to go that far.

00:02:45.940 --> 00:02:47.840
But let's bring another 0 down.

00:02:47.840 --> 00:02:51.790
16 goes into 40 two times.

00:02:51.790 --> 00:02:54.975
2 times 16 is 32.

00:02:54.975 --> 00:02:57.160
If you subtract, you get 8.

00:02:57.160 --> 00:02:59.960
And you could bring
down another 0.

00:02:59.960 --> 00:03:03.890
And we have 16 goes into 80.

00:03:03.890 --> 00:03:08.630
Let's see, 16 goes
into 80 five times.

00:03:08.630 --> 00:03:11.060
5 times 16 is 80.

00:03:11.060 --> 00:03:14.120
You subtract, you have no
remainder, and you're done.

00:03:14.120 --> 00:03:17.790
4/16 is the same thing as 0.25.

00:03:17.790 --> 00:03:21.570
Now, 0.25 is the same thing
as twenty-five hundredths.

00:03:21.570 --> 00:03:26.010
Or, this is the same
thing as 25/100,

00:03:26.010 --> 00:03:29.897
which is the same thing as 25%.


Dividing a whole number by a decimal

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:11.935
Let's divide 518 by-- so we're
going to divide it by 0.7.

00:00:11.935 --> 00:00:15.040
So we're dividing this
whole number by a decimal.

00:00:15.040 --> 00:00:21.352
So we could also
write this as 518

00:00:21.352 --> 00:00:23.560
divided-- let me write a
little bit bigger than that,

00:00:23.560 --> 00:00:25.143
since we have to do
some work with it.

00:00:25.143 --> 00:00:29.710
518 divided by-- I'll
do the division sign

00:00:29.710 --> 00:00:37.780
in white-- divided by 0.7.

00:00:37.780 --> 00:00:40.384
So the first thing we do, since
we have this decimal here,

00:00:40.384 --> 00:00:41.550
we're dividing by a decimal.

00:00:41.550 --> 00:00:43.796
So try to turn this into
a whole number somehow.

00:00:43.796 --> 00:00:45.920
Well, the best way to turn
this into a whole number

00:00:45.920 --> 00:00:49.550
is to multiply this by 10, which
is essentially multiplying,

00:00:49.550 --> 00:00:52.900
shifting the decimal
point over to the right.

00:00:52.900 --> 00:00:55.150
So this would become a 7.

00:00:55.150 --> 00:00:57.630
But we can't just do that only
for what we're dividing by.

00:00:57.630 --> 00:01:00.650
We also have to do
that to the 518,

00:01:00.650 --> 00:01:02.240
so that a value does not change.

00:01:02.240 --> 00:01:04.840
So we need to multiply
both of these times 10.

00:01:04.840 --> 00:01:08.370
So if we move the decimal
over to the right with the 0.7

00:01:08.370 --> 00:01:11.380
to turn into a 7, we also
need to move the decimal over

00:01:11.380 --> 00:01:13.214
to the right for 518.

00:01:13.214 --> 00:01:14.630
Now, you're probably
saying, well,

00:01:14.630 --> 00:01:16.690
I don't see a decimal in 518.

00:01:16.690 --> 00:01:17.540
Well, there is one.

00:01:17.540 --> 00:01:21.170
You just didn't have to write
it, because it's 518.00--

00:01:21.170 --> 00:01:23.690
and we can add as many
zeroes as we want.

00:01:23.690 --> 00:01:29.950
So if we move the decimal to
the right, it becomes 5,180.

00:01:29.950 --> 00:01:34.630
So really what we're saying
is 518 divided by 0.7

00:01:34.630 --> 00:01:49.220
is the same thing as
5,180 divided by 7.

00:01:49.220 --> 00:01:52.550
Notice all we did by moving the
decimal one place to the right,

00:01:52.550 --> 00:01:54.690
is we multiplied
both of these numbers

00:01:54.690 --> 00:01:57.100
by 10, which is
not going to change

00:01:57.100 --> 00:01:58.970
the actual value of the decimal.

00:01:58.970 --> 00:02:00.520
One other way of
thinking about this,

00:02:00.520 --> 00:02:02.280
if you wanted to write
this as a fraction,

00:02:02.280 --> 00:02:11.170
this is the same
thing as 518 over 0.7.

00:02:11.170 --> 00:02:14.440
You multiply both the numerator
and denominator by 10,

00:02:14.440 --> 00:02:25.440
you will get 5,180 over 7.

00:02:25.440 --> 00:02:26.940
So let's clean this
up a little bit,

00:02:26.940 --> 00:02:28.930
just so we remember what we did.

00:02:28.930 --> 00:02:31.130
So we moved the decimal
over to the right, one.

00:02:31.130 --> 00:02:33.320
So now this is just a 7.

00:02:33.320 --> 00:02:34.922
The decimal is there.

00:02:34.922 --> 00:02:37.380
In fact, we really don't have
to write the decimal anymore.

00:02:37.380 --> 00:02:40.760
It's just a 7.0-- you
could imagine 7.0,

00:02:40.760 --> 00:02:42.920
so we can just
write this as a 7.

