Adding and Subtracting Positive and Negative Numbers (2024)

Numbers Can be Positive or Negative

This is the Number Line:

Adding and Subtracting Positive and Negative Numbers (1)
Negative Numbers (−)Positive Numbers (+)

Adding and Subtracting Positive and Negative Numbers (2)Adding and Subtracting Positive and Negative Numbers (3)
"−" 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):

Adding and Subtracting Positive and Negative Numbers (4)

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.

Adding and Subtracting Positive and Negative Numbers (5)

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.

Adding and Subtracting Positive and Negative Numbers (6)

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:

Adding and Subtracting Positive and Negative Numbers (7)

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

Adding and Subtracting Positive and Negative Numbers (8)

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 and Subtracting Positive and Negative Numbers (9)

Adding a Positive is Addition

Positive and Negative Together ...

Adding and Subtracting Positive and Negative Numbers (10)

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 ...

Adding and Subtracting Positive and Negative Numbers (11)

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:

RuleExample
+(+)Two like signs become a positive signAdding and Subtracting Positive and Negative Numbers (12)3+(+2) = 3 + 2 = 5
−(−)6−(−3) = 6 + 3 = 9
+(−) Two unlike signs become a negative signAdding and Subtracting Positive and Negative Numbers (13)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

Adding and Subtracting Positive and Negative Numbers (14)
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

Adding and Subtracting Positive and Negative Numbers (15)
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!

Adding and Subtracting Positive and Negative Numbers (16)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:

Adding and Subtracting Positive and Negative Numbers (17)

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

Adding and Subtracting Positive and Negative Numbers (18)

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?
We add 3 back again!

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

Adding and Subtracting Positive and Negative Numbers (19)

So when we subtract a negative, we gain points
(i.e. the same as adding points).


Adding and Subtracting Positive and Negative Numbers (20)

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

Adding and Subtracting Positive and Negative Numbers (21)


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.

Adding and Subtracting Positive and Negative Numbers (22)

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

Adding and Subtracting Positive and Negative Numbers (2024)
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