Real Number System

All positive and negative numbers including zero are called integers. It is usually denoted by I or Z.

I or Z = { ... –3, –2, –1, 0, 1, 2, 3 …}

Here, –1, –2, –3 … are called negative integers whereas 1, 2, 3 … are called positive integers and 0 is taken as neutral.

The absolute value of an integer is its numerical value regardless of its sign. The absolute value of an integer n is denoted as |n|.

          Therefore, |−10| = 10, |−2| = 2, |0| = 0, |7| = 7 etc.

The opposite of an integer is the integer with its sign reversed. The opposite of integer a is −a and the opposite of integer −b is +b or b.

          Thus, opposite of 5 is −5, opposite of −8 is 8.

Addition and subtraction of integers

Integers are closed under addition and subtraction.

For two integers, a and b, a + b and a – b are integers. 

For example, (–14) + 3 = –11

(–7) – (–2) = –7 + 2 = –5

Addition is commutative for integers. For integers, a and b, a + b =b + a

For example, (–7) + 5 = 5 + (–7) = –2

Subtraction is not commutative for integers.

For example, (–7) – (4) = –11

4 – (–7) = 11

(–7) – (4) ≠ (4) – (–7)

Addition is associative for integers. For integers, a, b, and c,

a + (b + c) = (a + b) + c

For example, (–7) + (4 + (–3)) = ((–7) + 4) + (–3) = –6

Subtraction is not associative for integers. When 0 is added to any integer, say a, the same integer is obtained. Therefore, 0 is the additive identity of integers.

a + 0 = a = 0 + a

When –a is added to any integer a, 0 is obtained. Therefore, –a is the additive inverse…