Question

How do you evaluate `(-41)+(-40)`?

Answer 1

`(-41)+(-40)=-81`.

Explanation:

When dealing with adding and subtracting negative numbers, a good way to think of it is to picture yourself standing on a number line. To perform the operation of adding `(+)`, you face right, towards the positive/increasing numbers. Subtraction `(-)` means facing left, towards the negative/decreasing way. Now that you're oriented the correct way, the sign of the number (positive/negative) will tell you whether you walk forwards or backwards.

In this example, we've been asked to start at `-41` and add `-40`. So we place ourselves on the number line at `-41`. Since we are going to add something, we face right. The number we're adding, though, is negative, so we walk backwards by 41 steps, further into the negatives. We're going farther away from `0`, so the distances of "41 to the left of 0" and "40 more to the left" combine to put us at a total of "81 to the left of 0". Thus, our answer is `-81`.

(Incidentally, you can use this idea to show yourself that adding a negative value is the same as subtracting that positive value.)

Bonus:

If you've heard of (or seen) vectors before, they are another useful way to think about this. Vectors are arrows that have a length and a direction.

---> This vector has a length of 3, and is facing right. It represents +3.
<-- This vector has length 2, but is facing left. It represents -2.

Adding two vectors means taking the 2nd one's tail, and placing it at the 1st one's tip, without changing its direction. (Remember, vectors have a direction, so if we change the direction, we change the vector, and we don't want to do that.) The sum is the vector you get when you start from the 1st tail and go to the 2nd tip.

Suppose we want to add `5+3` using vectors. We would place the tail of the 3-vector at the tip of the 5-vector, like this:

`color(blue)(5: ----->)`
`color(violet)(3: --->)`

`5+3=`
`color(blue)(----->)`
`"                          "color(violet)(--->)`
`-------->`
`=8`

Simple, right? Now, suppose we want to add `5+("-"3)`. That would mean putting the tail of the -3 vector at the tip of the 5 vector, and walking from the first tail to the second tip:

`color(blue)(5: ----->)`
`color(violet)("-"3: <--)`

`5+("-"3)=`
`color(blue)(----->)`
`"         "color(violet)(<--)`
`-->`
`=2`

I won't be able to draw vectors with length 40 or 41 here, but hopefully you get the idea. If you like the vector notation, try to use it to explain subtraction.

Answer 2

-81

Explanation:

Add together 2 negative numbers and the answer is negative

Add together 2 positive numbers and the answer is positive

We have the first condition. If you like think of add as 'put together'

So we are putting together `-41" and "-40`

Thus `(-41)+(-40)" is the same as "-41-40 = -81`