Key tax terms

WEBVTT
Kind: captions
Language: en

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- [Instructor] What we're going to do here

00:00:01.680 --> 00:00:04.740
is a little bit of a case
study in doing taxes.

00:00:04.740 --> 00:00:06.810
So we have a situation where someone

00:00:06.810 --> 00:00:11.190
is bringing in $50,000
in this current tax year

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in gross income.

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So this is everything from their salaries,

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tips they might make,

00:00:15.930 --> 00:00:18.210
if they act as a contractor,
payments for that.

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It could be dividend
income if they own stocks.

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It could be interest
on bank accounts or CDs

00:00:25.230 --> 00:00:26.063
that they might have.

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And so when you add up all
of that, we get to $50,000.

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Now you aren't taxed on your
gross income just as is.

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Typically, you're going to adjust it

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to figure out what your taxable income is

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and then you'll go and look
at the various tax brackets

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and calculate your taxes.

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And just so you know,

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these are not the real tax
brackets in the United States.

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These have never been
the real tax brackets

00:00:50.280 --> 00:00:51.330
in the United States.

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I simplified it, one,
to make the numbers easy

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so I can do it with
back-of-the-envelope math,

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but also these are constantly
changing in the real world

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and I don't wanna have to
keep redoing this video

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every time they change the tax brackets.

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But it's gonna give you the
general idea of what's going on.

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So these are made up tax
brackets and tax rates

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and this is also a made
up standard deduction.

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We're gonna talk in a little bit

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about what a standard deduction is.

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So going back to the
case study of this person

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in this taxable year,
gross income $50,000.

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And then they have what we're calling

00:01:24.240 --> 00:01:26.460
above-the-line deductions.

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And above-the-line deductions,

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you can view these as
deductions to your gross income

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that you can make no matter what.

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The history of why it's
called above-the-line,

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it's the first form of
the typical tax form,

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so let me say above the line.

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Above the line.

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You can make these types of
deductions no matter what.

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This isn't an exhaustive list

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but these are ones that are
actually reasonably typical,

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things like a 401K contribution.

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This is a pre-tax contribution
to a retirement account,

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to a defined contribution
retirement account so to speak,

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where you put a certain
amount of money every year

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and that money is pre-tax.

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So it gets taken down
from your gross income

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to figure out what your
actual taxable income

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is going to be.

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Student loan interest is
also typically deductible

00:02:13.170 --> 00:02:14.910
from your gross income.

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And when we deduct those,

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we are left with our
adjusted gross income.

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Let me write here, add
adjusted, I lost a T there,

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adjusted gross income, which in this case

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is going to be, well it's gonna be $50,000

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minus 2,000, minus 3,000,

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so that gets us to $45,000.

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Now we're not ready to
just take this $45,000

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and figure out how that will be taxed

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based on this schedule here,

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this chart right over here.

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We're not ready to make more deductions,

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and we can view these as
below-the-line deductions.

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And this is where it gets interesting.

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You could go the path
of itemized deductions.

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So that's looking at things like,

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okay, did you make any
donations to charity?

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Or do you have any mortgage
interest on your home mortgage?

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And so let me write these,

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these are itemized.

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Itemized.

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So if you itemized your deductions

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you'd say, "Okay, I have a total of $9,000

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of itemized deductions right over here."

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I would take that from the $45,000,

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and I could have a
situation where my taxable,

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taxable income would be 45,000 minus nine

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which is going to be, lemme
draw a little line over here.

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This is going to be
$36,000 of taxable income.

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Now you could just take
that and figure out

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what you're now going to have to pay.

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But there's another
interesting alternative path

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that you should know about.

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As opposed to itemizing your deductions,

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your below-the-line
deductions, so to speak,

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where you have to list them all out,

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your other option is to just
take the standard deduction.

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Now in this situation, the
standard deduction is $10,000.

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So let's see what happens if instead

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of itemizing our deductions,

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we were to just take out the
$10,000 standard deduction.

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Well, in that situation,
our taxable income

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is going to be 45,000 minus 10,000,

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which is going to be $35,000.

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Now, generally speaking,
unless you just like

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giving money to the government,

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you wanna minimize your taxable income.

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You wanna do that legally.

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And in this situation

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when we pick the standard deduction route,

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we actually have a lower taxable income.

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So this is the path that you
are going to want to take.

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And now that we've calculated
this taxable income,

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this $35,000, we are ready
to now actually calculate

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based on this schedule of
the actual tax brackets.

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So one way to think about
it is the first 10,000

00:04:55.500 --> 00:04:58.080
right over here, we are
going to have no tax.

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So let me put that here.

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So we have 10K, no tax, no tax.

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Now the next $10,000 right over here

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is going to be taxed at 10%.

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So the next 10K is going
to be taxed at 10%.

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So now, sorry, my dog barked.

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So now we've accounted for
20K, we have $15,000 left,

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so we go into the next bracket.

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The next bracket actually
accounts for the next 20,000.

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We don't have that much,

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but if I just write over here, say okay,

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my next 15K, which is all
I have, falls over here,

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so I'm going to say the next 15K

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is going to be taxed at 20%.

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And this is federal taxes that
we are talking about here.

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You're gonna have to do it another time

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for your state taxes.

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But now let's calculate our total tax.

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First $10,000, no tax.

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So we don't have to worry about that.

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Now, $10,000 being taxed at 10%,

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10% of 10,000 is $1,000.

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And then $15,000 being taxed at 20%,

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that's going to be $3,000.

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And so you add these up

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and you get a total federal
tax in this situation

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in this made up tax
brackets that I just did,

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in this made up example of $4,000.

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But hopefully that gives you an idea

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of how taxes are actually calculated

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and a little bit of the terminology.


Associative and commutative properties of addition with negatives

WEBVTT
Kind: captions
Language: en

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- [Instructor] What we're
going to do in this video

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is evaluate this pretty hairy expression

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and we could just try to do it.

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We could go from left to right,

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but it feels like there might
be a simpler way to do it.

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I'm adding 13 here and
then I'm subtracting 13.

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I'm, I have a negative five here

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and then I have a positive five.

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You might be tempted to say,

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"Well, maybe I could change the order of,

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that I'm adding and subtracting things."

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Well, if we were just
adding a bunch of numbers

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you could change the order.

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For example, we know that nine
plus seven is equal to 16,

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and if I just change the order

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the commutative property tells us,

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well, seven plus nine,
it applies to addition.

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If I swap the order here it's
still going to be equal to 16,

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but that's not true if
I do nine minus seven

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or seven minus nine.

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If I change the order, I'm
not getting the same result.

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The commutative property does
not apply to subtraction.

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This top expression is positive two.

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This bottom expression is negative two.

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So I can't just use the
commutative property here

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to change the order with which
I'm adding and subtracting

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because I have subtraction here.

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But what if I could
rewrite this expression,

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so it only involves addition?

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"How can you do that, Sal?"

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You are probably thinking.

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And the key realization is
when you subtract a number

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it's the same thing as
adding the opposite.

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For example, if I have six minus three,

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that is the same thing
as adding the opposite

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of positive three,
which is negative three.

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Or if I had six minus negative three,

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subtracting a number is the same thing

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as adding its opposite.

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So what I could do is all of these places

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where I'm subtracting a number,

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instead I could just rewrite
it so I'm adding the opposite.

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So let me do that.

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So I can rewrite this expression
as negative 5 plus 13.

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So far, I'm only adding.

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Here, I'm subtracting all of a sudden.

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Minus 21, subtracting a number.

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Well, I can rewrite that
as adding its opposite.

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So subtracting, lemme just another color.

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Subtracting a number is the same thing

00:02:10.560 --> 00:02:13.830
as adding its opposite.

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All right, let me keep going.

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Then I'm adding a five.

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Remember, I'm just trying to make this,

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so I'm just adding a bunch of things

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instead of adding and subtracting.

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I have the plus 21.

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Now again, I am subtracting a number,

00:02:27.720 --> 00:02:30.660
so I can rewrite that as adding,

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I'm subtracting a positive 13.

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I can rewrite that as
adding a negative 13.

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And then last but not least,

00:02:38.880 --> 00:02:40.800
over here I am subtracting again.

00:02:40.800 --> 00:02:44.040
I'm subtracting a number,
so I can rewrite that

00:02:44.040 --> 00:02:46.740
as adding the opposite of this number.

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So I'm now adding positive 11.

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Subtracting negative 11

00:02:50.790 --> 00:02:53.238
is the same thing as adding positive 11.

00:02:53.238 --> 00:02:54.960
Now, why did I do this?

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Well, now I can use a
commutative property.

00:02:56.880 --> 00:02:59.910
All I'm doing is I'm adding
a bunch of numbers now

00:02:59.910 --> 00:03:02.795
so I can swap the order
in with, with which I add.

00:03:02.795 --> 00:03:04.440
So I could now rewrite it.

00:03:04.440 --> 00:03:07.320
Let's see, I have a negative five,

00:03:07.320 --> 00:03:11.610
and now let me add this
positive five next.

00:03:11.610 --> 00:03:15.360
So add the positive five, and then I have,

00:03:15.360 --> 00:03:18.723
I'm adding a positive 13,

00:03:20.070 --> 00:03:24.033
and to that I can add the negative 13.

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Remember, the only reason
why I can now swap the order

00:03:27.270 --> 00:03:29.753
is 'cause I'm only adding
a bunch of integers.

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Next, I have this negative 21.

00:03:35.430 --> 00:03:37.020
So let me circle that.

00:03:37.020 --> 00:03:42.020
I'm adding a negative
21, adding negative 21,

00:03:42.210 --> 00:03:45.150
and then I could add the positive 21,

00:03:45.150 --> 00:03:48.033
which is right over there.

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And then last but not least, I add 11.

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Now, why was all of this super useful?

00:03:54.446 --> 00:03:56.280
Well, now look what happens.

00:03:56.280 --> 00:03:58.380
Things start to simplify a lot.

00:03:58.380 --> 00:04:00.540
Not only when I'm doing addition

00:04:00.540 --> 00:04:03.810
can I use a commutative
property, can I change the order,

00:04:03.810 --> 00:04:06.840
but I can also use the
associative property very easily.

00:04:06.840 --> 00:04:08.490
So I could start to say, all right,

00:04:08.490 --> 00:04:12.330
let me add these two first and
I could also add these two.

00:04:12.330 --> 00:04:14.430
I can pair these up, and that's useful

00:04:14.430 --> 00:04:16.140
because these cancel out with each other.

00:04:16.140 --> 00:04:17.430
They're opposites.

00:04:17.430 --> 00:04:20.640
If I take a negative five
plus a positive five,

00:04:20.640 --> 00:04:21.900
that's a zero.

00:04:21.900 --> 00:04:25.890
A 13 plus a negative 13, that's a zero.