00:02:42.920 --> 00:02:47.120
And then the 518, the
decimal is now out here.

00:02:47.120 --> 00:02:50.600
So this is 5,180.

00:02:50.600 --> 00:02:53.590
And let's increase the
sign right over here.

00:02:53.590 --> 00:02:56.650
Now, this is just a straight-up
long division problem.

00:02:56.650 --> 00:02:59.060
How many times does 7 go into 5?

00:02:59.060 --> 00:03:00.365
Well, it goes 0 times.

00:03:00.365 --> 00:03:04.340
0 times 7-- actually, let's
just cut to the chase.

00:03:04.340 --> 00:03:06.070
7 doesn't go into 5.

00:03:06.070 --> 00:03:08.150
It does go into 51.

00:03:08.150 --> 00:03:10.600
7 times 7 is 49.

00:03:10.600 --> 00:03:12.050
So it goes 7 times.

00:03:12.050 --> 00:03:14.510
7 times 7 is 49.

00:03:14.510 --> 00:03:18.060
Subtract 51 minus 49 is 2.

00:03:18.060 --> 00:03:22.780
And now we can
bring down this 8.

00:03:22.780 --> 00:03:26.490
7 goes into 28 four times.

00:03:26.490 --> 00:03:29.080
4 times 7 is 28.

00:03:29.080 --> 00:03:31.570
Subtract, you get a 0.

00:03:31.570 --> 00:03:33.210
Now we can bring down another 0.

00:03:33.210 --> 00:03:35.860
We want at least get
to the decimal place.

00:03:35.860 --> 00:03:37.660
So we bring down another
0 right over here.

00:03:37.660 --> 00:03:39.160
When I say get to
the decimal place,

00:03:39.160 --> 00:03:40.610
we could put the
decimal place up

00:03:40.610 --> 00:03:41.985
here too, just to
make sure we're

00:03:41.985 --> 00:03:45.025
keeping track of the
right place values

00:03:45.025 --> 00:03:47.150
or that we can have the
decimal in the right place.

00:03:47.150 --> 00:03:48.660
So notice, I'm very particular.

00:03:48.660 --> 00:03:53.010
When I'm doing 7 goes into 51,
I put the 7 right above the 1

00:03:53.010 --> 00:03:54.130
in the 51's place.

00:03:54.130 --> 00:03:56.520
When I'm saying 7
goes into 28, I'm

00:03:56.520 --> 00:04:00.520
putting the 4 right above
the 8 in this one's place

00:04:00.520 --> 00:04:01.900
when we're doing the division.

00:04:01.900 --> 00:04:03.960
So now we say how many
times does 7 go into 0?

00:04:03.960 --> 00:04:05.560
Well, it goes 0 times.

00:04:05.560 --> 00:04:07.490
0 times 7 is 0.

00:04:07.490 --> 00:04:09.684
Subtract, you have no remainder.

00:04:09.684 --> 00:04:12.100
So we could keep going, and
we'll just keep getting zeroes

00:04:12.100 --> 00:04:12.880
like this.

00:04:12.880 --> 00:04:16.598
But we see that this
is equal to 740.


Subtracting decimals: example 1

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.920
Let's try to subtract
9.57 minus 8.09.

00:00:04.920 --> 00:00:07.890
So try to pause this video
and figure this out first

00:00:07.890 --> 00:00:10.411
before we work
through it together.

00:00:10.411 --> 00:00:10.910
All right.

00:00:10.910 --> 00:00:12.700
Well, let's just rewrite it.

00:00:12.700 --> 00:00:14.200
Let's rewrite it.

00:00:14.200 --> 00:00:17.836
And when I rewrite it, I
like to line up the decimals.

00:00:17.836 --> 00:00:19.210
This one it's a
little intuitive.

00:00:19.210 --> 00:00:23.800
We have 8.09.

00:00:23.800 --> 00:00:27.000
Just like that, and now
we're ready to subtract.

00:00:27.000 --> 00:00:29.729
And we want to subtract 9
hundredths from 7 hundredths.

00:00:29.729 --> 00:00:31.270
Well, we don't have
enough hundredths

00:00:31.270 --> 00:00:32.640
up here so let's move over here.

00:00:32.640 --> 00:00:35.150
Let's see if we can do some
regrouping so we always

00:00:35.150 --> 00:00:37.060
have a higher number on top.

00:00:37.060 --> 00:00:42.210
So over here we want to
subtract 0 tenths from 5 tenths

00:00:42.210 --> 00:00:43.970
so we have enough
tenths over here.

00:00:43.970 --> 00:00:45.460
So let's regroup.

00:00:45.460 --> 00:00:47.840
So instead of 5 tenths,
I'm going to have 4 tenths,

00:00:47.840 --> 00:00:50.630
and then I'm going to give
that other tenth, which

00:00:50.630 --> 00:00:53.120
is the same thing as 10
hundredths over here,

00:00:53.120 --> 00:00:56.250
so this becomes 17 hundredths.

00:00:56.250 --> 00:01:00.830
17 minus 9 is 8.

00:01:00.830 --> 00:01:03.600
4 minus 0 is 4.

