# 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:

** Integer Addition and Subtraction**
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__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**
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__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**
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**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