00:04:25.890 --> 00:04:29.370
A negative 21 plus 21, that's a zero.

00:04:29.370 --> 00:04:31.080
And so what am I left with?

00:04:31.080 --> 00:04:36.080
I am just left with a positive 11.

00:04:36.510 --> 00:04:39.870
So hopefully you see that
if I can rewrite subtraction

00:04:39.870 --> 00:04:42.510
as adding the opposite, I
can now use the commutative

00:04:42.510 --> 00:04:45.570
and the associative properties
to really simplify things

00:04:45.570 --> 00:04:47.010
which is really useful the rest

00:04:47.010 --> 00:04:48.810
of your mathematical careers.

00:04:48.810 --> 00:04:50.257
I encourage you in your own time,

00:04:50.257 --> 00:04:53.190
go left to right with
this original expression

00:04:53.190 --> 00:04:55.920
and you'll see that you
get this exact same result.

00:04:55.920 --> 00:04:58.570
It's just going to actually
take you a lot more time.


Graphing negative number addition and subtraction expressions

WEBVTT
Kind: captions
Language: en

00:00:00.150 --> 00:00:01.470
- [Instructor] In this
video, we're going to add

00:00:01.470 --> 00:00:04.830
and subtract negative
numbers on a number line,

00:00:04.830 --> 00:00:07.110
and the important thing to realize is,

00:00:07.110 --> 00:00:10.440
if you are adding a positive number,

00:00:10.440 --> 00:00:12.600
you start at some point on the number line

00:00:12.600 --> 00:00:16.380
and you move that many units to the right.

00:00:16.380 --> 00:00:18.780
If you are adding a negative number,

00:00:18.780 --> 00:00:20.850
you start at wherever you're starting

00:00:20.850 --> 00:00:22.980
and then you move that
many units to the left,

00:00:22.980 --> 00:00:25.560
whatever the absolute value
of that negative number is,

00:00:25.560 --> 00:00:27.840
and if you're subtracting either of them,

00:00:27.840 --> 00:00:29.580
you do the opposite,

00:00:29.580 --> 00:00:31.800
so we're going to see
a few examples of that.

00:00:31.800 --> 00:00:33.390
So let's start with this first example,

00:00:33.390 --> 00:00:35.820
negative 10 plus negative six.

00:00:35.820 --> 00:00:38.400
So we're going to start at
negative 10 right over here,

00:00:38.400 --> 00:00:40.320
so let's look at that on the number line.

00:00:40.320 --> 00:00:42.840
That's negative 10 right over there.

00:00:42.840 --> 00:00:44.490
We're gonna start over there,

00:00:44.490 --> 00:00:47.430
and then we are adding positive six.

00:00:47.430 --> 00:00:48.690
So what do we do?

00:00:48.690 --> 00:00:51.750
Well we start here, and as I mentioned,

00:00:51.750 --> 00:00:54.570
we're going to go six units to the right

00:00:54.570 --> 00:00:55.980
because it's a positive six,

00:00:55.980 --> 00:01:00.450
so one, two, three, four, five, six.

00:01:00.450 --> 00:01:04.410
So we're going to go right over there.

00:01:04.410 --> 00:01:07.440
We started at the negative 10,

00:01:07.440 --> 00:01:09.180
and since we're adding positive six,

00:01:09.180 --> 00:01:11.310
we go six units to the right,

00:01:11.310 --> 00:01:15.660
and we end up right over
here at negative four,

00:01:15.660 --> 00:01:18.630
so this is equal to negative four.

00:01:18.630 --> 00:01:19.710
Now let's do this one.

00:01:19.710 --> 00:01:20.970
Where are we starting?

00:01:20.970 --> 00:01:23.520
We are starting at negative
eight, so that's negative 10,

00:01:23.520 --> 00:01:27.990
negative nine, negative
eight is right over there.

00:01:27.990 --> 00:01:31.170
Now, we're going to subtract negative two,

00:01:31.170 --> 00:01:33.000
so let's be very careful here.

00:01:33.000 --> 00:01:35.400
If we were adding negative two,

00:01:35.400 --> 00:01:39.570
we would go two units
to the left, like that,

00:01:39.570 --> 00:01:41.520
but we're subtracting negative two,

00:01:41.520 --> 00:01:43.140
so we're going to do the opposite.

00:01:43.140 --> 00:01:44.130
We're going to instead,

00:01:44.130 --> 00:01:46.200
instead of going two units to the left,

00:01:46.200 --> 00:01:49.890
we're going to go two units to the right.

00:01:49.890 --> 00:01:53.160
So we're gonna go one,
two units to the right,

00:01:53.160 --> 00:01:55.440
and we are going to end
up right over there,

00:01:55.440 --> 00:01:58.920
so that's negative seven, negative six.

00:01:58.920 --> 00:02:01.110
So this is equal to negative six.

00:02:01.110 --> 00:02:02.850
Remember, if we were adding negative two,

00:02:02.850 --> 00:02:04.620
we would've gone two units to the left,

00:02:04.620 --> 00:02:05.580
but when you subtract,

00:02:05.580 --> 00:02:08.430
you do the opposite of what
you would've otherwise done,

00:02:08.430 --> 00:02:11.010
so now we're going two units to the right,

00:02:11.010 --> 00:02:11.970
even though it's a negative two,

00:02:11.970 --> 00:02:14.340
'cause we're subtracting negative two.

00:02:14.340 --> 00:02:15.173
All right.

00:02:15.173 --> 00:02:17.880
We're starting at four
in the third example.

00:02:17.880 --> 00:02:21.990
So, we're starting
right over here at four,

00:02:21.990 --> 00:02:24.450
and we're adding negative seven.

00:02:24.450 --> 00:02:27.240
So, negative seven,
you're just going to move

00:02:27.240 --> 00:02:29.160
the absolute value of that to the left.

00:02:29.160 --> 00:02:30.950
So the absolute value of
negative seven is just seven,

00:02:30.950 --> 00:02:34.170
so you're gonna move
seven units to the left,

00:02:34.170 --> 00:02:35.550
and we're not subtracting that

00:02:35.550 --> 00:02:38.250
so we're just going to just
move seven units to the left.

00:02:38.250 --> 00:02:39.420
We're not gonna do the opposite of that

00:02:39.420 --> 00:02:40.740
or anything like that,

00:02:40.740 --> 00:02:42.240
so we just go seven units to the left.

00:02:42.240 --> 00:02:45.060
We're adding negative seven so one, two,

00:02:45.060 --> 00:02:47.430
three, four, five, six, seven.

00:02:47.430 --> 00:02:49.440
We end up right over here.

00:02:49.440 --> 00:02:52.260
So we have positive four minus seven.

00:02:52.260 --> 00:02:53.610
We've gone seven units to the left

00:02:53.610 --> 00:02:55.830
and now we're at, let's see, this is zero,

00:02:55.830 --> 00:02:59.130
negative one, negative
two, negative three,

00:02:59.130 --> 00:03:02.490
so that is equal to negative three.

00:03:02.490 --> 00:03:04.560
Now this last one, try
to do it on your own

00:03:04.560 --> 00:03:06.630
before we do it together.

00:03:06.630 --> 00:03:07.463
All right.

00:03:07.463 --> 00:03:09.150
Now some of you might
be tempted to say oh,

00:03:09.150 --> 00:03:12.270
five and negative five, aren't
those additive inverses?

00:03:12.270 --> 00:03:13.920
Don't those just cancel out?

00:03:13.920 --> 00:03:15.210
Well they would if you were adding.

00:03:15.210 --> 00:03:18.720
Five plus negative five is equal to zero,

00:03:18.720 --> 00:03:21.330
but here we're doing
five minus negative five.

00:03:21.330 --> 00:03:23.430
So let's just do it step by step.

00:03:23.430 --> 00:03:25.710
We're starting at five.

00:03:25.710 --> 00:03:29.100
Now if we were adding
negative five to that,

00:03:29.100 --> 00:03:30.930
we would go five units to the left

00:03:30.930 --> 00:03:33.240
and we would end up at zero,

00:03:33.240 --> 00:03:35.130
but we are not adding negative five.

00:03:35.130 --> 00:03:37.320
We are subtracting negative five,

00:03:37.320 --> 00:03:39.450
so instead of going
five units to the left,

00:03:39.450 --> 00:03:41.700
we're going to go five units to the right,

00:03:41.700 --> 00:03:45.210
so one, two, three, four, five.

00:03:45.210 --> 00:03:47.343
We end up right over there,

00:03:48.240 --> 00:03:53.133
and so we end up at 10, and we are done.


Adding the opposite with number lines

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.060
- [Instructor] So this
number lined diagram here,

00:00:03.060 --> 00:00:06.750
it looks like I'm adding
or subtracting two numbers.

00:00:06.750 --> 00:00:10.020
I'm starting with what
looks like a positive nine,

00:00:10.020 --> 00:00:14.370
I'm starting at zero and
going nine units to the right.

00:00:14.370 --> 00:00:16.380
So that's a positive nine.

00:00:16.380 --> 00:00:18.420
And to that, it looks like

00:00:18.420 --> 00:00:20.250
I might be adding or
subtracting something.

00:00:20.250 --> 00:00:22.050
And we have this arrow that starts

00:00:22.050 --> 00:00:24.000
exactly at the tip of that nine

00:00:24.000 --> 00:00:25.230
and then it goes, let's see,

00:00:25.230 --> 00:00:28.440
one, two, three, four units to the left.

00:00:28.440 --> 00:00:31.830
So I could think about
this arrow right over here

00:00:31.830 --> 00:00:33.540
as just subtracting four,

00:00:33.540 --> 00:00:35.850
so it could just be nine minus four,

00:00:35.850 --> 00:00:38.010
which is of course equal
to where we end up;

00:00:38.010 --> 00:00:41.760
which is equal to this
five right over here.

00:00:41.760 --> 00:00:45.630
So that's one way to represent

00:00:45.630 --> 00:00:48.960
what is going on as an addition
or subtraction equation.

00:00:48.960 --> 00:00:50.340
But what's another way?

00:00:50.340 --> 00:00:52.410
We could start with the nine

00:00:52.410 --> 00:00:56.220
And is there anything that
we could add to the nine

00:00:56.220 --> 00:00:59.460
that would get us the same result?

00:00:59.460 --> 00:01:02.370
Well, we know when we
add negative numbers,

00:01:02.370 --> 00:01:05.460
we also move that many units to the left.

00:01:05.460 --> 00:01:09.600
So you could also view this
purple arrow right over here,

00:01:09.600 --> 00:01:12.420
instead of viewing it as
subtracting a positive four,

00:01:12.420 --> 00:01:16.410
you could view it as
adding a negative four,

00:01:16.410 --> 00:01:18.450
adding a negative four.