00:01:03.600 --> 00:01:06.970
And then I have 9 minus 8 is 1.

00:01:06.970 --> 00:01:09.330
So this is going to be 1.48.


Comparing decimals: ordering from smallest to biggest

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:02.650
Let's once again see
if we can order now

00:00:02.650 --> 00:00:05.620
a different set of decimals
from least to greatest,

00:00:05.620 --> 00:00:07.760
and once again I encourage
you to pause this video

00:00:07.760 --> 00:00:10.245
and try to do this on your own.

00:00:10.245 --> 00:00:12.370
So let's go to the most
significant place, the ones

00:00:12.370 --> 00:00:13.060
place here.

00:00:13.060 --> 00:00:15.600
None of these have any ones.

00:00:15.600 --> 00:00:18.040
So then we can go to the
next most significant place,

00:00:18.040 --> 00:00:19.370
which is the tenths place.

00:00:19.370 --> 00:00:21.120
This has five tenths.

00:00:21.120 --> 00:00:22.810
This has six tenths.

00:00:22.810 --> 00:00:24.460
This has one tenth.

00:00:24.460 --> 00:00:26.050
This has five tenths.

00:00:26.050 --> 00:00:27.372
This has one tenth.

00:00:27.372 --> 00:00:29.580
So if we just look at the
tenths place, the ones that

00:00:29.580 --> 00:00:32.619
have the fewest tenths--
this has only one tenth,

00:00:32.619 --> 00:00:37.040
this one only has one tenth,
this one has five tenths,

00:00:37.040 --> 00:00:40.480
this one has five tenths, and
then this one has six tenths.

00:00:40.480 --> 00:00:43.650
So I've ordered it by what's
going on in the tenths place.

00:00:43.650 --> 00:00:46.124
Now, both of these have
the same number of tenths.

00:00:46.124 --> 00:00:47.540
Let's move to the
hundredths place

00:00:47.540 --> 00:00:49.970
to figure out which
of these is larger.

00:00:49.970 --> 00:00:51.880
This one has six hundredths.

00:00:51.880 --> 00:00:55.122
This has five hundredths,
so this one is larger.

00:00:55.122 --> 00:00:56.080
It has more hundredths.

00:00:56.080 --> 00:00:58.816
Same number of tenths,
more hundredths.

00:00:58.816 --> 00:01:00.190
And hundredths
are obviously more

00:01:00.190 --> 00:01:02.060
significant than
thousandths, so it

00:01:02.060 --> 00:01:04.260
doesn't matter which one
has the more thousandths.

00:01:04.260 --> 00:01:06.320
It matters that this
one has more tenths,

00:01:06.320 --> 00:01:08.760
and actually this one has
more thousandths as well.

00:01:08.760 --> 00:01:10.309
But now let's go
look at these two.

00:01:10.309 --> 00:01:11.850
These have the same
number of tenths.

00:01:11.850 --> 00:01:13.650
They both have five tenths.

00:01:13.650 --> 00:01:15.830
But this one has six
hundredths, while this one only

00:01:15.830 --> 00:01:19.039
has two hundredths,
so this one is larger.

00:01:19.039 --> 00:01:20.580
And then finally,
this one of course,

00:01:20.580 --> 00:01:22.860
had six tenths, so this
one had the most tenths.

00:01:22.860 --> 00:01:26.330
So we don't even have to look
at the other places here.

00:01:26.330 --> 00:01:28.140
And we're done.


Decimals: writing it out as a number

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:03.030
I want to write this
expression here as a decimal.

00:00:03.030 --> 00:00:06.000
And the first question
that might pop in your head

00:00:06.000 --> 00:00:10.740
is do I multiply 4 times 1,000,
then add 3, then multiply

00:00:10.740 --> 00:00:15.020
1,000, then add 67, then
multiply by 1 over 1,000?

00:00:15.020 --> 00:00:18.069
Or do I do the
multiplication first?

00:00:18.069 --> 00:00:19.610
To answer that
question, we just have

00:00:19.610 --> 00:00:21.980
to remind ourselves about
order of operations.

00:00:21.980 --> 00:00:24.396
Order of operations, if you
have a bunch of multiplication

00:00:24.396 --> 00:00:26.190
and addition in a
row like this, you

00:00:26.190 --> 00:00:30.260
will do the
multiplication first.

00:00:30.260 --> 00:00:33.187
Let me put some parentheses
here to remind us of that.

00:00:33.187 --> 00:00:35.270
So let's figure out what
each of these expressions

00:00:35.270 --> 00:00:37.400
in the parentheses
actually represent.

00:00:37.400 --> 00:00:39.420
What's 4 times 1 million?

00:00:39.420 --> 00:00:41.870
Well, that's 4 million.

00:00:41.870 --> 00:00:44.230
What's 3 times 1,000?

00:00:44.230 --> 00:00:48.090
Well that's 3,000.

00:00:48.090 --> 00:00:52.096
What is 67 times 1 thousandth?

00:00:52.096 --> 00:00:54.220
And there's a bunch of ways
of thinking about this.

