Adding Negative Numbers

Find the sum of negative 15 plus
negative 46 plus negative 29.

To do this, let's
just first visualize

what each of these
numbers look like.

So I'm going to draw a
number line for each of them.

Negative 15 might look
something like this.

So if this is 0 and let's
say that this is negative 15,

I could represent
negative 15 as--

if I'd like to-- an arrow that
points from 0 to negative 15.

The length of the arrow is
the absolute value of this.

It's the distance from 0.

So the length here is 15.

And the negative says that
we are pointing to the left.

So the absolute value is 15.

That's the length of that arrow.

Let's do the same
thing for negative 46.

So once again, let me
draw my number line.

0 is going to be
right over there.

And negative 46 is going
to be someplace over here.

Negative 46.

Notice, same exact idea.

The distance between negative
46 and 0-- or another way

to think about that is the
absolute value of negative 46--

this distance right over
here is going to be 46.

And its direction
is to the left.

That is why we get to
the number negative 46.

The negative really tells you
whether you're to the left

or to the right of 0,
but the absolute value

says how far are you to
the left or the right of 0.

And then, finally, let's do
the same thing for negative 29.

So once again, let me
draw my number line.

I want to use the yellow again.

So draw my number line.

Let's say this is 0.

Then negative 29 is going
to be roughly over here.

I'm estimating it.

And once again, negative 29
is exactly 29 away from 0.

So this length right over here
is 29, and it is to the left.

That is why it is negative 29.

If it was positive 29, it
would be 29 to the right of 0.

So we've represented
all of these numbers.

And you can see what their
absolute values are like.

And now let's think about
what happens if you add them.

So one way to think about
adding these numbers

is if you added
these arrows, if you

put this arrow on
top of this arrow

or to the left of this
arrow, if you started where

this arrow leaves off-- and
then you put this green arrow,

and then you put
the orange arrow.

So let's do that.

Let's draw that.

It's going to be a
longer arrow now.

So we're going to start at 0.

First, we have the negative 15.

So we're going to move
15 spaces to the left

to get us to negative 15.

Then we're going to go
46 spaces to the left

to get us to negative
15 plus negative 46.

So let me draw that.

So then we're going
to go-- and we're

going to figure out what
that number is in a second.

So then we're going
to go 46 to the left.

That's about that far.

Really just this arrow,
I'm now placing it.

I'm starting off where
the negative 15 left off,

and then I'm putting
that arrow after that.

And we don't know yet what
number this gets us to.

We're going to have to
do a little math here.

And that's actually the
point of this problem.

But we know what the
length of this arrow is.

The length of this arrow is 46.

It is 46 to the
left, and we know

the length of this
magenta arrow is 15.

And then, finally, we
have this orange arrow.

The orange arrow, which we
know has a length of 29.

Although, it's 29 to the left.

It looks like that.

It has a length of 29.

So when all is said
and done, where are we?

Where do we end up
on our number line?

Well, the total length
that we are to the left

is going to be 15
plus 46 plus 29.

But it's going to
be to the left,

so it's going to be negative.

So what we can really view
this as, since all of these

have the same sign,
is we could say,

look, this is the same
thing as the absolute value

of negative 15, which is
15, plus the absolute value

of negative 46, which is
46, plus the absolute value

of negative 29,
which is 29, but then

take the negative value of it.

So let's just add that.

15 plus 46 plus 29 is equal
to, let's see-- 5 plus 6 is 11.

11 plus 9 is 20.

Carry the 2.

2 plus 1 is 3.

3 plus 4 is 7.

7 plus 2 is 9.

So you get 90.

So this entire
length here is 90.

The entire length, if you add
up the arrows, you get a 90.

But it is not 90 to the right.

So if it was 90 to the right,
these would all be positives.

And then we would just
get a positive 90.

But this is 90 to the left.

So when you add these guys up,
you don't get to positive 90.

You get to negative 90.

So one way to think about it
is these are all the same sign.

So this is going to be
equal to the negative

of the absolute
value of negative 15

plus the absolute
value of negative 46

plus the absolute
value of negative 29.

And the reason why we're
writing this here-- this

might look fancy,
but this is really

just the length of
this purple arrow.

This absolute value
of negative 46,

that's really just the
length of the green arrow.

46.

And this is just the
length of the orange arrow.

So this is 15 plus 46
plus 29, gave us 90,

but it is to the left.

So that is why it
is negative 90.