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In this video I want to familiarize ourselves

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with negative numbers.

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And also learn a bit of how do we add and subtract them.

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And when you first encounter them,

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they look like this deep and mysterious thing.

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When we first count things, we're counting positive numbers.

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What does a negative number even mean?

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But when we think about it, you probably have encountered

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negative numbers in your everyday life.

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And let me just give you a few examples.

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So before I give the example, the general idea is

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a negative number is any number less than zero.

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Less than zero.

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And if that sounds strange and abstract to you,

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let's just think about it in a couple of different contexts.

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If I have... if we're measuring the temperature ...

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(and it can be in Celsius or Farenheit,

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but let's just say we're measuring it in Celsius),

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and so let me draw a little scale

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that we can measure the temperature on.

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So let's say that this is 0° Celsius,

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that is 1° Celsius, 2° Celsius, 3° Celsius.

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Now, let's say it's a pretty chilly day

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and it's currently 3° Celsius.

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And someone who predicts the future

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tells you that it is going to get 4° colder the next day.

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So how cold will it be? How can you represent that coldness?

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Well, if it only got 1° colder it would be at 2°,

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but we know we have to go 4° colder.

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If we got 2° colder, we would be at 1°.

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If we got 3° colder, we would be at 0°.

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But 3° isn't enough, we have to get 4° colder,

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so we actually have to go one more below zero.

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And that 1 below 0 we call that "negative 1".

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And so you can kind of see that the number line,

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as you go to the right of zero increases in positive values,

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as you go to the left of zero you're going to have -1, -2, -3.

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And you're going to have-depending on how you think about it-

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you're going to have larger negative numbers.

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But I want to make it very clear: -3 is LESS than -1.

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There is less heat in the air at -3° than at -1°.

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It is colder---there is less temperature there.

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So let me just make it very clear: -100 is much smaller than -1.

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You might look at 100 and you might look at 1 and

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your gut reaction might be that 100 is much larger than 1.

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But when you think about it,

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-100 means there is a lack of something.

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-100: if it's -100° there is a lack of heat,

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so there is much less heat here than if we had -1°.

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Let me give you another example.

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Let's say in my bank account today I have $10.

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Now, let's say I go out there

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(because I feel good about my $10),

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and let's say I go and spend $30.

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And, for the sake of argument

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let's say I have a very flexible bank,

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one that lets me spend more money than I have

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(and these actually exist!).

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So I spend $30.

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So what's my bank account going to look like?

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So let me draw a number line here.

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And you may already have an intuitive response:

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I will owe the bank some money.

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Tomorrow, what is my bank account?

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You might immediately say, "if I have $10 and I spend $30,

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there's $20 that had to come from some place."

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And that $20 is coming from the bank.

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So I'm going to owe the bank $20.

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And so, in my bank account,

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to show how much I have, I could say $10 - $30 is -$20.

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So in my bank account tomorrow, I'm going to have -$20.

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So, if I say I have -$20, that means that I owe the bank.

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---I don't even have it.

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Not only do I have nothing, I owe something.

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It's going in reverse.

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Here, I have something to spend...

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if my $10 in my bank account means the bank owes me $10.

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I have $10 that I can use to go spend.

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Now, all of a sudden I owe the bank.

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I've gone the other direction.

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If we use a number line here

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it should hopefully make more sense.

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So that is 0.

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I'm starting off with $10,

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and spending $30 means I'm moving 30 spaces to the left.

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So if I move 10 spaces to the left---

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if I only spend $10 I'll be back at $0.

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If I spend another $10, I'll be at -$10.

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If I spend another $10 after that, I will be at -$20.

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So, each of these distances, I spent $10 I'd be at $0.

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Another $10 I'd be at -$10.

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Another $10 I would be at -$20.

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So this whole distance here is how much I spent.

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I spent $30.

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So the general idea when you spend or if you subtract,

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or getting colder, you would move to the left.

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The numbers would get smaller.

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And we now know they can get even smaller than 0.

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They can go to -1, -2---they can even go to -1.5, -1.6.

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The more and more negative, the more you lose.

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If you're adding, if I go and get my paycheck,

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I will move to the right of the number line.

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Now, with that out of the way,

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let's just do a couple more pure math problems.

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What it means, if we were to say...

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Let's say, 3 - 4.

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So once again,

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this is exactly the situation we did with the temperature.

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We're starting with 3 and we're subtracting 4,

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so we're going to move 4 to the left.

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We go 1, 2, 3, 4.

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That gets us to -1.

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And when you're starting to do this

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you really understand what a negative number means.

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I really encourage you to visualize the number line

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and really move along it depending on

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whether you're adding or subtracting.

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Let's do a couple more.

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Let's say I have 2 - 8

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(and we'll think about more ways to do this in future videos),

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but once again, you just want to do the number line.

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You have a 0 here.

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We're at (let me draw the spacing a little bit).

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We have 0 here...we're at 1... 2.

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If we're subtracting 8,

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that means we're going to move 8 to the left.

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So we're going to go 1 to the left, 2 to the left.

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So, we've gone 2 to the left to get to 0.

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We have to move how many more to the left?

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We've already moved 2 to the left,

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to get to 8, we have to move 6 more to the left.

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So we're going to have to move 1-2-3-4-5-6 more to the left.

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Well, where is that going to put us?

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Well, we were at 0.

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This is -1, -2, -3, -4, -5, -6.

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So, 2 - 8 is -6.

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2-2 would be 0.

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When you're subtracting 8 you're subtracting another 6.

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So we go to -6, we go 6 below 0.

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Let me do one more example.

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(and this will be a little less conventional

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but hopefully it will make sense).

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Let us take... (and I'll do this in a new color)...

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Let me take -4 - 2.

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So we're starting at a negative number

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and then we're subtracting from that.

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Now, if this seems confusing just remember the number line!

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So this is 0 right here.

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This is -1, -2, -3, -4. So that's where we're starting.

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Now we're going to subtract 2 from -4,

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so we're going to move 2 to the left.

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So if we subtract 1 we'll be at -5.

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If we subtract another 1 we are going to be at -6.

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So this is -6.

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Let's do another interesting thing.

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Let's start at -3 ... let's say we have -3.

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Instead of subtracting something from that, let's add 2.

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So where would this put us on the number line?

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So we're starting at -3 and we're adding 2.

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So we're going to move to the right.

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So you add 1, you become -2

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But if you add another 1 (which we have to do),

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you become -1.

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You move 2 to the right.

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So, -3 + 2 is -1.

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And you can see for yourself,

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this all fits our traditional notion of adding and subtracting.

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If we start at -1 and we subtract 2, we should get -3.

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Kind of reverses this thing up here.

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-3 +2 gets us there.

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And if we start there and we subtract 2

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we should get back to -3.

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And we see that happens.

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If you start at -1, right over here,

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and you subtract 2, you move 2 to the left.

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You get back to -3.

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So hopefully this starts to give you a sense of what it means

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to deal with or add and subtract negative numbers.

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But we are going to give a lot more examples in the next video.

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And we're actually going to see what it means

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to subtract a negative number.