00:00:54.220 --> 00:00:56.540
Actually, let me write
them all over here.

00:00:56.540 --> 00:00:59.510
Well, I'll do the most
obvious one right over here.

00:00:59.510 --> 00:01:05.340
67 times 1 thousandth
is 67 thousandths.

00:01:05.340 --> 00:01:09.730
And we can represent this
literally as 67/1000.

00:01:09.730 --> 00:01:19.270
Or we could represent this as
60 over 1,000 plus 7 over 1,000.

00:01:19.270 --> 00:01:21.680
And what's 60 over 1,000?

00:01:21.680 --> 00:01:26.680
Well, 60 over 1,000
is 6 hundredths.

00:01:26.680 --> 00:01:28.820
So let me put that
in a different color.

00:01:28.820 --> 00:01:32.010
6 hundredths, 7 thousandths.

00:01:32.010 --> 00:01:34.090
So you could view this
is 67 thousandths.

00:01:34.090 --> 00:01:39.180
Or you could view this is 6
hundredths and 7 thousandths.

00:01:39.180 --> 00:01:42.760
Either way, let's add all
of these things together.

00:01:42.760 --> 00:01:44.810
So we have 4 million.

00:01:44.810 --> 00:01:48.080
So the 4 in the millions place
literally represents 4 million.

00:01:48.080 --> 00:01:50.410
Then we have no
hundred thousands.

00:01:50.410 --> 00:01:52.860
We have no ten thousands.

00:01:52.860 --> 00:01:54.590
But then we have 3,000.

00:01:54.590 --> 00:01:56.910
So the 3 is in the
thousands place.

00:01:56.910 --> 00:02:00.070
Let me put a comma here, so
we can keep track of things.

00:02:00.070 --> 00:02:02.700
And then we have no hundreds.

00:02:02.700 --> 00:02:04.220
We have no tens.

00:02:04.220 --> 00:02:06.170
We have no ones.

00:02:06.170 --> 00:02:08.680
We don't even have any tenths.

00:02:08.680 --> 00:02:10.620
But we do have some hundredths.

00:02:10.620 --> 00:02:11.896
We have 6 hundredths.

00:02:15.120 --> 00:02:17.350
And then we have
some thousandths.

00:02:17.350 --> 00:02:19.060
We have 7 thousandths.

00:02:19.060 --> 00:02:23.380
So we put that 7 in
the thousandths place.

00:02:23.380 --> 00:02:29.640
So we could read this 4 million,
3 thousand, 6 hundredths

00:02:29.640 --> 00:02:31.890
and 7 thousandths.

00:02:31.890 --> 00:02:36.213
Or we could read it as
4 million, 3 thousand,

00:02:36.213 --> 00:02:39.304
and 67 thousandths.


Finding factors and multiples

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.890
Which of the following
numbers is a factor of 154?

00:00:04.890 --> 00:00:07.240
So when a number is going
to be a factor of 154

00:00:07.240 --> 00:00:10.020
is if we can divide
that number into 154

00:00:10.020 --> 00:00:11.554
and not have a remainder.

00:00:11.554 --> 00:00:13.470
Or another way of thinking
about it-- a number

00:00:13.470 --> 00:00:19.250
is a factor of 154 if 154 is
a multiple of that number.

00:00:19.250 --> 00:00:22.060
So let's look at each of these
and see which of these we

00:00:22.060 --> 00:00:24.420
can rule out or say is a factor.

00:00:24.420 --> 00:00:27.309
So does 3 divide
evenly into 154?

00:00:27.309 --> 00:00:28.850
Or, another way of
thinking about it,

00:00:28.850 --> 00:00:32.172
is 154 a multiple of 3?

00:00:32.172 --> 00:00:34.130
Well you'll later learn
that you could actually

00:00:34.130 --> 00:00:36.640
test whether something
is divisible by 3

00:00:36.640 --> 00:00:37.890
by adding up the digits.

00:00:37.890 --> 00:00:39.870
And if that's divisible
by 3, then it's

00:00:39.870 --> 00:00:41.760
going to be divisible by 3.

00:00:41.760 --> 00:00:43.690
And so you see
here, 1 plus 5 is 6.

00:00:43.690 --> 00:00:46.280
6 plus 4 is 10.

00:00:46.280 --> 00:00:48.431
10 is not divisible by 3.

00:00:48.431 --> 00:00:50.430
But if you didn't want
to do that little trick--

00:00:50.430 --> 00:00:51.480
and we have other
videos where we

00:00:51.480 --> 00:00:53.350
go into more detail
about that trick--

00:00:53.350 --> 00:00:58.170
you can actually just
divide 3 into 154.

00:00:58.170 --> 00:00:59.260
3 doesn't go into 1.

00:00:59.260 --> 00:01:02.060
It does go into 15 five times.

00:01:02.060 --> 00:01:04.170
5 times 3 is 15.

00:01:04.170 --> 00:01:05.190
Subtract.

00:01:05.190 --> 00:01:06.710
Then we have 0.

00:01:06.710 --> 00:01:08.910
Then you bring down a 4.

00:01:08.910 --> 00:01:10.960
3 goes into 4 one time.

