How to Add and Subtract Positive and Negative Numbers

## No Sign Means Positive

If a number has **no sign** it usually means that it is a **positive**number.

Example: **5** is really **+5**

## Balloons and Weights

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

* The balloons pull up (**positive**)
* And the weights drag down (**negative**)

Adding positive numbers is just simple addition.

the basket gets pulled upwards (positive)

"Positive 2 plus Positive 3 equals Positive 5"

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

Subtracting positive numbers is just simple subtraction.

the basket gets pulled downwards (negative)

"Positive 6 minus Positive 3 equals Positive 3"

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

Now let's see what adding and subtracting **negative** numbers looks like:

the basket gets pulled downwards (negative)

"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**:

* **(+6) − (+3) = (+3)**
* **(+6) + (−3) = (+3)**

In other words **subtracting a positive** is the same as **adding a negative**.

## Subtracting A Negative Number

the basket gets pulled upwards (positive)

### 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

## What Did We Find?

### Positive and Negative Together ...

Subtracting a Positive
*or*
*Adding a Negative*
*is*
**Subtraction**

### Subtracting a negative ...

Subtracting a Negative is the same as **Adding**

## 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 −

 + − ⇒ −
 − + ⇒ −
 − − ⇒ +

## A Bank Example

### 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.

## A Long Example You Might Like

### 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."

How do we "undo" a minus 3?
We **add** 3 back again!

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**