In this video I want to familiarize ourselves
with negative numbers.
And also learn a bit of how do we add and subtract them.
And when you first encounter them,
they look like this deep and mysterious thing.
When we first count things, we're counting positive numbers.
What does a negative number even mean?
But when we think about it, you probably have encountered
negative numbers in your everyday life.
And let me just give you a few examples.
So before I give the example, the general idea is
a negative number is any number less than zero.
Less than zero.
And if that sounds strange and abstract to you,
let's just think about it in a couple of different contexts.
If I have... if we're measuring the temperature ...
(and it can be in Celsius or Farenheit,
but let's just say we're measuring it in Celsius),
and so let me draw a little scale
that we can measure the temperature on.
So let's say that this is 0° Celsius,
that is 1° Celsius, 2° Celsius, 3° Celsius.
Now, let's say it's a pretty chilly day
and it's currently 3° Celsius.
And someone who predicts the future
tells you that it is going to get 4° colder the next day.
So how cold will it be? How can you represent that coldness?
Well, if it only got 1° colder it would be at 2°,
but we know we have to go 4° colder.
If we got 2° colder, we would be at 1°.
If we got 3° colder, we would be at 0°.
But 3° isn't enough, we have to get 4° colder,
so we actually have to go one more below zero.
And that 1 below 0 we call that "negative 1".
And so you can kind of see that the number line,
as you go to the right of zero increases in positive values,
as you go to the left of zero you're going to have -1, -2, -3.
And you're going to have-depending on how you think about it-
you're going to have larger negative numbers.
But I want to make it very clear: -3 is LESS than -1.
There is less heat in the air at -3° than at -1°.
It is colder---there is less temperature there.
So let me just make it very clear: -100 is much smaller than -1.
You might look at 100 and you might look at 1 and
your gut reaction might be that 100 is much larger than 1.
But when you think about it,
-100 means there is a lack of something.
-100: if it's -100° there is a lack of heat,
so there is much less heat here than if we had -1°.
Let me give you another example.
Let's say in my bank account today I have $10.
Now, let's say I go out there
(because I feel good about my $10),
and let's say I go and spend $30.
And, for the sake of argument
let's say I have a very flexible bank,
one that lets me spend more money than I have
(and these actually exist!).
So I spend $30.
So what's my bank account going to look like?
So let me draw a number line here.
And you may already have an intuitive response:
I will owe the bank some money.
Tomorrow, what is my bank account?
You might immediately say, "if I have $10 and I spend $30,
there's $20 that had to come from some place."
And that $20 is coming from the bank.
So I'm going to owe the bank $20.
And so, in my bank account,
to show how much I have, I could say $10 - $30 is -$20.
So in my bank account tomorrow, I'm going to have -$20.
So, if I say I have -$20, that means that I owe the bank.
---I don't even have it.
Not only do I have nothing, I owe something.
It's going in reverse.
Here, I have something to spend...
if my $10 in my bank account means the bank owes me $10.
I have $10 that I can use to go spend.
Now, all of a sudden I owe the bank.
I've gone the other direction.
If we use a number line here
it should hopefully make more sense.
So that is 0.
I'm starting off with $10,
and spending $30 means I'm moving 30 spaces to the left.
So if I move 10 spaces to the left---
if I only spend $10 I'll be back at $0.
If I spend another $10, I'll be at -$10.
If I spend another $10 after that, I will be at -$20.
So, each of these distances, I spent $10 I'd be at $0.
Another $10 I'd be at -$10.
Another $10 I would be at -$20.
So this whole distance here is how much I spent.
I spent $30.
So the general idea when you spend or if you subtract,
or getting colder, you would move to the left.
The numbers would get smaller.
And we now know they can get even smaller than 0.
They can go to -1, -2---they can even go to -1.5, -1.6.
The more and more negative, the more you lose.
If you're adding, if I go and get my paycheck,
I will move to the right of the number line.
Now, with that out of the way,
let's just do a couple more pure math problems.
What it means, if we were to say...
Let's say, 3 - 4.
So once again,
this is exactly the situation we did with the temperature.
We're starting with 3 and we're subtracting 4,
so we're going to move 4 to the left.
We go 1, 2, 3, 4.
That gets us to -1.
And when you're starting to do this
you really understand what a negative number means.
I really encourage you to visualize the number line
and really move along it depending on
whether you're adding or subtracting.
Let's do a couple more.
Let's say I have 2 - 8
(and we'll think about more ways to do this in future videos),
but once again, you just want to do the number line.
You have a 0 here.
We're at (let me draw the spacing a little bit).
We have 0 here...we're at 1... 2.
If we're subtracting 8,
that means we're going to move 8 to the left.
So we're going to go 1 to the left, 2 to the left.
So, we've gone 2 to the left to get to 0.
We have to move how many more to the left?
We've already moved 2 to the left,
to get to 8, we have to move 6 more to the left.
So we're going to have to move 1-2-3-4-5-6 more to the left.
Well, where is that going to put us?
Well, we were at 0.
This is -1, -2, -3, -4, -5, -6.
So, 2 - 8 is -6.
2-2 would be 0.
When you're subtracting 8 you're subtracting another 6.
So we go to -6, we go 6 below 0.
Let me do one more example.
(and this will be a little less conventional
but hopefully it will make sense).
Let us take... (and I'll do this in a new color)...
Let me take -4 - 2.
So we're starting at a negative number
and then we're subtracting from that.
Now, if this seems confusing just remember the number line!
So this is 0 right here.
This is -1, -2, -3, -4. So that's where we're starting.
Now we're going to subtract 2 from -4,
so we're going to move 2 to the left.
So if we subtract 1 we'll be at -5.
If we subtract another 1 we are going to be at -6.
So this is -6.
Let's do another interesting thing.
Let's start at -3 ... let's say we have -3.
Instead of subtracting something from that, let's add 2.
So where would this put us on the number line?
So we're starting at -3 and we're adding 2.
So we're going to move to the right.
So you add 1, you become -2
But if you add another 1 (which we have to do),
you become -1.
You move 2 to the right.
So, -3 + 2 is -1.
And you can see for yourself,
this all fits our traditional notion of adding and subtracting.
If we start at -1 and we subtract 2, we should get -3.
Kind of reverses this thing up here.
-3 +2 gets us there.
And if we start there and we subtract 2
we should get back to -3.
And we see that happens.
If you start at -1, right over here,
and you subtract 2, you move 2 to the left.
You get back to -3.
So hopefully this starts to give you a sense of what it means
to deal with or add and subtract negative numbers.
But we are going to give a lot more examples in the next video.
And we're actually going to see what it means
to subtract a negative number.