00:01:10.960 --> 00:01:12.520
1 times 3 is 3.

00:01:12.520 --> 00:01:15.690
Subtract, and you
have a remainder of 1.

00:01:15.690 --> 00:01:21.160
So 3 is clearly not a factor of
154, so we can rule that out.

00:01:21.160 --> 00:01:22.830
Now what about 5?

00:01:22.830 --> 00:01:24.900
Well, any multiple
of 5 is either

00:01:24.900 --> 00:01:27.900
going to have 5 or
0 in the ones place.

00:01:27.900 --> 00:01:31.290
You see that if we write 5
times 1 is 5, 5 times 2 is 10,

00:01:31.290 --> 00:01:37.150
5 times 3 is 15, 5 times 4 is
20, you either have a 5 or a 0

00:01:37.150 --> 00:01:38.400
in the ones place.

00:01:38.400 --> 00:01:40.960
This does not have 5
or a 0 the ones place,

00:01:40.960 --> 00:01:43.610
so it's not going to
be divisible by 5.

00:01:43.610 --> 00:01:45.130
5 is not a factor.

00:01:45.130 --> 00:01:49.070
154 is not a multiple of 5.

00:01:49.070 --> 00:01:50.044
Now 6 is interesting.

00:01:50.044 --> 00:01:51.210
You could do the same thing.

00:01:51.210 --> 00:01:53.890
You could try to
divide 6 into 154.

00:01:53.890 --> 00:01:55.927
But if something
is divisible by 6,

00:01:55.927 --> 00:01:58.010
it's definitely going to
be divisible by 3 as well

00:01:58.010 --> 00:01:59.950
because 6 is divisible by 3.

00:01:59.950 --> 00:02:02.080
So we can immediately
rule this one out as well.

00:02:02.080 --> 00:02:05.480
Because 154 is not
divisible by 3,

00:02:05.480 --> 00:02:07.590
it's also not going
to be divisible by 6.

00:02:07.590 --> 00:02:09.419
And you could try
it out if you like.

00:02:09.419 --> 00:02:11.540
And we could make the
same argument for 9.

00:02:11.540 --> 00:02:14.990
If something is divisible by 9,
it's going to be divisible by 3

00:02:14.990 --> 00:02:16.540
because 9 is divisible by 3.

00:02:16.540 --> 00:02:18.810
Well, it's not
divisible by 3, so we're

00:02:18.810 --> 00:02:20.241
going to rule out 9 as well.

00:02:20.241 --> 00:02:21.490
So we've ruled out everything.

00:02:21.490 --> 00:02:23.360
It looks like 14
is our only option,

00:02:23.360 --> 00:02:24.960
but let's actually verify it.

00:02:24.960 --> 00:02:29.940
Let's actually
divide 14 into 154.

00:02:29.940 --> 00:02:31.570
14 doesn't go into 1.

00:02:31.570 --> 00:02:34.360
It goes into 15
exactly one time.

00:02:34.360 --> 00:02:36.570
1 times 14 is 14.

00:02:36.570 --> 00:02:37.840
We subtract.

00:02:37.840 --> 00:02:39.060
We get 1.

00:02:39.060 --> 00:02:40.230
Bring down the 4.

00:02:40.230 --> 00:02:43.150
14 goes into 14 one time.

00:02:43.150 --> 00:02:44.660
1 times 14 is 14.

00:02:44.660 --> 00:02:46.790
And of course, we
have no remainder.

00:02:46.790 --> 00:02:50.750
So 14 goes into 154
exactly 11 times.

00:02:50.750 --> 00:02:53.280
Or 11 times 14 is 154.

00:02:53.280 --> 00:02:56.940
154 is a multiple of 14.

00:02:56.940 --> 00:02:58.490
Let's do one more.

00:02:58.490 --> 00:03:01.122
Which of the following
numbers is a multiple of 14?

00:03:01.122 --> 00:03:02.580
So now we have 14,
and we're trying

00:03:02.580 --> 00:03:04.610
to think of its multiples.

00:03:04.610 --> 00:03:06.600
So there's two
ways of doing this.

00:03:06.600 --> 00:03:09.824
You could go number by number
and try to divide 14 into them,

00:03:09.824 --> 00:03:11.240
or we could just
try to figure out

00:03:11.240 --> 00:03:13.620
what all of the multiples
of 14 actually are.

00:03:13.620 --> 00:03:14.620
So let's try to do that.

00:03:14.620 --> 00:03:16.390
Let's try to do that
second technique.

00:03:16.390 --> 00:03:20.810
14 times 1 is 14.

00:03:20.810 --> 00:03:21.795
You add another 14.

00:03:21.795 --> 00:03:24.010
14 times 2 is 28.

00:03:24.010 --> 00:03:25.274
Add another 14.

00:03:25.274 --> 00:03:26.190
Let's see, you add 10.

00:03:26.190 --> 00:03:27.150
You get to 38.

00:03:27.150 --> 00:03:28.600
Then you add 4 more.

00:03:28.600 --> 00:03:31.210
You get to 42.

