How to Add and Subtract Positive and Negative Numbers

Numbers Can be Positive or Negative
This is the Number Line:

No Sign Means Positive
If a number has no sign it usually means that it is a positivenumber.

Example: 5 is really +5

Balloons and Weights
Let us think about numbers as balloons (positive) and weights (negative):

Adding a Positive Number
Adding positive numbers is just simple addition.

Example: 2 + 3 = 5
is really saying

"Positive 2 plus Positive 3 equals Positive 5"

We could write it as (+2) + (+3) = (+5)

Subtracting A Positive Number
Subtracting positive numbers is just simple subtraction.

Example: 6 − 3 = 3
is really saying

"Positive 6 minus Positive 3 equals Positive 3"

We could write it as (+6) − (+3) = (+3)

Adding A Negative Number
Now let's see what adding and subtracting negative numbers looks like:

Example: 6 + (−3) = 3
is really saying

"Positive 6 plus Negative 3 equals Positive 3"

We could write it as (+6) + (−3) = (+3)

The last two examples showed us that taking away balloons (subtracting a positive) or adding weights (adding a negative) both make the basket go down.

So these have the same result: In other words subtracting a positive is the same as adding a negative.
 * (+6) − (+3) = (+3)
 * (+6) + (−3) = (+3)

Example: What is 6 − (−3) ?
6−(−3) = 6 + 3 = 9

Yes indeed! Subtracting a Negative is the same as adding!

Two Negatives make a Positive

Adding a positive number is simple addition ...
Adding a Positive is Addition

Positive and Negative Together ...
Subtracting a Positive or Adding a Negative is Subtraction

Example: What is 6 − (+3) ?
6−(+3) = 6 − 3 = 3

Example: What is 5 + (−7) ?
5+(−7) = 5 − 7 = −2

Subtracting a negative ...
Subtracting a Negative is the same as Adding

Example: What is 14 − (−4) ?
14−(−4) = 14 + 4 = 18

The Rules:
It can all be put into two rules: They are "like signs" when they are like each other (in other words: the same).

So, all you have to remember is:

Two like signs become a positive sign

Two unlike signs become a negative sign

Example: What is 5+(−2) ?
+(−) are unlike signs (they are not the same), so they become a negative sign.

5+(−2) = 5 − 2 = 3

Example: What is 25−(−4) ?
−(−) are like signs, so they become a positive sign.

25−(−4) = 25+4 = 29

Example: What is −6+(+3) ?
+(+) are like signs, so they become a positive sign.

−6+(+3) = −6 + 3 = -3

Start at −6 on the number line, move forward 3, and you end up at −3

A Common Sense Explanation
And there is a "common sense" explanation:

If I say "Eat!" I am encouraging you to eat (positive)

If I say "Do not eat!" I am saying the opposite (negative).

Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).

So, two negatives make a positive, and if that satisfies you, then you are done!

Another Common Sense Explanation
A friend is +, an enemy is −

Example: Last year the bank subtracted $10 from your account by mistake, and they want to fix it.
So the bank must take away a negative $10.

Let's say your current balance is $80, so you will then have:

$80−(−$10) = $80 + $10 = $90

So you get $10 more in your account.

Ally's Points
Ally can be naughty or nice. So Ally's parents have said

"If you are nice we will add 3 points (+3).  If you are naughty, we take away 3 points (−3).  When you reach 30 Points you get a toy." So when we subtract a negative, we gain points (i.e. the same as adding points). So Subtracting a Negative is the same as Adding See: both "15 − (+3)" and "15 + (−3)" result in 12.

So:

It doesn't matter if you subtract positive points or add negative points, you still end up losing points.

So Subtracting a Positive or Adding a Negative is Subtraction