00:01:18.450 --> 00:01:19.830
Now, this is interesting.

00:01:19.830 --> 00:01:21.570
Because in this case,

00:01:21.570 --> 00:01:24.180
when I subtract a positive number,

00:01:24.180 --> 00:01:28.590
it's the same thing as
adding the opposite of it.

00:01:28.590 --> 00:01:30.990
Now will that work the other way around?

00:01:30.990 --> 00:01:32.610
If I subtract a negative number,

00:01:32.610 --> 00:01:34.910
is that the same thing
as adding the opposite?

00:01:36.150 --> 00:01:40.950
Let's say we had negative
five minus negative three.

00:01:40.950 --> 00:01:42.570
What is that going to be equal to?

00:01:42.570 --> 00:01:45.210
And let's think about
this with integer chips.

00:01:45.210 --> 00:01:49.260
So we could start with five
negative integer chips,

00:01:49.260 --> 00:01:54.260
one, two, three, four and five.

00:01:55.290 --> 00:01:56.760
And now I'm going to take away

00:01:56.760 --> 00:01:58.980
three of those negative integer chips.

00:01:58.980 --> 00:02:01.980
So I'm gonna take away
one, two, three of them.

00:02:01.980 --> 00:02:03.510
And so what am I left with?

00:02:03.510 --> 00:02:07.140
I am left over here with negative two.

00:02:07.140 --> 00:02:09.420
So if we think about it on a number line,

00:02:09.420 --> 00:02:10.710
let me do that right over here.

00:02:10.710 --> 00:02:12.273
So if I have a number line,

00:02:13.200 --> 00:02:16.920
and let's say that this is zero here

00:02:16.920 --> 00:02:21.920
and I'm gonna go negative
one, two, three, four, five.

00:02:22.020 --> 00:02:23.910
So negative five,

00:02:23.910 --> 00:02:25.860
so this negative five right over here,

00:02:25.860 --> 00:02:28.380
I can represent, I can start at zero,

00:02:28.380 --> 00:02:31.530
and I could go five units to the left.

00:02:31.530 --> 00:02:34.410
And I know when I subtract negative three,

00:02:34.410 --> 00:02:36.570
I'm gonna end up at negative two.

00:02:36.570 --> 00:02:39.060
So I know I'm gonna end up,

00:02:39.060 --> 00:02:40.890
I'm gonna end up, I'll do
this in a different color.

00:02:40.890 --> 00:02:43.260
I'm gonna end up right over here.

00:02:43.260 --> 00:02:45.859
So subtracting negative three needs to be

00:02:45.859 --> 00:02:50.790
the same thing as moving
three units to the right.

00:02:50.790 --> 00:02:52.440
If I was adding negative three,

00:02:52.440 --> 00:02:53.520
I'd go three units to the left.

00:02:53.520 --> 00:02:55.740
But subtracting negative three must be

00:02:55.740 --> 00:02:58.440
the same thing as going
three units to the right.

00:02:58.440 --> 00:02:59.790
Well, what also is the same thing

00:02:59.790 --> 00:03:01.710
as going three units to the right?

00:03:01.710 --> 00:03:03.960
Well that's the same
thing as adding three.

00:03:03.960 --> 00:03:06.870
So this is going to be the same thing.

00:03:06.870 --> 00:03:09.360
So negative five minus negative three

00:03:09.360 --> 00:03:13.890
is actually the same thing
as negative five plus three.

00:03:13.890 --> 00:03:15.753
This purple arrow right over here,

00:03:16.860 --> 00:03:20.970
you could view it as
subtracting negative three

00:03:20.970 --> 00:03:22.080
or you could view it

00:03:22.080 --> 00:03:25.560
as adding the opposite of negative three.

00:03:25.560 --> 00:03:28.170
So it looks like when
you subtract a number

00:03:28.170 --> 00:03:30.870
it's the same thing as
adding the opposite.

00:03:30.870 --> 00:03:32.940
When you subtract a number

00:03:32.940 --> 00:03:37.110
it's the same thing as
adding the opposite.

00:03:37.110 --> 00:03:38.790
Now that's really useful,

00:03:38.790 --> 00:03:41.370
because now we can even
think about doing things

00:03:41.370 --> 00:03:43.980
with things that aren't even integers.

00:03:43.980 --> 00:03:46.470
We can start thinking about
just rational numbers.

00:03:46.470 --> 00:03:49.110
For example, negative fractions.

00:03:49.110 --> 00:03:51.630
So if we take that principle
we just came up with,

00:03:51.630 --> 00:03:52.463
and if we say, okay,

00:03:52.463 --> 00:03:57.120
well what's three minus
negative two fifths?

00:03:57.120 --> 00:03:59.940
Well, if we take that principle
that subtracting a number

00:03:59.940 --> 00:04:04.230
is the same thing as adding its opposite.

00:04:04.230 --> 00:04:07.050
subtracting negative two
fifths is the same thing

00:04:07.050 --> 00:04:09.450
as adding the opposite
of negative two fifths.

00:04:09.450 --> 00:04:12.060
So it's adding two fives.

00:04:12.060 --> 00:04:14.490
Well, we have seen this
type of thing before

00:04:14.490 --> 00:04:15.940
and we could even do
that on a number line.

00:04:15.940 --> 00:04:17.703
Lemme do that right over here.

00:04:18.660 --> 00:04:21.900
So, let's say that this is

00:04:21.900 --> 00:04:23.520
zero,

00:04:23.520 --> 00:04:24.750
one,

00:04:24.750 --> 00:04:26.340
two,

00:04:26.340 --> 00:04:27.173
three,

00:04:28.110 --> 00:04:30.750
four and five.

00:04:30.750 --> 00:04:35.160
So we're starting here with
a three in either case.

00:04:35.160 --> 00:04:37.983
So that's three units to
the right, a positive three.

00:04:38.940 --> 00:04:41.160
And so subtracting a negative two-fifths,

00:04:41.160 --> 00:04:43.980
we're saying is the same
thing as adding two-fifths.

00:04:43.980 --> 00:04:45.472
So adding two-fifths,

00:04:45.472 --> 00:04:48.510
we are just moving
two-fifths to the right.

00:04:48.510 --> 00:04:50.430
So it's one-fifth and two-fifths.

00:04:50.430 --> 00:04:52.950
So we're just going to do that.

00:04:52.950 --> 00:04:55.412
Subtracting a number is the same thing

00:04:55.412 --> 00:04:57.063
as adding its opposite.


Adding with integer chips

WEBVTT
Kind: captions
Language: en

00:00:00.180 --> 00:00:01.803
- [Instructor] Let's say
I wanted to figure out

00:00:01.803 --> 00:00:06.360
what negative two plus
negative four is equal to.

00:00:06.360 --> 00:00:08.580
There are a bunch of ways
of thinking about them,

00:00:08.580 --> 00:00:10.050
but what we're gonna do in this video is

00:00:10.050 --> 00:00:13.320
think about it using something
called integer chips.

00:00:13.320 --> 00:00:14.610
So with integer chips

00:00:14.610 --> 00:00:17.370
if I have a chip that looks like this,

00:00:17.370 --> 00:00:20.336
a positive sign with a circle around it,

00:00:20.336 --> 00:00:23.118
that is equal to a positive one.

00:00:23.118 --> 00:00:25.264
And if I have a negative sign

00:00:25.264 --> 00:00:27.930
with a circle around it,
that's a negative integer chip

00:00:27.930 --> 00:00:29.580
and that's equal to negative one.

00:00:29.580 --> 00:00:32.820
And of course if I have exactly
one positive integer chip

00:00:32.820 --> 00:00:35.640
and one negative integer chip,

00:00:35.640 --> 00:00:37.590
well then they're going to cancel out

00:00:37.590 --> 00:00:40.740
because positive one plus negative one is

00:00:40.740 --> 00:00:42.630
of course equal to zero.

00:00:42.630 --> 00:00:44.490
So let's use that to figure out this

00:00:44.490 --> 00:00:46.320
and several other problems.

00:00:46.320 --> 00:00:47.760
So negative two,

00:00:47.760 --> 00:00:50.430
how could I represent
that with integer chips?

00:00:50.430 --> 00:00:51.570
Well, that's going to be two

00:00:51.570 --> 00:00:53.190
of these negative integer chips.

00:00:53.190 --> 00:00:56.010
So negative one and then negative two.

00:00:56.010 --> 00:00:58.290
And then negative four is going to be four

00:00:58.290 --> 00:00:59.790
of these negative integer chips.

00:00:59.790 --> 00:01:02.400
So negative one, negative two,

00:01:02.400 --> 00:01:05.100
negative three, and negative four.

00:01:05.100 --> 00:01:08.760
So now how many total negative
integer chips do I have?

00:01:08.760 --> 00:01:11.700
I have 1, 2, 3, 4, 5, 6.

00:01:11.700 --> 00:01:14.040
So that's going to be six
negative integer chips

00:01:14.040 --> 00:01:16.530
or negative six.

00:01:16.530 --> 00:01:18.360
Let's do another example.

00:01:18.360 --> 00:01:23.130
Let's say I now have positive
seven plus negative six.

00:01:23.130 --> 00:01:25.410
How can I represent
this with integer chips?

00:01:25.410 --> 00:01:27.930
Pause this video and think about it.

00:01:27.930 --> 00:01:29.220
Well, positive seven,

00:01:29.220 --> 00:01:30.780
you could view that as seven

00:01:30.780 --> 00:01:32.940
of these positive integer chips.

00:01:32.940 --> 00:01:37.940
So 1, 2, 3, 4, 5, 6, and 7.

00:01:41.490 --> 00:01:43.950
And then negative six,
you can represent that

00:01:43.950 --> 00:01:46.950
as six of these negative integer chips.

00:01:46.950 --> 00:01:50.413
So we have negative one,
negative two, negative three

00:01:50.413 --> 00:01:54.570
negative four, negative
five, and negative six.

00:01:54.570 --> 00:01:57.510
Now we can use the fact
that every positive integer,

00:01:57.510 --> 00:01:59.940
if you have a positive and
a negative integer chip,

00:01:59.940 --> 00:02:01.650
well positive one plus negative one

00:02:01.650 --> 00:02:03.360
is going to be equal to
zero, so they cancel out.

00:02:03.360 --> 00:02:04.530
So those cancel.

00:02:04.530 --> 00:02:05.670
Those also add up to zero.

00:02:05.670 --> 00:02:07.830
Those also add up to zero.
Those also add up to zero.

00:02:07.830 --> 00:02:09.870
Those add up to zero.
Those add up to zero.

00:02:09.870 --> 00:02:13.650
So you have a bunch of zeros
and then a positive one.

00:02:13.650 --> 00:02:16.320
So all you're left with
is a positive one here,

00:02:16.320 --> 00:02:19.350
so that's going to be equal to one.