00:03:31.210 --> 00:03:32.430
Then you add another 14.

00:03:32.430 --> 00:03:34.580
I haven't seen any of
these numbers show up yet.

00:03:34.580 --> 00:03:36.840
Add another 14 to this.

00:03:36.840 --> 00:03:40.440
You get to 56-- still
not quite there.

00:03:40.440 --> 00:03:42.006
Add another 14.

00:03:42.006 --> 00:03:43.630
Let's see, if you
add 4, you get to 60,

00:03:43.630 --> 00:03:44.754
and you have to add the 10.

00:03:44.754 --> 00:03:46.870
So then you get to 70.

00:03:46.870 --> 00:03:49.560
And it looks like we have
found one of these numbers.

00:03:49.560 --> 00:03:52.070
70 is a multiple of 14.

00:03:52.070 --> 00:03:59.268
14 times 1, 2, 3, 4,
5-- 14 times 5 is 70.


Absolute value word problems

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.280
The Statue of Liberty,
pictured above,

00:00:03.280 --> 00:00:06.120
is a gigantic sculpture
located on Liberty Island

00:00:06.120 --> 00:00:08.250
in New York, United States.

00:00:08.250 --> 00:00:11.430
The height of the
sculpture is 151 feet.

00:00:11.430 --> 00:00:14.420
So this distance right
over here is 151 feet.

00:00:14.420 --> 00:00:17.860
So it's like a 15 story
building or more, or about a 15

00:00:17.860 --> 00:00:18.840
story building.

00:00:18.840 --> 00:00:21.690
The distance from the
ground to the torch is 305.

00:00:21.690 --> 00:00:24.910
So from here to
there is 305 feet.

00:00:24.910 --> 00:00:27.330
What is the absolute
value of the difference

00:00:27.330 --> 00:00:31.620
between 305 and 151?

00:00:31.620 --> 00:00:36.054
Well we just have to
subtract 151 from 305.

00:00:36.054 --> 00:00:37.970
And you might be able
to do that on your head.

00:00:37.970 --> 00:00:42.210
Let's see, if you add 150
to 151, you get to 301.

00:00:42.210 --> 00:00:44.864
So then you have to add
another 4, so it's 154.

00:00:44.864 --> 00:00:46.780
Or you could use a
calculator right over here.

00:00:46.780 --> 00:00:58.010
You could say 305 minus
151 is equal to 154,

00:00:58.010 --> 00:00:59.590
which is what we
did in our head.

00:00:59.590 --> 00:01:02.374
And the absolute
value of 154-- there's

00:01:02.374 --> 00:01:04.040
a couple of ways you
can think about it.

00:01:04.040 --> 00:01:05.498
You could learn to
just view it as,

00:01:05.498 --> 00:01:07.090
how far is that number from 0?

00:01:07.090 --> 00:01:11.330
And positive 154
is 154 away from 0.

00:01:11.330 --> 00:01:13.870
If this was negative
154 then that also

00:01:13.870 --> 00:01:15.942
would be 154 away from 0.

00:01:15.942 --> 00:01:17.400
The other way to
think about it is,

00:01:17.400 --> 00:01:18.820
whether it's
positive or negative,

00:01:18.820 --> 00:01:20.240
the absolute value,
you just take

00:01:20.240 --> 00:01:21.670
the positive version of it.

00:01:21.670 --> 00:01:27.310
So either way we say 154.

00:01:27.310 --> 00:01:30.680
Let's do a couple more of these.

00:01:30.680 --> 00:01:31.990
I like these pictures.

00:01:31.990 --> 00:01:34.670
The average weight of a
car in the United States

00:01:34.670 --> 00:01:36.670
is roughly 4,000 pounds.

00:01:36.670 --> 00:01:39.780
The dinosaur Ankylosaurus--
artist rendering above--

00:01:39.780 --> 00:01:42.710
is thought to have weighed
up to 13,000 pounds.

00:01:42.710 --> 00:01:44.700
What is the absolute
value of the difference

00:01:44.700 --> 00:01:45.967
between these two weights?

00:01:45.967 --> 00:01:48.300
And you could calculate the
difference between these two

00:01:48.300 --> 00:01:48.860
weights.

00:01:48.860 --> 00:01:51.880
You could take the smaller
and subtract from the larger.

00:01:51.880 --> 00:01:54.290
So if you take
4,000 from 13,000,

00:01:54.290 --> 00:01:56.400
the difference would
be 9,000 pounds.

00:01:56.400 --> 00:01:58.714
The absolute value
of 9,000 is 9,000.

00:01:58.714 --> 00:02:00.130
You could have
gone the other way.

00:02:00.130 --> 00:02:02.090
You could've subtracted
13,000 from 4,000,

00:02:02.090 --> 00:02:04.230
and you would have
gotten negative 9,000.

00:02:04.230 --> 00:02:07.430
But the absolute value of
negative 9,000 is still 9,000.

00:02:07.430 --> 00:02:09.960
So we just want to think
about the absolute difference

00:02:09.960 --> 00:02:10.727
between these two.

