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Math Integers

Set of Integers is a combination of Whole numbers and negative numbers.
..... -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 .....

Properties of Integers:
  1. Integer Addition and Subtraction
  2. Integer Multiplication
  3. Integer Division

Integer Addition and Subtraction - Back to top

Closure under Addition:
Given two integers i and j, i+j is also an integer.
Example: 2 and 3 are integers. 2 + 3 = 5 is also an integer.

Closure under Subtraction:
Given two integers i and j, i-j is also an integer.

Commutative property of integer addition:
Given two integers i and j. i + j = j + i.
Example: Given two integers 2 and 3. 2 + 3 = 3 + 2 = 5
Subtraction is not commutative for integers.
Example: 2-3 = -1 is not equal to 3-2 = 1.

Associative property of integer addition:
Given three integers i,j and k. i + (j + k) = (i + j) + k.
Example: Consider three integers, 2,3 and 5.
2 + (3 + 5) = 2 + 8 = 10.
(2 + 3) + 5 = 5 + 5 = 10.

Associative identity of integer addition:
When we add zero to integer, we get the same integer back again.
For an integer i. i + 0 = 0 + i.
Example: Consider integer 3.
3 + 0 = 0 + 3 = 3

Integer Multiplication - Back to top

Multiplication with a negative integer:
Given two integers i and j
i x (-j) = -(i x j)
Example: 2 x (-3) = -(2 x 3) = -6

Multiplication of two negative integers:
Mutiplying two negative integers results in a positive integer.
(-i) x (-j) = i x j
Example: (-2) x (-3) = 2 x 3 = 6

Multiplication of three negative integers:
(-i) x (-j) x (-k) = -(i x j x k)
Example: (-2) x (-3) x (-4) = -(2 x 3 x 4) = -24

Multiplication Properties:
Clousure under multiplication. Given two integers i and j. i x j is also an integer.
Example: Consider two integers 2 and 3. 2 x 3 = 6 is also an integer.

Commutative property of integer multiplication:
Given two integers i and j.
Then, i x j = j x i.
Example: Consider two integers 2 and 3. 2 x 3 = 3 x 2 = 6.

Multiplying by zero:
Multiplying any integer by zero results in another zero.
Given an integer i, then
i x 0 = 0 x i = 0
Example: Given integer 3, then 3 x 0 = 0 x 3 = 0.

Mutiplicative identify for integers:
One is the multiplicative identify for integers.
Given an integer i, then
i x 1 = 1 x i = i
Example: Given integer 4, then 4 x 1 = 1 x 4 = 4.

Multiplying by -1:
We can multiply an integer with -1 to get it's counterpart on the other side of the number line.
Given an integer i, then:
i x -1 = -1 x i = -i
Example: Given integer 2, then 2 x -1 = -1 x 2 = -2.
Given integer -3, then -3 x -1 = -1 x -3 = 3.

Associative Property for Integer Multiplication:
Given three integers i, j and k, then:
i x (j x k) = (i x j) x k
Example: Given integer 2, 3 and 5. Then,
2 x (3 x 5) = 2 x 15 = 30.
(2 x 3) x 5 = 6 x 5 = 30.

Distributive Property for Integer Multiplication:
Given three integers i, j and k, then:
i x (j + k) = (i x j) + (i x k)
Example: Given integer 2, 3 and 7. Then,
2 x (3 + 7) = 2 x 10 = 20.
(2 x 3) + (2 x 7) = 6 + 14 = 20.

Integer Division - Back to top

Division is inverse operation on multiplication.
Example: 2 x 3 = 6
So, 6 / 3 = 2
and, 6 / 2 = 3

Division by negative integer:
Consider two integers i and j.
i / (-j) = - (i / j)
Example: 50 / (-5) = -(50/5) = -5

Division by zero:
Division by zero is invalid operation. The number that is obtained is meaningless.
If you divide an integer x by 0

Division by one:
When we divide integer by one, we get the same integer in return.
i / 1 = 1
Example: 5 / 1 = 5

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