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)

FAQs

What is the rule for adding and subtracting positive and negative numbers? ›

Addition of Integers
Type of NumbersOperationResult
Positive + PositiveAddPositive (+)
Negative + NegativeAddNegative (-)
Positive + Negative*SubtractPositive (+)
Negative + Positive*SubtractNegative (-)

When you add a positive and a negative What is the answer? ›

To get the sum of a negative and a positive number, use the sign of the larger number and subtract. For example: (–7) + 4 = –3.

What is the answer when you subtract a positive from a negative? ›

Thus, positive minus negative equals positive.

What happens if you add a positive and a negative number? ›

Adding Positive and Negative Numbers

If the positive number is bigger than the absolute value of the negative number, then the solution will be a positive number. If the positive number is smaller than the absolute value of the negative number, then the solution will be a negative number.

What are the four rules of integers? ›

RULE 1: The product of a positive integer and a negative integer is negative. RULE 2: The product of two positive integers is positive. RULE 3: The product of two negative integers is positive. RULE 1: The quotient of a positive integer and a negative integer is negative.

Do two negatives make a positive when adding and subtracting? ›

If you subtract a negative number, the two negatives combine to make a positive. −10−(−10) is not −20. Instead, you can think of it as turning one of the negative signs upright, to cross over the other, and make a plus. The sum would then be −10+10 = 0.

Why is subtracting a negative number the same as adding a positive? ›

Subtracting a number is the same as adding its opposite. So subtracting a positive number is like adding a negative number - you move to the left on the number line. Subtracting a negative number is like adding a positive number - you move to the right on the number line.

What do a positive and a negative make? ›

If one is positive and the other negative, the answer will be negative. When more integers are involved work the calculations from left to right.

What is the sum of a positive and negative number? ›

(iii) The sum of a negative integer and a positive integer is always a negative integer.

How do you add and subtract positive and negative decimals? ›

Adding and Subtracting Two Signed Decimals

When adding a positive and a negative decimal, subtract the number with a smaller absolute value from the number with the greatest absolute value, and the sign of the final answer is the same as the number whose absolute value is the greatest.

What is the bodmas rule? ›

The Bodmas rule follows the order of the BODMAS acronym ie B – Brackets, O – Order of powers or roots, D – Division, M – Multiplication A – Addition, and S – Subtraction. Mathematical expressions with multiple operators need to be solved from left to right in the order of BODMAS.

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