00:02:10.727 --> 00:02:12.310
We don't want to
think about which one

00:02:12.310 --> 00:02:13.940
we're subtracting
from whichever one.

00:02:13.940 --> 00:02:15.856
Whatever we're subtracting
from the other one,

00:02:15.856 --> 00:02:17.870
we just find the
absolute difference.

00:02:17.870 --> 00:02:21.155
So whether the difference
was negative 9,000 or 9,000,

00:02:21.155 --> 00:02:23.280
depending on which one you
subtract from the other,

00:02:23.280 --> 00:02:24.970
the absolute value
of either of those

00:02:24.970 --> 00:02:29.160
is going to be 9,000 pounds.

00:02:29.160 --> 00:02:31.280
Let's do one more.

00:02:31.280 --> 00:02:32.610
No picture this time.

00:02:32.610 --> 00:02:34.010
I was getting used to that.

00:02:34.010 --> 00:02:35.917
The Panama Canal
is a canal cutting

00:02:35.917 --> 00:02:38.250
through the country of Panama
that conveniently connects

00:02:38.250 --> 00:02:39.570
the Atlantic and Pacific Oceans.

00:02:39.570 --> 00:02:41.028
What's nice about
these is we learn

00:02:41.028 --> 00:02:42.850
other things other
than mathematics here.

00:02:42.850 --> 00:02:45.940
France began constructing
the canal in the year 1881

00:02:45.940 --> 00:02:48.060
and, after the French
abandoned construction,

00:02:48.060 --> 00:02:51.460
the United States finished
construction in 1914.

00:02:51.460 --> 00:02:54.780
What is the absolute value of
the difference between the year

00:02:54.780 --> 00:02:56.670
construction of the
Panama Canal started

00:02:56.670 --> 00:02:58.350
and the year that it ended?

00:02:58.350 --> 00:03:03.250
Well, there's a couple
ways you could do it.

00:03:03.250 --> 00:03:05.750
You could take the larger year
and subtract the smaller year

00:03:05.750 --> 00:03:06.700
from it.

00:03:06.700 --> 00:03:08.330
We have our calculator here.

00:03:08.330 --> 00:03:14.070
You could literally
say 1914 minus 1881,

00:03:14.070 --> 00:03:17.470
and you get 33 years.

00:03:17.470 --> 00:03:20.156
And the absolute value
of 33 is just 33.

00:03:20.156 --> 00:03:22.030
Or you could've done it
the other way around.

00:03:22.030 --> 00:03:23.005
Since we're taking
absolute value,

00:03:23.005 --> 00:03:25.500
it doesn't matter what we
subtract from the other ones.

00:03:25.500 --> 00:03:30.025
You could do on 1881
minus 1914, and you

00:03:30.025 --> 00:03:33.760
would've gotten negative 33.

00:03:33.760 --> 00:03:36.260
But the absolute value of
either of these numbers

00:03:36.260 --> 00:03:39.180
is, you're saying, how far
is 33 or negative 33 from 0?

00:03:39.180 --> 00:03:40.759
Well, they're both
33 away from zero.

00:03:40.759 --> 00:03:43.300
Or you could say, well, what's
the positive version of either

00:03:43.300 --> 00:03:43.800
of these?

00:03:43.800 --> 00:03:44.880
Well, it's 33.

00:03:44.880 --> 00:03:47.310
So the absolute value of
the difference in the years

00:03:47.310 --> 00:03:50.070
is 33 years.

00:03:50.070 --> 00:03:51.850
And we're done.


Adding and subtracting negative numbers

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.570
Let's have some practice
adding and subtracting

00:00:02.570 --> 00:00:03.420
negative numbers.

00:00:03.420 --> 00:00:08.360
So the first example I want
to look at is 2 minus 3.

00:00:08.360 --> 00:00:11.050
So right now I'm just
subtracting a positive number

00:00:11.050 --> 00:00:13.370
from another positive
number, but you might already

00:00:13.370 --> 00:00:16.021
see that I'm subtracting a
larger number from a smaller

00:00:16.021 --> 00:00:16.520
number.

00:00:16.520 --> 00:00:18.370
So I'm probably, or
I will, definitely

00:00:18.370 --> 00:00:19.710
end up with a negative number.

00:00:19.710 --> 00:00:21.584
But let's just think
about this a little bit.

00:00:21.584 --> 00:00:23.530
And I'm going to do
it with a number line.

00:00:23.530 --> 00:00:26.230
So there's my number
line right over there.

00:00:26.230 --> 00:00:32.080
Now this is 0, this is 1,
this is 2, this is negative 1,

00:00:32.080 --> 00:00:34.030
this is negative 2.

00:00:34.030 --> 00:00:36.220
We could view this
as starting at 2.

00:00:36.220 --> 00:00:38.280
So this is 2 right over
here, and then we're

00:00:38.280 --> 00:00:41.150
going to subtract 3 from that 2.

00:00:41.150 --> 00:00:44.260
So we're going to move 3 to
the left on the number line.

00:00:44.260 --> 00:00:48.420
So we're going to move
3 to the left, 1, 2, 3.

00:00:48.420 --> 00:00:51.330
And that gets us to negative 1.

