## Integers

### Integers are a set of numbers that include all whole numbers, positive, negative, and zero

Integers are a set of numbers that include all whole numbers, positive, negative, and zero. They can be represented on the number line and are denoted by the symbol “Z”. The set of integers can be written as {…, -3, -2, -1, 0, 1, 2, 3, …}.

Some key properties of integers include:

1. Closure: When two integers are added, subtracted, or multiplied, the result is always an integer. For example, (-4) + 7 = 3, (-8) – (-3) = -5, and (-2) × 6 = -12.

2. Commutativity and Associativity: Addition and multiplication of integers are commutative and associative. This means that changing the order of the integers being added or multiplied does not affect the result. For example, 5 + 3 = 3 + 5, and (2 × 3) × 4 = 2 × (3 × 4).

3. Identity elements: The identity element for addition in the set of integers is 0. Adding 0 to any integer does not change its value. Similarly, the identity element for multiplication is 1. Multiplying any integer by 1 does not change its value.

4. Inverse elements: For every integer, there is an inverse element that, when added to the integer, gives the identity element (0). For example, the inverse of 5 is -5 since 5 + (-5) = 0. Similarly, the inverse of -3 is 3.

5. Distributive Property: The distributive property holds for integers. This means that when you multiply an integer by the sum or difference of two other integers, you can distribute the multiplication to both of them separately. For example, 2 × (4 + 3) = 2 × 4 + 2 × 3.

Integers are widely used in various mathematical concepts such as algebra, number theory, and arithmetic operations. Understanding the properties and operations of integers is important for solving math problems and applications in real-world scenarios.

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