Adding And Subtracting Positive And Negative Numbers - Math Is Fun

L2I_-PCESlY How to Add and Subtract Positive and Negative Numbers

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 right and negative goes 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

Last, 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 Negative is 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:

+(+)	Two like signs become a positive sign:
−(−)
+(−)	Two unlike signs become a negative sign
−(+)

They are "like signs" when they are like each other (in other words: the same).

And we can use them like this:

Like Signs
+(+)	3+(+2) = 3 + 2 = 5
−(−)	6−(−3) = 6 + 3 = 9
Unlike Signs
+(−)	7+(−2) = 7 − 2 = 5
−(+)	8−(+2) = 8 − 2 = 6

So just remember:

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 Final 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? We add 3 back again!

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: 12 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.

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