00:00:51.330 --> 00:00:52.940
This is equal to negative 1.

00:00:52.940 --> 00:00:54.810
Now let's mix it up
a little bit more.

00:00:54.810 --> 00:01:01.800
Let's imagine what would happen
if we had negative 2 minus 3.

00:01:01.800 --> 00:01:03.500
So this was positive 2 minus 3.

00:01:03.500 --> 00:01:06.290
Now let's think about
negative 2 minus 3.

00:01:06.290 --> 00:01:10.350
So once again, let's
draw our number line.

00:01:10.350 --> 00:01:12.350
And I'll put 0 over here.

00:01:12.350 --> 00:01:17.800
So this is 0, this is 1, this
is negative 1, negative 2,

00:01:17.800 --> 00:01:23.030
negative 3, negative 4,
negative 5, negative 6, and I

00:01:23.030 --> 00:01:24.380
could keep going.

00:01:24.380 --> 00:01:28.460
But we're starting
at negative 2,

00:01:28.460 --> 00:01:30.160
and then we're
subtracting 3 again.

00:01:30.160 --> 00:01:31.826
So once again, we're
going to move three

00:01:31.826 --> 00:01:33.490
to the left of negative 2.

00:01:33.490 --> 00:01:36.640
So we go 1, 2, 3.

00:01:36.640 --> 00:01:39.230
We end up at negative 5.

00:01:39.230 --> 00:01:40.380
So this is negative 5.

00:01:40.380 --> 00:01:42.770
So notice in both
situations we subtracted 3,

00:01:42.770 --> 00:01:44.960
we moved 3 to the left
on the number line.

00:01:44.960 --> 00:01:47.920
It's just here we started
2 to the right of 0.

00:01:47.920 --> 00:01:51.800
Here we started 2
to the left of 0.

00:01:51.800 --> 00:01:53.330
This is negative 2.

00:01:53.330 --> 00:01:57.640
Let's do another example
with these same numbers.

00:01:57.640 --> 00:02:05.170
Let's imagine negative 2 plus 3.

00:02:05.170 --> 00:02:06.830
I encourage you to
pause this video,

00:02:06.830 --> 00:02:10.060
and try to think about
this on your own.

00:02:10.060 --> 00:02:12.254
So we could draw
the number line--

00:02:12.254 --> 00:02:14.420
I could draw a straighter
number line than that-- so

00:02:14.420 --> 00:02:16.170
draw the number line again.

00:02:16.170 --> 00:02:18.460
And let's say that
this is negative 2,

00:02:18.460 --> 00:02:23.102
negative 1, 0, 1, and 2 again.

00:02:23.102 --> 00:02:24.560
We're starting at
negative 2, we're

00:02:24.560 --> 00:02:27.724
starting 2 to the left of zero.

00:02:27.724 --> 00:02:30.140
So we're starting at negative
2, and we're going to add 3.

00:02:30.140 --> 00:02:31.598
So we're going to
go 3 to the right

00:02:31.598 --> 00:02:40.670
now, 1, 2, 3, and we
end up at positive 1.

00:02:40.670 --> 00:02:47.450
Now let's think about
2, so positive 2,

00:02:47.450 --> 00:02:50.440
and we're going to
subtract a negative 3.

00:02:54.909 --> 00:02:56.950
And other videos we've
already talked about this.

00:02:56.950 --> 00:02:59.158
In fact, there's a video
explaining why this actually

00:02:59.158 --> 00:02:59.840
makes sense.

00:02:59.840 --> 00:03:01.390
But when you
subtract a negative,

00:03:01.390 --> 00:03:04.560
this is the same thing
as adding the positive.

00:03:04.560 --> 00:03:09.460
So 2 minus negative 3 is
the exact same thing as 2

00:03:09.460 --> 00:03:13.030
plus 2 plus positive 3.

00:03:13.030 --> 00:03:15.060
These two statements
are equivalent,

00:03:15.060 --> 00:03:18.280
and this just boils down
to, this right over here,

00:03:18.280 --> 00:03:21.290
is just going to be 5.

00:03:21.290 --> 00:03:23.390
Now, let's mix it up
a little bit more.

00:03:23.390 --> 00:03:31.280
Let's imagine negative
2 minus negative 3.

00:03:34.510 --> 00:03:36.890
Now, this might seem
really intimidating to have

00:03:36.890 --> 00:03:38.810
all of these negatives
in place here,

00:03:38.810 --> 00:03:41.130
but you just have to
remember subtracting

00:03:41.130 --> 00:03:45.210
a negative, like this, is
going to get you a positive.

00:03:45.210 --> 00:03:52.090
So this is the exact same
thing as negative 2 plus 3,

00:03:52.090 --> 00:03:54.720
and negative 2 plus
3, we've already

00:03:54.720 --> 00:03:55.890
seen it right over here.

00:03:55.890 --> 00:03:59.310
You start at negative 2, you
start 2 to the left of 0,

00:03:59.310 --> 00:04:01.080
and then we're going
to go 3 to the right,

00:04:01.080 --> 00:04:05.678
we're adding 3, 1, 2, 3.