00:02:19.350 --> 00:02:21.120
Let's do another example.

00:02:21.120 --> 00:02:23.580
It's really helpful to see
these different scenarios.

00:02:23.580 --> 00:02:26.040
Negative five plus five,

00:02:26.040 --> 00:02:27.690
you might already suspect what that is.

00:02:27.690 --> 00:02:28.650
And pause that video.

00:02:28.650 --> 00:02:31.530
Actually try to represent
it with integer chips.

00:02:31.530 --> 00:02:32.850
Well, negative five,

00:02:32.850 --> 00:02:36.780
that's five of these
negative integer chips.

00:02:36.780 --> 00:02:39.000
So that's five right there.

00:02:39.000 --> 00:02:39.990
And then positive five

00:02:39.990 --> 00:02:44.990
would be five of the positive
integer chips, 1, 2, 3, 4, 5.

00:02:46.740 --> 00:02:48.175
And what's going to happen here?

00:02:48.175 --> 00:02:49.980
Well, those add up to zero.

00:02:49.980 --> 00:02:52.320
Those add up to zero.
Those add up to zero.

00:02:52.320 --> 00:02:54.330
Those add up to zero.
Those add up to zero.

00:02:54.330 --> 00:02:55.855
So you're just adding a bunch of zeros,

00:02:55.855 --> 00:03:00.480
which is of course going
to give you a big zero.

00:03:00.480 --> 00:03:01.794
Last but not least,

00:03:01.794 --> 00:03:06.510
let's add three and negative eight.

00:03:06.510 --> 00:03:09.930
And let's represent that
again with integer chips.

00:03:09.930 --> 00:03:14.930
So positive three is 1, 2,
3, positive integer chips.

00:03:16.230 --> 00:03:18.480
Negative eight is, let's see,

00:03:18.480 --> 00:03:23.480
1, 2, 3, 4, 5, 6, 7, 8

00:03:24.780 --> 00:03:27.450
of these negative integer chips.

00:03:27.450 --> 00:03:30.150
And these cancel.

00:03:30.150 --> 00:03:31.950
These cancel, lemme scratch
that out a little bit better.

00:03:31.950 --> 00:03:34.830
Those cancel, those
cancel, and those cancel.

00:03:34.830 --> 00:03:36.540
So you have zero plus zero plus zero

00:03:36.540 --> 00:03:38.310
plus, what do you have left over?

00:03:38.310 --> 00:03:40.620
Five of these negative integer chips.

00:03:40.620 --> 00:03:42.120
Five negative integer chips

00:03:42.120 --> 00:03:44.700
that's going to be equal to negative five.

00:03:44.700 --> 00:03:45.533
And we're done.


Adding and subtracting fractions with negatives

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:01.500
- [Instructor] Let's say we wanted

00:00:01.500 --> 00:00:06.500
to figure out what 3/7 minus
negative 7/3 minus 11/3 is.

00:00:11.760 --> 00:00:13.710
Pause this video and see
if you can have a go at it

00:00:13.710 --> 00:00:15.003
before we do it together.

00:00:16.200 --> 00:00:19.830
All right? Now let's
work on this together.

00:00:19.830 --> 00:00:22.527
And you might be tempted to deal

00:00:22.527 --> 00:00:26.010
with the negative 7/3 and the 11/3 first

00:00:26.010 --> 00:00:28.530
because they already have
a common denominator.

00:00:28.530 --> 00:00:31.890
But you have to realize that subtraction,

00:00:31.890 --> 00:00:33.630
you can't use the associative property.

00:00:33.630 --> 00:00:36.330
It's not the this, for example

00:00:36.330 --> 00:00:38.190
which is what you would typically do first

00:00:38.190 --> 00:00:43.190
is not the same thing
as this right over here.

00:00:43.230 --> 00:00:45.840
So you have to be very,
very, very careful.

00:00:45.840 --> 00:00:48.720
But what we could do is
rewrite this, instead

00:00:48.720 --> 00:00:52.020
of saying, minus something,
minus something else

00:00:52.020 --> 00:00:54.000
we could rewrite it in terms of addition.

00:00:54.000 --> 00:00:54.990
What do I mean by that?

00:00:54.990 --> 00:00:57.840
Well, if I have 3/7, I'll start with that.

00:00:57.840 --> 00:01:00.150
Subtracting something is the same thing

00:01:00.150 --> 00:01:03.600
as adding that something's opposite.

00:01:03.600 --> 00:01:06.930
So subtracting negative
7/3 is the same thing

00:01:06.930 --> 00:01:10.500
as adding the opposite of negative 7/3

00:01:10.500 --> 00:01:13.110
which is just positive 7/3.

00:01:13.110 --> 00:01:16.590
And subtracting 11/3 is the same thing

00:01:16.590 --> 00:01:21.590
as adding the opposite of
11/3 which is negative 11/3.

00:01:22.710 --> 00:01:25.770
Now addition, you can use
the associative property,

00:01:25.770 --> 00:01:29.610
you could add these two first

00:01:29.610 --> 00:01:32.400
or you could add these two first.

00:01:32.400 --> 00:01:33.870
And I like adding these two first

00:01:33.870 --> 00:01:36.600
because they have the same denominator.

00:01:36.600 --> 00:01:41.600
So if I have 7/3 plus negative 11/3,

00:01:41.910 --> 00:01:43.503
what is that going to get me?

00:01:44.340 --> 00:01:45.870
Well, we have a common denominator.

00:01:45.870 --> 00:01:47.040
We could rewrite it like this.

00:01:47.040 --> 00:01:51.330
3/7 plus common denominator of 3.

00:01:51.330 --> 00:01:56.070
We could write 7 plus
negative 11 in the numerator.

00:01:56.070 --> 00:02:01.070
And so 7 plus negative 11 is
the same thing as 7 minus 11.

00:02:01.620 --> 00:02:03.240
Because subtracting
something's the same thing

00:02:03.240 --> 00:02:04.620
as adding its opposite.

00:02:04.620 --> 00:02:06.870
So if we're adding negative 11, same thing

00:02:06.870 --> 00:02:08.100
as subtracting 11.

00:02:08.100 --> 00:02:10.950
So 7 plus negative 11, you might,

00:02:10.950 --> 00:02:12.000
we could get a number line out,

00:02:12.000 --> 00:02:13.830
but hopefully you've
gotten some practice now.

00:02:13.830 --> 00:02:15.960
That is going to be negative 4.

00:02:15.960 --> 00:02:17.760
That is negative 4.

00:02:17.760 --> 00:02:22.760
And so now we have 3/7 plus negative 4/3.

00:02:23.910 --> 00:02:27.630
And so now we definitely need
to find a common denominator.

00:02:27.630 --> 00:02:28.463
So let me rewrite this.

00:02:28.463 --> 00:02:32.640
This is equal to 3/7
plus, lemme write this,

00:02:32.640 --> 00:02:34.200
plus negative 4.

00:02:34.200 --> 00:02:38.280
Actually yeah, this is
fine plus negative 4/3.

00:02:38.280 --> 00:02:42.480
Or I could write this as even
negative 4/3, either way.

00:02:42.480 --> 00:02:44.910
But if we wanna have a common denominator,

00:02:44.910 --> 00:02:47.100
it looks like 21 is going

00:02:47.100 --> 00:02:50.490
to be the least common
multiple of 7 and 3.

00:02:50.490 --> 00:02:52.140
So let's rewrite each of these

00:02:52.140 --> 00:02:55.650
as something over 21, 3/7.

00:02:55.650 --> 00:02:58.080
To go from 7 to 21, you multiply by 3.

00:02:58.080 --> 00:03:00.420
So 3 times 3 is 9.

00:03:00.420 --> 00:03:03.840
And then to go from 3
to 21, we multiply by 7.

00:03:03.840 --> 00:03:07.590
So if we have negative 4 times 7,

00:03:07.590 --> 00:03:10.080
that is negative 28.

00:03:10.080 --> 00:03:11.400
And so this is going to be equal

00:03:11.400 --> 00:03:16.400
to 9 plus negative 28/21, which is,

00:03:17.920 --> 00:03:21.707
this is the same thing as 9 minus 28/21,

00:03:21.707 --> 00:03:23.850
'cause subtracting a
number is the same thing

00:03:23.850 --> 00:03:25.860
as adding its opposite.

00:03:25.860 --> 00:03:30.060
And so this gets us, let's
see if 9 minus 9 is 0

00:03:30.060 --> 00:03:33.180
and then we're gonna have
19 more to go below 0.

00:03:33.180 --> 00:03:36.540
So this is negative 19/21

00:03:36.540 --> 00:03:41.540
or we could write that as negative 19/21.

00:03:42.360 --> 00:03:43.953
And we are done.


Equivalent expressions with negative numbers

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:02.580
- [Teacher] We're asked which
of the following expressions

00:00:02.580 --> 00:00:07.580
are equivalent to two minus
9.4 plus zero plus 3.71?

00:00:08.487 --> 00:00:10.980
And we need to pick two answers.

00:00:10.980 --> 00:00:11.813
So pause this video

00:00:11.813 --> 00:00:12.930
and see if you can have a go at it

00:00:12.930 --> 00:00:14.313
before we do this together.

00:00:15.810 --> 00:00:18.510
All right, now let's
look through the choices.

00:00:18.510 --> 00:00:23.280
So this first choice, we have 9.4.

00:00:23.280 --> 00:00:25.740
Here we subtracted 9.4.

00:00:25.740 --> 00:00:27.510
Now when you subtract a number,

00:00:27.510 --> 00:00:31.050
you can also view it
as adding its opposite.

00:00:31.050 --> 00:00:36.050
So we could rewrite this top
thing as two plus negative 9.4.

00:00:37.170 --> 00:00:38.850
Notice when we subtract 9.4,

00:00:38.850 --> 00:00:42.000
that's the same thing
as adding negative 9.4

00:00:42.000 --> 00:00:44.700
so we could write it this way.

00:00:44.700 --> 00:00:46.200
And when you are adding numbers,

00:00:46.200 --> 00:00:48.060
and that's actually this property

00:00:48.060 --> 00:00:50.160
or this principle that we
just used, it's really useful

00:00:50.160 --> 00:00:52.200
because when you're
adding numbers like this,

00:00:52.200 --> 00:00:54.900
it doesn't matter what
order you add them in.

00:00:54.900 --> 00:00:56.970
When you have some subtraction involved,

00:00:56.970 --> 00:00:58.830
now all of a sudden the order does matter.

00:00:58.830 --> 00:01:00.150
But when you're adding like this,

00:01:00.150 --> 00:01:02.430
it doesn't matter the order
that you are doing it with.

00:01:02.430 --> 00:01:04.530
So we can swap these
numbers any way we want.

