The four basic arithmetic operations associated with integers are:
- Addition of Integers
- Subtraction of Integer
- Multiplication of Integers
- Division of Integers
Answer: There are some rules for adding, subtracting, multiplying, and dividing positive and negative numbers.
Before we start learning these methods of integer operations, we need to remember a few things. If there is no sign in front of a number, it means that the number is positive.
Explanation:
The following content shows the rules for adding, subtracting, multiplying, and dividing positive and negative numbers.
Adding Integers Rule:
Case 1: Signs are the same
If the signs are the same, add and keep the same sign.
- (+) + (+) = Add the numbers and the answer is positive
Example : 2 + 5 = 7
- (‐) + (‐) = Add the numbers and the answer is negative
Example : (-5) + (-4) = -9
Case 2: Signs are different
If the signs are different, subtract the numbers and use the sign of the larger number.
- (+) + (‐) = Subtract the numbers and take the sign of the bigger number.
Example: 7 + (-3) = 4
- (‐) + (+) = Subtract the numbers and take the sign of the bigger number.
Example: (-9) + 6 = -3
Subtracting Integers Rule:
To subtract a number from another number, the sign of the number (which is to be subtracted) should be changed and then this number with the changed sign should be added to the first number.
- (+) - (+) = Change the sign of the number to be subtracted and add them up. The result takes the sign of the greater number.
Example: (+6) – (+2)
= (+6) + (-2) = 6 - 2 = 4
- (-) - (-) = Change the sign of the number to be subtracted and add them up.The result takes the sign of the greater number.
Example: (-9) – (-6)
= (-9) + (+6) = -9 + 6 = -3
- (+) - (-) = Change the sign of the number to be subtracted and add them up.
Example: (+5) – (-3)
= (+5) +(+3) = 5 + 3 = 8
- (-) - (+) = Change the sign of the number to be subtracted and add them up. Result is always negative
Example: (-7) – (+2)
= (-7) + (-2) = -7 - 2 = -9
Multiplying and Dividing Integers Rule:
Case 1: Signs are same
If the signs are the same, the answer is always positive.
- (+) × (+) = +
Example: 5 × 4 = 20
- (+) ÷ (+) = +
Example: 16 ÷ 4 = 4
- (‐) × (‐) = +
Example: (-7) × (-9) = 63
- (‐) ÷ (‐) = +
Example: (-20) ÷ (-2) = 10
Case 2: Signs are different
If the signs are different, the answer is always negative.
- (+) × (‐) = ‐
Example: 6 × (-10) = -60
- (+) ÷ (‐) = ‐
Example: 30 ÷ (-15) = -2
- (‐) × (+) = ‐
Example: -3 × 11 = 33
- (‐) ÷ (+) = ‐
Example: -25 ÷ 5 = -5
Thus, these are the rules to add, subtract, multiply and divide positive and negative numbers.
How to Add and Subtract
Positive and Negative Numbers
Numbers Can be Positive or Negative
This is the Number Line:
| Negative Numbers (−) | Positive Numbers (+) |
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 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:
| 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:
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