Video transcript
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 actually
give the example, the general idea is
a negative number is any number less than the 0. And if that sounds strange
and abstract to you, let's just think about it in a
couple of different contexts. If we're measuring the
temperature-- and it could be in Celsius
or Fahrenheit. Let's just assume we're
measuring it in Celsius. And so, let me
draw a little scale that we can measure
the temperature on. And so, let's say this
is 0 degrees Celsius. That is 1 degree Celsius,
2 degrees Celsius, 3 degrees Celsius. Now let's say that it's
a pretty chilly day, and it's currently
3 degrees Celsius. And someone who
predicts the future tells you that it
is going to get 4 degrees colder the next day. So how cold will it be? How can you represent
that coldness? Well, if it only
got 1 degree colder, we would be at 2 degrees. But we know we have to
go 4 degrees colder. If we got 2 degrees colder,
we would be at 1 degree. If we got 3 degrees
colder, we would be at 0. But 3 isn't enough. We have to get 4 degrees colder. So we have to actually
go one more below the 0. And that one 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 0 on the number line, it increases in positive values. As you go to the left
of the number line, you're going to have negative 1. Then you're going
to have negative 2. And you're going
to have negative 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-- negative 3 is less than negative 1. There is less heat in
the air then negative 1. It is colder. There's less temperature there. So let me just
make it very clear. Negative 100 is much
smaller than negative 1. You might look at 100,
and you might look at 1, and your gut reaction
is to say, wow, 100 is a much larger
number than 1. But when you think
about it, negative 100 means there's a
lack of something. If it's negative 100 degrees,
there's a lack of heat. So there's much less heat here
then if we had negative 1. Let me give you another example. Let's say in my bank
account today, I have $10. Let me do this in a new color. So let's say 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 might already have
an intuitive response. I will owe the bank some money. So let me write this over here. So tomorrow, what
is my bank account? So you might immediately
say, look, if I have $10, and I spend $30,
there's $20 that had to come from
someplace, and that $20 is coming from the bank. So I'm going to
owe the bank $20. And so, in my bank account,
to sure how much I have, I could say $10 minus $30
is actually negative $20. So in my bank account tomorrow,
I'm going to have negative $20. So if I say I have negative $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 is in my bank, that
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 in the
other direction. And if we use a
number line here, it should hopefully make
a little bit of 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 negative $10. If I spend another
$10 after that, I will be at negative $20. So each of these distances--
I spent $10, I'd be at 0. Another $10, I'd
be at negative $10. Another $10, I would
be at negative $20. And this whole distance right
here is how much I spent. I spent $30. So the general idea, when
you spend or if you subtract or if we're getting
colder, you would move to the left
of the number line. The numbers would get smaller. And we now know they
can get smaller than 0. They can go to
negative 1, negative 2, they can even go to
negative 1.5, negative 1.6. You're getting 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 of more
pure math problems. And just think
about what it means if we were to say 3 minus 4. So once again, this is
exactly the situation that we did here
with the temperature. We're starting at 3,
and we're subtracting 4. So we're going to
move 4 to the left. We go 1, 2, 3, 4. So that gets us to negative 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 and subtracting. Now let's do a couple more. Let's say I have 2 minus 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. Let me draw the
spacing a little bit. So 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 the left. So we've gone 2 to
the left to get to 0. We have to move how
many more to the left? Well, we've already
moved 2 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 a negative 1, negative
2, negative 3, negative 4, negative 5, negative 6. So 2 minus 8 is negative 6. 2 minus 2 would be 0. When you're subtracting 8,
you're subtracting another 6. So we'd go up to negative 6. We go 6 below 0. Let me do one more example. And this one will be a little
less conventional for you, but hopefully it
will make sense. And I'll do this in a new color. Let me take negative 4 minus 2. So we're starting at
a negative number, and then we're
subtracting from that. And if this seems confusing,
just remember the number line. So this is 0 right here. This is negative 1, negative
2, negative 3, negative 4. So that's where we're starting. Now we're going to
subtract 2 from negative 4. So we're going to
move 2 to the left. So if we subtract 1, we
will be at negative 5. If we subtract another 1, we
are going to be at negative 6. We are going to
be at negative 6. So this is negative 6. Let's do another
interesting thing. Let's start at negative 3. Let's say we have negative 3. Instead of subtracting something
from that, let's add 2 to it. So where would this put
us on the number line? So we're starting at negative
3, and we're adding 2, so we're going to
move to the right. So you add 1, you
become negative 2. But if you add another
1, which we have to do, you become negative 1. You move 2 to the right. So negative 3 plus
negative 2 is negative 1. And you can see for
yourself but this all fits our traditional notion
of adding and subtracting. If we start at negative
1, and we subtract 2, we should get negative 3--
kind of reverse of this thing up here. Negative 3 plus 2 gets us there. Then if we start
then, we subtract 2, we should get back to negative
3, and we see that happens. If you start at negative
1 right over here, and you subtract 2--
you move 2 to the left-- you get back to negative 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're 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.