00:01:04.530 --> 00:01:05.640
We could write this, for example,

00:01:05.640 --> 00:01:10.640
as negative 9.4 plus two plus 3.71.

00:01:11.130 --> 00:01:12.150
We could ignore the zero

00:01:12.150 --> 00:01:13.560
or put that anywhere obviously.

00:01:13.560 --> 00:01:16.470
Plus zero, something like that.

00:01:16.470 --> 00:01:18.480
But what we see over here,

00:01:18.480 --> 00:01:21.360
they don't have a negative 9.4.

00:01:21.360 --> 00:01:24.210
They have a positive 9.4.

00:01:24.210 --> 00:01:27.000
So I really don't know how I
can reconcile this first thing

00:01:27.000 --> 00:01:27.833
in this right over here,

00:01:27.833 --> 00:01:29.550
so I'm gonna rule that one out.

00:01:29.550 --> 00:01:32.580
Now this one over here, you have a two.

00:01:32.580 --> 00:01:34.740
We have a two there.

00:01:34.740 --> 00:01:37.800
We have a 3.71.

00:01:37.800 --> 00:01:39.300
We have a 3.71 there

00:01:39.300 --> 00:01:41.190
and both of those things
are just being added.

00:01:41.190 --> 00:01:42.540
And now we have a,

00:01:42.540 --> 00:01:44.960
and now from that we are subtracting 9.4.

00:01:45.960 --> 00:01:49.500
And so here they are subtracting 9.4

00:01:49.500 --> 00:01:51.900
and it is indeed the case here,

00:01:51.900 --> 00:01:54.300
you have to be very careful
with subtraction if,

00:01:55.470 --> 00:01:58.620
we wouldn't wanna make it
9.4 minus something else.

00:01:58.620 --> 00:02:01.470
But as long as we're
only subtracting the 9.4,

00:02:01.470 --> 00:02:03.810
then the swapping of this order does work

00:02:03.810 --> 00:02:06.360
and you could even verify that
with the numbers over here.

00:02:06.360 --> 00:02:09.990
You could do two minus 9.4 plus 3.71

00:02:09.990 --> 00:02:12.990
or you could do two plus 3.71 minus 9.4.

00:02:12.990 --> 00:02:15.000
So I like this choice.

00:02:15.000 --> 00:02:16.050
Now let's look at this one.

00:02:16.050 --> 00:02:19.470
This has 3.71, 3.71.

00:02:19.470 --> 00:02:22.020
It has the two, it has the two,

00:02:22.020 --> 00:02:24.810
and now it's adding 9.4.

00:02:24.810 --> 00:02:26.310
Now we already said we could rewrite

00:02:26.310 --> 00:02:30.240
subtracting 9.4 as adding negative 9.4,

00:02:30.240 --> 00:02:33.120
but we can't rewrite
that as just adding 9.4.

00:02:33.120 --> 00:02:34.080
If this was like this,

00:02:34.080 --> 00:02:35.760
if it had a negative there,
then this would work,

00:02:35.760 --> 00:02:37.080
but it's not that.

00:02:37.080 --> 00:02:39.570
And so I'm going to rule that one out

00:02:39.570 --> 00:02:42.480
and I'm guessing this one's
going to work, but let's see.

00:02:42.480 --> 00:02:44.250
So this one put parentheses around it,

00:02:44.250 --> 00:02:46.890
so it's really just telling
us what we want to do first.

00:02:46.890 --> 00:02:49.260
So it says first do the two minus 9.4

00:02:49.260 --> 00:02:50.880
which if you just went left to right,

00:02:50.880 --> 00:02:52.230
you would've done anyway.

00:02:52.230 --> 00:02:54.810
And then plus zero plus 3.71

00:02:54.810 --> 00:02:57.240
and zero plus 3.71,

00:02:57.240 --> 00:03:00.510
obviously that's just going to be 3.71.

00:03:00.510 --> 00:03:04.530
So of course, this does
seem very reasonable to do.

00:03:04.530 --> 00:03:09.510
That you could go two minus 9.4 plus 3.71.

00:03:09.510 --> 00:03:12.900
Two minus 9.4, the zero
doesn't matter here,

00:03:12.900 --> 00:03:14.430
and then plus 3.71.

00:03:14.430 --> 00:03:17.070
So I like that choice.

00:03:17.070 --> 00:03:18.320
Let's do another example.

00:03:20.310 --> 00:03:21.990
So once again, we wanna come up

00:03:21.990 --> 00:03:24.990
with an equivalent with
equivalent expressions,

00:03:24.990 --> 00:03:26.880
so let's look at this one.

00:03:26.880 --> 00:03:28.410
Actually before I even
look at the choices,

00:03:28.410 --> 00:03:30.480
let's just recognize that
if you subtract a number,

00:03:30.480 --> 00:03:35.370
it's the same thing as
adding its opposite.

00:03:35.370 --> 00:03:37.620
So for example, this could be rewritten

00:03:37.620 --> 00:03:42.150
as negative 6/5 plus 1/2

00:03:42.150 --> 00:03:45.180
plus the opposite of negative 8/5

00:03:45.180 --> 00:03:47.280
which is positive 8/5.

00:03:47.280 --> 00:03:48.510
That's another way we could rewrite it.

00:03:48.510 --> 00:03:49.830
That might be helpful.

00:03:49.830 --> 00:03:54.830
Let's see, over here we have 1/2

00:03:55.080 --> 00:03:57.090
which we see right over there.

00:03:57.090 --> 00:03:59.430
We have plus 8/5

00:03:59.430 --> 00:04:01.260
and interestingly that looks
like the one that we did

00:04:01.260 --> 00:04:03.960
in the second version right over here,

00:04:03.960 --> 00:04:06.720
and then we have minus 6/5.

00:04:06.720 --> 00:04:10.170
So is minus 6/5 the same
thing as this over here?

00:04:10.170 --> 00:04:11.003
Well, let's think about it.

00:04:11.003 --> 00:04:12.180
What we're really doing when we write

00:04:12.180 --> 00:04:14.400
this negative 6/5 out front,

00:04:14.400 --> 00:04:16.620
that's the same thing as,

00:04:16.620 --> 00:04:19.860
we could do that as plus negative 6/5

00:04:19.860 --> 00:04:21.990
and that's going to be the same thing

00:04:21.990 --> 00:04:23.880
as subtracting the opposite of it.

00:04:23.880 --> 00:04:25.410
Or another way of thinking about it

00:04:25.410 --> 00:04:29.520
is subtracting positive 6/5.

00:04:29.520 --> 00:04:34.520
Let me write it this way,
subtracting positive 6/5.

00:04:34.620 --> 00:04:35.580
It's actually hard to tell

00:04:35.580 --> 00:04:37.260
because I put that I have,

00:04:37.260 --> 00:04:38.880
actually let me write it this way.

00:04:38.880 --> 00:04:41.460
If we just took this version

00:04:41.460 --> 00:04:44.310
and if we were to use
a commutative property

00:04:44.310 --> 00:04:45.780
to swap the orders,

00:04:45.780 --> 00:04:50.190
we could write this as 1/5 plus 8/5

00:04:50.190 --> 00:04:53.670
and then plus negative 6/5.

00:04:53.670 --> 00:04:55.320
All I did is swap the order

00:04:55.320 --> 00:04:57.840
and now we know that this
last part could be rewritten.

00:04:57.840 --> 00:05:00.240
So it's 1/5 plus 8/5.

00:05:00.240 --> 00:05:02.190
Instead of plus negative 6/5,

00:05:02.190 --> 00:05:04.110
that can be rewritten as negative

00:05:04.110 --> 00:05:07.530
or minus I should say, minus positive 6/5.

00:05:07.530 --> 00:05:08.520
Why can I do that?

00:05:08.520 --> 00:05:10.260
Because subtracting a number

00:05:10.260 --> 00:05:13.440
is the same thing as adding its opposite.

00:05:13.440 --> 00:05:15.210
And so all of these are equivalent

00:05:15.210 --> 00:05:16.650
and this is exactly what they have

00:05:16.650 --> 00:05:18.270
written here, this last scenario.

00:05:18.270 --> 00:05:20.820
So I like that one.

00:05:20.820 --> 00:05:25.350
Now let's see, this one has a negative 6/5

00:05:25.350 --> 00:05:28.860
and it has a plus 8/5 and a plus 1/2.

00:05:28.860 --> 00:05:30.600
So all they did is they took this version

00:05:30.600 --> 00:05:32.310
which we knew is equivalent

00:05:32.310 --> 00:05:35.250
and they swapped the
order that we are doing.

00:05:35.250 --> 00:05:38.940
They actually swapped the
order of the 1/2 and the 8/5.

00:05:38.940 --> 00:05:41.040
And when you're dealing
with addition like this,

00:05:41.040 --> 00:05:43.780
you can move things around,
the commutative property.

00:05:43.780 --> 00:05:46.770
You can do your addition
in different orders

00:05:46.770 --> 00:05:47.603
and that's all they're doing.

00:05:47.603 --> 00:05:48.720
Actually this is not just commutative,

00:05:48.720 --> 00:05:49.710
this is also associative.

00:05:49.710 --> 00:05:52.110
They're literally putting
that parentheses there

00:05:52.110 --> 00:05:53.550
and say hey, let's just do,

00:05:53.550 --> 00:05:56.040
and let's just do the
those first two first.

00:05:56.040 --> 00:05:57.360
Let's swap these two

00:05:57.360 --> 00:05:59.550
and then of course, even if
you're going left to right,

00:05:59.550 --> 00:06:01.230
you would do these first two

00:06:01.230 --> 00:06:03.090
even if the parentheses weren't there.

00:06:03.090 --> 00:06:07.260
So all of this is definitely
equivalent to negative 6/5,

00:06:07.260 --> 00:06:09.090
swapping the 1/2 and the 8/5.

00:06:09.090 --> 00:06:12.030
So 8/5 plus 1/2.

00:06:12.030 --> 00:06:13.980
So I like that choice too.

00:06:13.980 --> 00:06:15.930
And so we had two choices,

00:06:15.930 --> 00:06:18.390
so these are probably not going to work

00:06:18.390 --> 00:06:20.370
and let's see why they don't real fast.

00:06:20.370 --> 00:06:22.350
They both have the negative 6/5.

00:06:22.350 --> 00:06:24.483
This has the plus 1/2,

00:06:24.483 --> 00:06:27.841
but it has minus 8/5 instead
of minus negative 8/5.

00:06:27.841 --> 00:06:30.390
It could have plus 8/5 here,

00:06:30.390 --> 00:06:31.410
but not minus 8/5.

00:06:31.410 --> 00:06:33.000
So we rule that one out.

00:06:33.000 --> 00:06:35.790
Negative 6/5, and let's see.

