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# Integers

Integers are numbers that do not have decimal or fractional part. Integers include all counting numbers i.e. natural numbers, zero and negative of natural numbers. The set of integers is denoted by $\mathbb{Z}$.

Integers can be classified as follows:

• Positive Integers
• Negative Integers
• Zero

Positive Integers:  All the positive natural numbers come under the category of positive integers. These are denoted by $\mathbb{Z}^{+}$.
$\mathbb{Z}^{+}$ = 1, 2, 3, 4, .....

Negative Integers: Negative of all natural numbers come under the category of negative integers. These are denoted by $\mathbb{Z}^{-}$.
$\mathbb{Z}^{-}$ = -1, -2, -3, -4, .....

Zero: Zero is also included into integers.

Thus, we can say that the set of integers can represented as follows:

$\mathbb{Z}$ = {...., -4, -3, -2, -1, 0, 1, 2, 3, 4, ....}

Integers are the subset of real numbers. So, we can represent integers on the real number line. When we move on the right side of zero, we find positive integers and on moving on the left side of zero, we obtain negative integers. The image shown below illustrates integers on real number line:

The integers extends to the infinity on the both sides of zero.

Properties of Integers:
Integers follow the following algebraic properties:
1) Closure: If a, b $\in \mathbb{Z}$, then a + b $\in \mathbb{Z}$ and a x b $\in \mathbb{Z}$
.
2) Associativity: If a, b, c $\in \mathbb{Z}$, then (a + b) + c = a + (b + c) and (a x b) x c = a x (b x c).
3) Commutativity:
If a, b $\in \mathbb{Z}$, then a + b = b + a and a x b = b x a.
4) Distributivity:
If a, b, c $\in \mathbb{Z}$, then a x (b + c) = a x b + a x c and (a + b) x c = a x c + b x c.
5) Identity:

• Additive Identity: If a $\in \mathbb{Z}$, then 0 is additive identity, such that a + 0 = a.
• Multiplicative Identity: If a $\in \mathbb{Z}$, then 1 is multiplicative identity, such that a x 1 = a.

6) Additive Inverse: If a $\in \mathbb{Z}$, then -a is additive inverse, such that a + (-a) = 0.

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