## Numbers Can be Positive or Negative

This is the Number Line:

Negative Numbers (−) | Positive Numbers (+) |

"−" is the negative sign. | "+" is the positive sign |

## No Sign Means Positive

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

Example: **5** is really **+5**

## Play with it!

On the Number Line positive goes to the right and negative to the left.

Try the sliders below and see what happens:

numbers/images/number-line-add.js?sub=n

## Balloons and Weights

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

This basket has balloons and weights tied to it:

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

## Adding a Positive Number

Adding positive numbers is just simple addition.

We can add balloons (we are **adding positive** value)

the basket gets pulled upwards (positive)

### 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.

We can take away balloons (we are **subtracting positive** value)

the basket gets pulled downwards (negative)

### 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:

We can add weights (we are **adding negative** values)

the basket gets pulled downwards (negative)

### 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**:

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

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

## Subtracting A Negative Number

Lastly, we can take away weights (we are **subtracting negative** values)

the basket gets pulled upwards (positive)

### Example: What is 6 − (−3) ?

6−(−3) = 6 **+** 3 = 9

Yes indeed! Subtracting a Negativeis the same as adding!

Two Negatives make a Positive

## What Did We Find?

### 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**:

Rule | Example | ||||
---|---|---|---|---|---|

+(+) | Two like signs become a positive sign | 3+(+2) = 3 + 2 = 5 | |||

−(−) | 6−(−3) = 6 + 3 = 9 | ||||

+(−) | Two unlike signs become a negative sign | 7+(−2) = 7 − 2 = 5 | |||

−(+) | 8−(+2) = 8 − 2 = 6 | ||||

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

## Starting Negative

What if we **start** with a negative number?

Using The Number Line can help:

### Example: What is −3+(+2) ?

+(+) are **like** signs, so they become a **positive sign**.

−3+(+2) = −3 **+** 2

Start at −3 on the number line,

move forward 2 and you end up at −1

−3+(+2) = −3 **+** 2 = −1

### Example: What is −3+(−2) ?

+(−) are **unlike** signs, so they become a **negative sign**.

−3+(−2) = −3 **−** 2

Start at −3 on the number line,

move back 2 and you end up at −5

−3+(−2) = −3 **−** 2 = −5

## Now Play With It!

Try playing Casey Runner, you need to know the rules of positive and negative to succeed! |

## 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 friend of a friend is my friend | |

+ − ⇒ − | a friend of an enemy is my enemy | |

− + ⇒ − | an enemy of a friend is my enemy | |

− − ⇒ + | an enemy of an enemy is my friend |

## 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."*

Ally starts the day with 9 Points: | 9 | |

Ally's Mom discovers spilt milk: | 9 − 3 = 6 | |

Then Dad confesses he spilt the milk and writes "undo". How do we "undo" a minus 3? | ||

So Mom calculates: | 6 − (−3) = 6 + 3 = 9 |

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**

A few days later. Ally has 12 points. | ||

Mom adds 3 points because Ally's room is clean. | 12 + 3 = 15 | |

Dad says "I cleaned that room" and writes "undo" on the chart. Mom calculates: | 15 − (+3) = 12 | |

Dad sees Ally brushing the dog. Writes "+3" on the chart. Mom calculates: | 12 + (+3) = 15 | |

Ally throws a stone against the window. Dad writes "−3" on the chart. Mom calculates: | 15 + (−3) = 12 |

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**

## Try These Exercises ...

Now try This Worksheet, and see how you go.

And also try these questions:

11715, 11716, 11717, 11718, 11719, 11720, 11721, 3445, 3446

Number Line Using The Number Line Whole Numbers and Integers Multiplying Negatives Number Index Algebra Index