00:06:35.790 --> 00:06:38.760
Then it's adding negative 8/5

00:06:38.760 --> 00:06:41.190
instead of subtracting negative 8/5.

00:06:41.190 --> 00:06:42.330
So I don't like that,

00:06:42.330 --> 00:06:44.670
and then it subtracts 1/2
instead of adding 1/2,

00:06:44.670 --> 00:06:48.930
so it's no good on multiple dimensions.

00:06:48.930 --> 00:06:51.540
Let's do one last example.

00:06:51.540 --> 00:06:53.430
Which of the expressions are equivalent?

00:06:53.430 --> 00:06:56.460
As always pause and see if
you can figure that out.

00:06:56.460 --> 00:06:59.652
So let's see, so actually
before I look at the choices,

00:06:59.652 --> 00:07:02.580
I'm gonna see if there's
different ways to rewrite this.

00:07:02.580 --> 00:07:07.580
We could rewrite this as 1.7 minus 8.33.

00:07:07.920 --> 00:07:09.300
Why can I do that again?

00:07:09.300 --> 00:07:10.620
Because subtracting a number

00:07:10.620 --> 00:07:13.530
is the same thing as adding its opposite,

00:07:13.530 --> 00:07:15.930
and then I could say plus.

00:07:15.930 --> 00:07:17.860
I could write this as 8.33

00:07:19.020 --> 00:07:20.820
and if I wanted to I
could even write this,

00:07:20.820 --> 00:07:23.580
instead of minus 1.95,
I could even write that

00:07:23.580 --> 00:07:28.580
as plus negative 1.95 if I choose

00:07:29.070 --> 00:07:30.780
'cause once again, subtracting a number

00:07:30.780 --> 00:07:33.660
is the same thing as adding its opposite.

00:07:33.660 --> 00:07:35.670
Now I'm tempted to try to compute this

00:07:35.670 --> 00:07:37.953
because we could just,

00:07:39.090 --> 00:07:41.640
when we're doing addition like this

00:07:41.640 --> 00:07:42.840
and I could write it like this

00:07:42.840 --> 00:07:44.340
if I want everything to be addition.

00:07:44.340 --> 00:07:49.340
1.7 plus negative 8.33

00:07:49.560 --> 00:07:54.510
plus 8.33 plus negative 1.95.

00:07:54.510 --> 00:07:56.160
When everything is expressed in addition,

00:07:56.160 --> 00:07:57.810
we know that the order doesn't matter.

00:07:57.810 --> 00:07:59.580
The commutative property tells us that.

00:07:59.580 --> 00:08:02.200
And even the associative property tells us

00:08:03.210 --> 00:08:04.980
how we put the parentheses doesn't matter,

00:08:04.980 --> 00:08:06.750
and so we could actually do that first

00:08:06.750 --> 00:08:08.910
and though these two would cancel out.

00:08:08.910 --> 00:08:10.320
Actually maybe we will have to do that.

00:08:10.320 --> 00:08:11.400
That choice looks like that,

00:08:11.400 --> 00:08:13.650
but let's just go step by step.

00:08:13.650 --> 00:08:15.690
So this has 1.7 and 1.7.

00:08:15.690 --> 00:08:17.130
Let me do this in a different color.

00:08:17.130 --> 00:08:20.100
One point, that's not a different color.

00:08:20.100 --> 00:08:25.100
1.7, 1.7, they're subtracting 8.33.

00:08:27.030 --> 00:08:29.100
That's subtracting 8.33 here,

00:08:29.100 --> 00:08:32.100
so that's equivalent to
this version right here.

00:08:32.100 --> 00:08:34.470
Then they're subtracting it again.

00:08:34.470 --> 00:08:36.420
No, this does not look good

00:08:36.420 --> 00:08:37.770
because even when we try to compute it,

00:08:37.770 --> 00:08:41.100
we know that these two terms
will cancel out each other,

00:08:41.100 --> 00:08:43.440
so I don't like that one.

00:08:43.440 --> 00:08:45.000
Let's see, this one.

00:08:45.000 --> 00:08:48.300
If we take this one here

00:08:48.300 --> 00:08:49.980
and we were to reorder it

00:08:49.980 --> 00:08:53.910
because this is all addition
of different integers here,

00:08:53.910 --> 00:08:55.200
but if we were to reorder it,

00:08:55.200 --> 00:08:59.833
we could reorder it as
negative 8.33, negative 8.33,

00:09:01.230 --> 00:09:05.310
plus 8.33, plus 8.33,

00:09:05.310 --> 00:09:08.370
plus 1.7, plus 1.7,

00:09:08.370 --> 00:09:12.450
and then we have plus negative 1.95

00:09:12.450 --> 00:09:17.053
which we know we can write
as minus 1.95, minus 1.95,

00:09:18.030 --> 00:09:19.320
because subtracting a number

00:09:19.320 --> 00:09:21.300
is the same thing as adding its opposite,

00:09:21.300 --> 00:09:23.190
and that's what they have here.

00:09:23.190 --> 00:09:24.210
They put these parentheses.

00:09:24.210 --> 00:09:27.540
But if you just went left
to right, I should say,

00:09:27.540 --> 00:09:30.750
with this one, you're going
to get the same thing.

00:09:30.750 --> 00:09:33.960
So I like that choice there.

00:09:33.960 --> 00:09:37.470
Now this choice, if you just
canceled these two terms out,

00:09:37.470 --> 00:09:42.150
you're left with a 1.7 plus negative 1.95

00:09:42.150 --> 00:09:46.320
which we already know is the
same thing as minus 1.95,

00:09:46.320 --> 00:09:48.540
so I like this choice as well.

00:09:48.540 --> 00:09:50.550
Now why won't this one work?

00:09:50.550 --> 00:09:52.683
Let's see, 1.95.

00:09:55.710 --> 00:09:57.300
They're trying to change the order,

00:09:57.300 --> 00:09:59.970
but you can add a negative 1.95

00:09:59.970 --> 00:10:00.960
and then put it out front.

00:10:00.960 --> 00:10:02.760
But this is a positive 1.95,

00:10:02.760 --> 00:10:05.550
so I'm already not liking this.

00:10:05.550 --> 00:10:07.110
And then they're subtracting 1.7.

00:10:07.110 --> 00:10:09.540
There's no reason why
you should subtract 1.7,

00:10:09.540 --> 00:10:12.033
so this is definitely not looking good.


Subtracting integers  find the missing value

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:01.500
- [Instructor] So, if I were to ask you,

00:00:01.500 --> 00:00:02.430
or if I were to tell you

00:00:02.430 --> 00:00:07.430
that -3 minus blank is equal to -4,

00:00:07.620 --> 00:00:11.343
can you pause this video and
figure out what this blank is?

00:00:12.900 --> 00:00:15.120
All right, now let's do this together,

00:00:15.120 --> 00:00:17.460
and I'm gonna do this by
drawing out a number line

00:00:17.460 --> 00:00:19.140
because that's what my head tries to do

00:00:19.140 --> 00:00:20.430
with things like this.

00:00:20.430 --> 00:00:21.870
Let me draw a straighter line than that.

00:00:21.870 --> 00:00:24.330
All right, that's a
pretty good number line.

00:00:24.330 --> 00:00:27.270
Let's see, I'm gonna be dealing with -3

00:00:27.270 --> 00:00:29.520
and I'm gonna subtract
something of it to get to -4,

00:00:29.520 --> 00:00:31.440
so let me focus on the negative end here.

00:00:31.440 --> 00:00:35.400
So, let's say this is
zero, that could be +1.

00:00:35.400 --> 00:00:40.007
Then, I have -1, -2, -3,

00:00:40.007 --> 00:00:44.010
- 4, -5, let's go -6.

00:00:44.010 --> 00:00:45.270
That's the other end of it.

00:00:45.270 --> 00:00:48.952
So, this is -1, -2, -3,

00:00:48.952 --> 00:00:51.840
- 4, and -5.

00:00:51.840 --> 00:00:54.180
So, let's start at -3.

00:00:54.180 --> 00:00:57.990
So, -3 is that point on the number line

00:00:57.990 --> 00:01:01.740
and I want to end up at -4,

00:01:01.740 --> 00:01:04.140
that is that point on the number line.

00:01:04.140 --> 00:01:06.900
So, to go from -3 to -4,

00:01:06.900 --> 00:01:10.860
I have to go one step in
the leftward direction.

00:01:10.860 --> 00:01:13.230
So, one step in the leftward direction,

00:01:13.230 --> 00:01:16.290
you could either view
that as subtracting 1

00:01:16.290 --> 00:01:20.220
or you could view that as adding a -1.

00:01:20.220 --> 00:01:22.620
Now, we have, we're
already subtracting here,

00:01:22.620 --> 00:01:24.450
so the simplest thing to
do would just say, "Okay,

00:01:24.450 --> 00:01:27.450
this is the same thing as subtracting 1."

00:01:27.450 --> 00:01:32.430
And we're done, -3 minus
1 is indeed equal to -4.

00:01:32.430 --> 00:01:34.140
Let's do another example.

00:01:34.140 --> 00:01:37.030
Let's say we had -1

00:01:37.890 --> 00:01:42.890
is equal to -7, minus what?

00:01:43.350 --> 00:01:45.000
Pause a video and try
to work through that.

00:01:45.000 --> 00:01:46.050
And try to do a number line,

00:01:46.050 --> 00:01:47.403
I always find that useful.

00:01:48.660 --> 00:01:51.360
All right, let's work
through this together again,

00:01:51.360 --> 00:01:54.633
so let me draw a number line here.

00:01:55.560 --> 00:01:57.209
Let's see, I have a -1,

00:01:57.209 --> 00:01:58.383
I have a negative, I have a -7,

00:01:59.790 --> 00:02:01.770
so I'm gonna deal with
the negate end of things.

00:02:01.770 --> 00:02:06.332
So, let me make this zero,
I'll make that positive 1,

00:02:06.332 --> 00:02:09.249
- 1, -2, -3,

00:02:09.249 --> 00:02:14.249
- 4, -5, -6, -7.

00:02:14.940 --> 00:02:17.228
Let me write this, this
is -7 right over here,

00:02:17.228 --> 00:02:20.820
- 8, -9, -10.

00:02:20.820 --> 00:02:22.560
I think that's probably enough.

00:02:22.560 --> 00:02:23.820
So, let's see.

00:02:23.820 --> 00:02:26.820
We're saying -1 is equal
to -7 minus something,

00:02:26.820 --> 00:02:30.780
so essentially we're starting at -7 here,

00:02:30.780 --> 00:02:31.993
we're starting at -7,

00:02:33.270 --> 00:02:36.030
and we're subtracting something from that

00:02:36.030 --> 00:02:40.710
to end up over here at -1.

00:02:40.710 --> 00:02:43.920
This is -1 right over there.

