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

Adding and Subtracting Positive and Negative Numbers? ›

Adding two positive integers results in positive integers, whereas adding two negative integers will result in the sum with a negative sign. But, the addition of two different signed integers will result in subtraction only and the sign of the result will be the same as the larger number has.

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

When two signs are written next to each other, the rules for adding and subtracting numbers are: two signs that are different become a negative sign. two signs that are the same become a positive sign.

What is the formula for adding negative and positive numbers? ›

Alternatively, to find the sum of a positive and a negative integer, take the absolute value (“absolute value” means to remove any negative sign of a number, and make the number positive) of each integer and then subtract these values. Take the above example, 10 + (-15); absolute value of 10 is 10 and -15 is 15.

What are the 2 rules of adding and subtracting integers? ›

Adding & Subtracting Integers
  • same signs = add the numbers, and keep the same sign (Ex: 3 + 2 = 5 or -3 + -2 = -5)
  • different signs = subtract the numbers, and keep the sign of the bigger number (Ex: -3 + 2 = -1 or 3 + -2 = 1)
Sep 29, 2020

What are the four operations with positive and negative numbers? ›

The four operations are addition (+), subtraction (–), multiplication (×) and division (÷). When adding and subtracting negative numbers, it may help to have a number line. Prior Knowledge: Be able to perform the four operations with positive numbers.

How do you get better at adding and subtracting integers? ›

The most successful teaching strategy I have found with subtraction is to have them change all subtraction problems into addition problems by adding the opposite. For example, 5 - (-3) changes to 5 + 3, because you add the opposite of the second number. Another example, -4 - 6 changes to -4 + (-6).

What is the rule for adding and subtracting? ›

The rules to add and subtract numbers are given below: Addition of two positive numbers is always positive. Addition of two negative numbers is always negative. Subtraction of two positive numbers can be either positive or negative. Subtraction of two negative numbers can be either positive or negative.

Why is subtracting a negative the same as adding? ›

Another way to think of a negative sign is as an OPPOSITE sign. So when you are subtracting a negative, it's really telling you to do the opposite of subtracting, which ends up being adding.

When you add a positive and a negative number is the sum always negative? ›

That depends on their absolute values. If the negative number has a greater absolute value than the absolute value of the positive number, the sum will be negative. If the positive number has a greater absolute value than the absolute value of the negative number, the sum will be positive.

How to teach adding and subtracting negative numbers? ›

In order to add and subtract negative numbers: If you have two signs next to each other, change them to a single sign. If the signs are the same replace with a positive sign (+). If the signs are different, replace with a negative sign (−).

What is the rule for positive and negative 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.

What is an example of subtracting negative numbers? ›

Example Problem 1: Subtracting Two Negative Integers

So we have -9 - (-5) = -9 + 5. Then, we can use a number line to visualize the answer by starting at -9 and moving to the right 5 places. Using the number line, we can see that the final result is -4. So, -9 - (-5) = -4.

What are the four rules of integers? ›

Rules of Integers

Sum of two positive integers is an integer. Sum of two negative integers is an integer. Product of two positive integers is an integer. Product of two negative integers is an integer.

What is the bodmas rule? ›

According to BODMAS rule, the brackets have to be solved first followed by powers or roots (i.e. of), then Division, Multiplication, Addition, and at the end Subtraction. Solving any expression is considered correct only if the BODMAS rule or the PEDMAS rule is followed to solve it.

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