Adding integers with different signs

Find the sum 15 plus
negative 46 plus 29.

Let's just think about
this first part over here

where we have 15
plus negative 46.

And we'll worry about
the plus 29 later on.

So let's just do
15-- let me do that

in a different color--
plus negative 46.

I'll do it in that orange.

Let me draw a number line
here, just so we can properly

visualize what is going on.

That's my number line.

And we're starting at 15.

So let's draw a 0 over here.

And so we're starting at 15.

15 could be right over here.

So this is 15.

Let me draw a big fat arrow
to signify this is 15.

15 has an absolute value of
15, so the length of this arrow

would be 15.

Now we're adding
negative 46 to that 15.

This is equivalent
to 15 minus 46,

which means we are going to
move 46 spots to the left of 15.

A negative sign or
a minus means we're

moving to the left
on the number line.

So we're going to
move 46 to the left.

We're starting at 15, and we are
going to move 46 to the left.

So the length of this arrow
right here is going to be 46,

and we're moving to the left.

This is the negative 46
that we're adding to the 15.

We're going to end up
at some point over here.

That point is clearly 0.

Because we were 15 to the right.

Now we're going to
move 46 to the left.

So we're definitely going
to be to the left of 0.

It's definitely going
to be a negative number.

And we can even think
about the absolute value

of that negative number.

We can just visualize it.

This yellow arrow
has a length of 15.

This orange arrow
has a length of 46.

The blue arrow that
I'm about to draw,

which is the sum
of these two, is

going to have this
length right over here.

And just visually,
how can we figure out

the length of this
blue part if we

know the length of
this orange part

and we know the length
of this yellow part?

Well, it's just going
to be the difference.

It's just going to be the
difference of these two.

So the absolute
value of the sum is

going to be the difference
between this length, 46,

and 15.

Let me just figure
that out, 46 minus 15.

6 minus 5 is 1.

4 minus 1 is 3.

So the length of this
is going to be 31,

and it's going to be
31 to the left of 0.

So this is going to be
negative 31 right over here.

We know that this first part
over here is negative 31,

and then to that we
are going to add 29.

So we're going to add 29.

Let me do that in another color.

What does that mean?

That means that we're going
to start at negative 31,

and we're going to
move 29 to the right.

We're adding 29, so we're
going to move 29 to the right.

So maybe that gets
us right about there.

I'm trying to draw an
arrow of length 29.

I can draw a cleaner
looking arrow than that.

I'll do it right
over here, actually.

So then we're going to
move 29 over to the right.

That's the 29 part.

Now this is a positive
29, and so how

do we figure out what this is?

This is 29 right here
that we're adding.

This is going to land us right
over here on the number line.

So how do we figure out
what number that is?

Well, once again, we
can just visualize it.

And eventually you won't have
to draw number lines and stuff,

but I think it'll
be useful here.

We're starting at negative 31.

We're adding 29 to it, so it's
going to make it less negative.

But we're still
adding less than 31,

so we're not going to get
all the way back to 0.

We're still going to
have a negative number.

But how could we figure
out the absolute value

of that negative
number, its distance?

Well, once again, that little
white part right there,

that white part plus
this 29, is going

to equal 31, if you just
think of absolute value,

if we don't think
about the signs,

if we just think
about the length.

Or another way to think
about it, 31 minus 29

will give us the length
of that white part.

And of course, it's
going to be negative,

because the negative
number here is

larger than the positive number.

And we're adding the 2.

If we do 31 minus 29, you
could borrow and all of that,

but that's clearly just
going to be equal to 2.

You could say that's 11.

This is a 2.

You subtract, this
is equal to 2.

But since it's
negative 31 plus 29,

it's going to be a negative 2.

We're still going
to stay negative.

We haven't moved far to
the right enough to pass 0.

So this right here is
going to be negative 2.

Or another way to
think about it,

the length of this white
bar, the absolute value,

is going to be 2.

2 plus 29 is 31.

But we're operating to the
left of 0, so it's negative 31.

This is negative 2.

Anyway, hopefully you
found that useful.

And let me make it clear.

Our final answer is negative 2.