00:02:43.920 --> 00:02:44.760
So, let's just think about

00:02:44.760 --> 00:02:49.680
what the arrow needs to go
do to get from -7 to -1.

00:02:49.680 --> 00:02:53.190
Well, to do, that you're
going to have to go,

00:02:53.190 --> 00:02:56.250
it looks like six units to the right.

00:02:56.250 --> 00:02:58.620
One, two, three, four, five, six,

00:02:58.620 --> 00:03:00.840
we go six units to the right.

00:03:00.840 --> 00:03:01.920
Now, there's two ways

00:03:01.920 --> 00:03:03.840
to describe going six units to the right.

00:03:03.840 --> 00:03:06.810
You could say that is just plus 6,

00:03:06.810 --> 00:03:11.810
or you could view that that's
the same thing as minus -6.

00:03:11.820 --> 00:03:12.780
And we've seen that,

00:03:12.780 --> 00:03:15.270
subtracting a number is the same thing

00:03:15.270 --> 00:03:17.820
as adding its inverse.

00:03:17.820 --> 00:03:21.780
And since we already have
a a negative sign here

00:03:21.780 --> 00:03:23.370
or minus sign here,

00:03:23.370 --> 00:03:24.960
we might as well say, "Well,
this is the same thing

00:03:24.960 --> 00:03:28.920
as subtracting a -6,"

00:03:28.920 --> 00:03:31.230
a -6, I'll put in parentheses

00:03:31.230 --> 00:03:32.700
to make it a little bit cleaner,

00:03:32.700 --> 00:03:33.723
and we're done.

00:03:34.650 --> 00:03:38.010
Let's do another example,

00:03:38.010 --> 00:03:40.170
another example here.

00:03:40.170 --> 00:03:43.618
So, let's say we make,

00:03:43.618 --> 00:03:46.560
give ourselves a clean slate.

00:03:46.560 --> 00:03:48.330
Let's say we wanted to figure out

00:03:48.330 --> 00:03:52.830
so I have blank minus -5

00:03:52.830 --> 00:03:55.140
is equal to 13.

00:03:55.140 --> 00:03:56.440
How would you tackle that?

00:03:57.630 --> 00:03:59.880
Well, let me just draw
my number line again.

00:03:59.880 --> 00:04:02.220
I have my whole real
estate to use this time,

00:04:02.220 --> 00:04:03.330
so let me just go all the way there.

00:04:03.330 --> 00:04:06.390
And let's see, I'm dealing with something,

00:04:06.390 --> 00:04:11.390
it's gonna be roughly five
away, it feels like from 13.

00:04:11.400 --> 00:04:14.040
So, let me just make my number line,

00:04:14.040 --> 00:04:17.670
I'm just gonna say this is
zero, I have to get to 13,

00:04:17.670 --> 00:04:20.250
so actually let me give myself
a little bit more space.

00:04:20.250 --> 00:04:24.690
So, this is 0, 1, 2, 3,

00:04:24.690 --> 00:04:29.329
4, 5, 6, 7, 8, 9, 10,

00:04:29.329 --> 00:04:34.329
11, 12, 13, 14, 15, 16, 17, 18.

00:04:35.220 --> 00:04:37.170
I think that is sufficient.

00:04:37.170 --> 00:04:40.290
So, we don't know where we're starting,

00:04:40.290 --> 00:04:42.060
we know we wanna end up at 13.

00:04:42.060 --> 00:04:47.040
So, 13, let me put that
on our number line here,

00:04:47.040 --> 00:04:49.170
so it's 1, 2, 3, 4, 5,

00:04:49.170 --> 00:04:54.170
6, 7, 8, 9, 10, 11, 12, 13.

00:04:54.240 --> 00:04:57.630
So, 13 is right over here.

00:04:57.630 --> 00:05:00.120
And now how did we get to 13?

00:05:00.120 --> 00:05:04.080
Well, we are subtracting a -5.

00:05:04.080 --> 00:05:05.280
Now, as I just mentioned,

00:05:05.280 --> 00:05:07.860
subtracting a number is the same thing

00:05:07.860 --> 00:05:09.870
as adding the inverse of the number,

00:05:09.870 --> 00:05:14.100
so this is the same thing as adding +5

00:05:14.100 --> 00:05:16.410
is another way of thinking about that.

00:05:16.410 --> 00:05:18.480
So, or another way of
depicting either of these,

00:05:18.480 --> 00:05:21.240
subtracting -5 or adding +5

00:05:21.240 --> 00:05:23.670
would essentially be
you're starting someplace

00:05:23.670 --> 00:05:27.660
and you're going 5 units to the right.

00:05:27.660 --> 00:05:28.830
So, you started someplace

00:05:28.830 --> 00:05:32.430
and you're going 5 units to
the right to end up at 13.

00:05:32.430 --> 00:05:35.070
So, that means you started
5 units to the left,

00:05:35.070 --> 00:05:37.860
one, two, three, four, five.

00:05:37.860 --> 00:05:39.540
You started right over here,

00:05:39.540 --> 00:05:42.930
you go 5 units to the right

00:05:42.930 --> 00:05:45.150
and you ended up at 13.

00:05:45.150 --> 00:05:46.380
Well, this right over here,

00:05:46.380 --> 00:05:48.750
5 units to the left of 13,

00:05:48.750 --> 00:05:52.080
this is going to be 8.

00:05:52.080 --> 00:05:57.030
And it is indeed the case that 8 minus -5

00:05:57.030 --> 00:06:00.840
is equal to 13.

00:06:00.840 --> 00:06:02.340
Now, there's other ways

00:06:02.340 --> 00:06:05.400
that you might be able
to think about this.

00:06:05.400 --> 00:06:09.240
If something minus -5

00:06:09.240 --> 00:06:12.120
is equal to 13

00:06:12.120 --> 00:06:13.440
then another way you could think about it,

00:06:13.440 --> 00:06:17.100
if I were to tell you that
3 minus 2 is equal to 1,

00:06:17.100 --> 00:06:19.530
you could also say that
an equivalent statement

00:06:19.530 --> 00:06:22.200
is that 3 is equal to 1 plus 2.

00:06:22.200 --> 00:06:24.690
You could turn that subtraction equation

00:06:24.690 --> 00:06:26.730
into an addition equation.

00:06:26.730 --> 00:06:30.270
So, over here, if I say this
minus this is equal to that

00:06:30.270 --> 00:06:32.410
then that means that 13

00:06:34.260 --> 00:06:37.540
plus -5

00:06:38.640 --> 00:06:42.150
must be equal to blank,
our mystery number.

00:06:42.150 --> 00:06:44.580
And if I add 13 and -5,

00:06:44.580 --> 00:06:48.210
you might recognize that
as being equal to 8.

00:06:48.210 --> 00:06:49.043
So, either way,

00:06:49.043 --> 00:06:50.550
there's a lot of ways that
you could approach this,

00:06:50.550 --> 00:06:53.613
but they all get you to kind
of the same conceptual place.


Adding integers: find the missing value

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:02.370
- [Instructor] We are asked
to find the missing value,

00:00:02.370 --> 00:00:03.600
and they give us a hint.

00:00:03.600 --> 00:00:06.540
Use the number line to
find the missing value.

00:00:06.540 --> 00:00:11.070
And let's see, they say, 4
plus something is equal to -6.

00:00:11.070 --> 00:00:12.180
So why don't you pause this video

00:00:12.180 --> 00:00:13.740
and see if you can work through this

00:00:13.740 --> 00:00:16.080
before we do it ourselves.

00:00:16.080 --> 00:00:19.500
All right, so let's just start at 4.

00:00:19.500 --> 00:00:21.120
So there's a couple of
ways to think about it.

00:00:21.120 --> 00:00:23.370
We could just say where
is 4 on the number line.

00:00:23.370 --> 00:00:25.950
4 is here. One, two, three, four.

00:00:25.950 --> 00:00:27.480
We're going to start there,

00:00:27.480 --> 00:00:29.250
and we're going to add something

00:00:29.250 --> 00:00:33.030
to that 4 to get us to -6.

00:00:33.030 --> 00:00:34.920
To get us right there.

00:00:34.920 --> 00:00:36.270
So what are we going to add?

00:00:36.270 --> 00:00:38.340
Well, we're going to
be moving to the left.

00:00:38.340 --> 00:00:39.600
So we're going to be adding

00:00:39.600 --> 00:00:41.730
a negative number right over here.

00:00:41.730 --> 00:00:44.070
And we can see what that
negative number is going to be.

00:00:44.070 --> 00:00:46.050
How many units do we
have to move to the left?

00:00:46.050 --> 00:00:50.070
We have to move one,
two, three, four, five,

00:00:50.070 --> 00:00:55.070
six, seven, eight, nine, 10 to the left.

00:00:55.140 --> 00:00:58.050
So what do we add if we
want to move 10 to the left?

00:00:58.050 --> 00:01:00.540
We have to add a -10.

00:01:00.540 --> 00:01:03.360
So this is -10 right over here.

00:01:03.360 --> 00:01:04.410
Another way to think about it,

00:01:04.410 --> 00:01:06.780
if you're four to the right of zero,

00:01:06.780 --> 00:01:09.450
well, it's going take you
four just to get back to zero.

00:01:09.450 --> 00:01:10.590
That's four there.

00:01:10.590 --> 00:01:12.990
And it's going to take you another six

00:01:12.990 --> 00:01:15.300
to get you to -6.

00:01:15.300 --> 00:01:16.980
And then of course you're
moving to the left,

00:01:16.980 --> 00:01:19.380
so that's going to be a -10.

00:01:19.380 --> 00:01:22.650
Let's do another a example.

00:01:22.650 --> 00:01:24.690
So let me scroll down a bit

00:01:24.690 --> 00:01:27.270
so we can see that second example.

00:01:27.270 --> 00:01:29.880
So once again, find the missing value.

00:01:29.880 --> 00:01:32.480
And pause this video again
and try to work that out.

00:01:33.360 --> 00:01:35.100
All right, so there's a couple of ways

00:01:35.100 --> 00:01:35.940
we could think about it.

00:01:35.940 --> 00:01:39.900
We're saying -3 is equal
to something plus 5.

00:01:39.900 --> 00:01:43.198
So let's just see where
-3 is on this number line.

00:01:43.198 --> 00:01:45.990
- 3 is right over there.

00:01:45.990 --> 00:01:50.990
And something plus 5 is
going to be equal to -3.

00:01:51.750 --> 00:01:53.310
So plus 5, you could view it

00:01:53.310 --> 00:01:55.650
as you're going to start someplace,

00:01:55.650 --> 00:01:57.720
and you're going to move
five units to the right,

00:01:57.720 --> 00:02:01.290
something like that to
get to where you end up.

00:02:01.290 --> 00:02:04.230
So where could we start to
get five units to the right

00:02:04.230 --> 00:02:05.250
and be at -3?

00:02:05.250 --> 00:02:06.990
Well, if we just go
five units to the left,

00:02:06.990 --> 00:02:11.100
so one, two, three, four, five.

00:02:11.100 --> 00:02:12.750
If you start from this point,

00:02:12.750 --> 00:02:15.570
and you were to add 5,

00:02:15.570 --> 00:02:17.430
you were get to -3.

00:02:17.430 --> 00:02:19.680
And so what is this point right over here?

00:02:19.680 --> 00:02:20.760
Well, this right over here,

00:02:20.760 --> 00:02:25.020
let's see, this is -5, -6, -7, -8.

00:02:25.020 --> 00:02:28.470
So we have -8 plus 5 is equal to -3.

00:02:28.470 --> 00:02:30.570
Another way to think about it is,

00:02:30.570 --> 00:02:32.827
you could have used a
commutative property to say,

00:02:32.827 --> 00:02:34.680
"Hey, instead of blank plus 5,

00:02:34.680 --> 00:02:38.400
you could say this is -3
is equal to 5 plus blank.

00:02:38.400 --> 00:02:39.330
And so you could say,

00:02:39.330 --> 00:02:41.160
let's start at 5,

00:02:41.160 --> 00:02:42.180
let me do this in another color.

00:02:42.180 --> 00:02:43.803
Let's start at 5,

00:02:44.670 --> 00:02:48.120
and then what would I have
to add to it to get to -3?

00:02:48.120 --> 00:02:51.420
And so, well, I would have
to move to the left five

00:02:51.420 --> 00:02:54.840
and then three in order to get to -3.

00:02:54.840 --> 00:02:57.990
So you would essentially have to add -8.

00:02:57.990 --> 00:03:00.690
So that's another way that
you could think about that.


Adding the opposite with integer chips

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:02.670
- [Instructor] So let's
use integer chips again

00:00:02.670 --> 00:00:04.800
to start exploring a little bit more

00:00:04.800 --> 00:00:06.480
when we deal with negative numbers.

00:00:06.480 --> 00:00:07.920
So let's say we wanted to compute

00:00:07.920 --> 00:00:10.680
what negative one minus seven is.

00:00:10.680 --> 00:00:12.060
See if you can pause this video

00:00:12.060 --> 00:00:14.163
and figure that out using integer chips.

00:00:15.180 --> 00:00:16.770
Well, let's do this together.

00:00:16.770 --> 00:00:18.660
So we're starting with negative one.

00:00:18.660 --> 00:00:22.680
We could represent negative
one as just one integer,

00:00:22.680 --> 00:00:25.320
one negative integer chip,

00:00:25.320 --> 00:00:29.070
but we need to subtract seven,
positive seven from that.

00:00:29.070 --> 00:00:32.580
We have no positive integer chips here,

00:00:32.580 --> 00:00:34.350
so, and we need to have at least seven

00:00:34.350 --> 00:00:37.380
positive integer chips in
order to subtract seven.

00:00:37.380 --> 00:00:39.990
So how could we get some
positive integer chips?

00:00:39.990 --> 00:00:41.850
Well, we can just add pairs of negative

00:00:41.850 --> 00:00:43.080
and positive integer chips.

00:00:43.080 --> 00:00:45.480
So if I add one negative integer chip

00:00:45.480 --> 00:00:48.990
and I add one positive
integer chip, just like that,

00:00:48.990 --> 00:00:51.210
this is still negative one over here

00:00:51.210 --> 00:00:52.770
because these two integer chips

00:00:52.770 --> 00:00:55.170
are going to cancel each other out.

00:00:55.170 --> 00:00:57.360
So let me just do that seven times.

00:00:57.360 --> 00:00:58.470
So let me just do this.

00:00:58.470 --> 00:01:02.820
So that's two, three, four, five,

00:01:02.820 --> 00:01:04.890
six and seven.

00:01:04.890 --> 00:01:06.990
And then I just have to
add the corresponding

00:01:06.990 --> 00:01:11.990
positive integer chips three, four, five,

00:01:12.240 --> 00:01:13.710
six and seven.

00:01:13.710 --> 00:01:15.090
So notice, what I just wrote,

00:01:15.090 --> 00:01:18.240
this is just another way
of writing negative one.

00:01:18.240 --> 00:01:20.430
But I wrote it this way
because I can actually

00:01:20.430 --> 00:01:24.390
subtract out positive seven now from this.

00:01:24.390 --> 00:01:27.510
So now let's subtract out positive seven.

00:01:27.510 --> 00:01:31.290
So subtract out one, two, three, four,

00:01:31.290 --> 00:01:33.150
five, six, seven.

00:01:33.150 --> 00:01:35.820
And then what are we left with?

00:01:35.820 --> 00:01:40.710
Well, we're left with all of
this business right over here.

00:01:40.710 --> 00:01:41.640
And what is that?

00:01:41.640 --> 00:01:43.740
That's one, two, three, four,

00:01:43.740 --> 00:01:45.210
five, six, seven, eight.

00:01:45.210 --> 00:01:47.850
This is equal to negative eight.

00:01:47.850 --> 00:01:49.800
Now that's interesting by itself,

00:01:49.800 --> 00:01:52.020
but you might notice something.

00:01:52.020 --> 00:01:54.600
What I have left over
when I take a negative one

00:01:54.600 --> 00:01:57.900
and I subtract positive seven from that,

00:01:57.900 --> 00:02:00.000
I'm left with essentially the equivalent

00:02:00.000 --> 00:02:03.570
of negative one and negative seven.

00:02:03.570 --> 00:02:06.180
So another way of
writing what we just have

00:02:06.180 --> 00:02:10.920
left over here is I have negative one

00:02:10.920 --> 00:02:13.500
is that one negative integer
chip right over there.

00:02:13.500 --> 00:02:14.730
And then I have negative seven,

00:02:14.730 --> 00:02:18.030
these seven negative integer
chips right over there.

00:02:18.030 --> 00:02:20.490
So you could also view
this as the same thing

00:02:20.490 --> 00:02:25.490
as negative one plus negative seven.

00:02:26.040 --> 00:02:28.560
And so this makes us
think about something.

00:02:28.560 --> 00:02:32.370
Is it true that if I subtract a positive,

00:02:32.370 --> 00:02:34.740
that's the same thing
as adding the inverse

00:02:34.740 --> 00:02:38.430
of that positive, adding, in
the case of a positive seven,

00:02:38.430 --> 00:02:40.830
in the case of subtracting
a positive seven,

00:02:40.830 --> 00:02:44.940
that's gonna be the same thing
as adding a negative seven?

00:02:44.940 --> 00:02:45.870
Interesting.

00:02:45.870 --> 00:02:46.950
And let's see actually if it works

00:02:46.950 --> 00:02:49.170
the other way around.

00:02:49.170 --> 00:02:52.020
So let's see what happens
when we subtract a negative.

00:02:52.020 --> 00:02:56.163
So if we have negative
three minus negative five,

00:02:57.060 --> 00:03:01.410
maybe, maybe this is the
same thing as negative three

00:03:01.410 --> 00:03:04.770
plus the opposite of negative five,

00:03:04.770 --> 00:03:07.140
which would be positive five.

00:03:07.140 --> 00:03:09.360
Let's see if these two things actually

00:03:09.360 --> 00:03:10.530
amount to be the same thing.

00:03:10.530 --> 00:03:12.690
So let's just start with
this first one up here.

00:03:12.690 --> 00:03:14.430
We're gonna start with negative three,

00:03:14.430 --> 00:03:18.210
so that gives us three
negative integer chips.

00:03:18.210 --> 00:03:22.560
So negative one, negative
two, negative three.

00:03:22.560 --> 00:03:25.800
Now if we wanna subtract
out negative five,

00:03:25.800 --> 00:03:28.920
if we wanna take away five
negative integer chips,

00:03:28.920 --> 00:03:30.630
well we need more negative
integer chips here.

00:03:30.630 --> 00:03:33.510
We need at least two more
negative integer chips.

00:03:33.510 --> 00:03:36.240
So if we have two more
negative integer chips,

00:03:36.240 --> 00:03:37.950
we're not changing the value of that

00:03:37.950 --> 00:03:41.010
if we have two more
positive integer chips.

00:03:41.010 --> 00:03:43.320
What I have depicted here
is still negative three

00:03:43.320 --> 00:03:45.000
because that and that cancel out.

00:03:45.000 --> 00:03:47.910
And so this is still the
number negative three

00:03:47.910 --> 00:03:51.030
being represented, but
I added these two pairs

00:03:51.030 --> 00:03:55.080
because now I can subtract out
five negative integer chips.

00:03:55.080 --> 00:03:56.850
That's what negative five represents.

00:03:56.850 --> 00:03:59.310
These top four negative integer
chips, there's five of 'em,

00:03:59.310 --> 00:04:01.320
I can take 'em all away.

00:04:01.320 --> 00:04:04.380
That's subtracting out a negative five.

00:04:04.380 --> 00:04:05.880
And what am I left with?

00:04:05.880 --> 00:04:09.630
What I'm left with is just these
two positive integer chips.

00:04:09.630 --> 00:04:11.913
So this is going to be
equal to positive two.

00:04:12.780 --> 00:04:16.770
Well, that's interesting
because that's kind of feeling

00:04:16.770 --> 00:04:19.020
very similar to what we have here.

00:04:19.020 --> 00:04:22.740
If we start with negative
three, so negative one,

00:04:22.740 --> 00:04:26.490
negative two, negative three,
and I add a positive five,

00:04:26.490 --> 00:04:28.020
so five positive integer chips,

00:04:28.020 --> 00:04:33.020
one, two, three, four and five.

00:04:33.330 --> 00:04:35.520
Well, we already know that
that cancels with that,

00:04:35.520 --> 00:04:37.680
that cancels with that,
that cancels with that.

00:04:37.680 --> 00:04:41.160
This is the equivalent of positive two.

00:04:41.160 --> 00:04:43.230
And what I just did here on both sides,

00:04:43.230 --> 00:04:45.780
this isn't a proof that
this will always work,

00:04:45.780 --> 00:04:47.640
but hopefully this gives you an intuition

00:04:47.640 --> 00:04:49.350
that it does seem to work.

00:04:49.350 --> 00:04:51.900
And I will tell you that without
giving you the full proof

00:04:51.900 --> 00:04:53.610
that it actually does always work,

00:04:53.610 --> 00:04:58.610
that it is actually the case
that if you subtract a number,

00:04:58.650 --> 00:05:02.610
it's the same thing as adding
the opposite of that number.

00:05:02.610 --> 00:05:05.880
If you subtract a number
it's the same thing

00:05:05.880 --> 00:05:09.393
as adding the opposite of that number.