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# Rational Numbers

Top
 Sub Topics We all know that real number is something which exists in nature as real. Rational number is a part of it. Rational number is all about expressing the numbers in Fractions or Decimals. Let us study more about it in this section.

## What is a Rational Number?

Rational numbers are those numbers which can be represented in the form of form. We generally use q to represent rational numbers in mathematical world.

Characteristics of Rational Numbers:

1. All the rational numbers are subset of Real Numbers or we can say all rational numbers lie in real line.
2. Countless rational numbers lie between two rational numbers.
3. There can be infinite numbers of rational numbers between two integers.
4. Any Integer can be represented as rational number.
5. Rational numbers are countable numbers as we can easily count them.

Rational numbers are very densely populated as mentioned above that there can be infinite rational number between two integers. We can also find many rational numbers between two numbers. We can also perform various Operations on Rational Numbers like addition, subtraction, division and multiplication can also be done.

## Real Numbers Definition

Let us consider a number like $\frac{\sqrt{-5}}{2}$ , $\sqrt{-7}$. These kind of numbers can be expressed in terms of i like $\sqrt{7}$ i, 2.5 i etc. where i represents the imaginary numbers. To differentiate the imaginary numbers from the numbers existing in real there came a concept of Real Numbers.

Real numbers are defined as those numbers which do not have i (imaginary numbers) as a part of it. It is a Set of both Rational and Irrational Numbers. It is represented by R.

Example: R = {-3, -2, -1, 0, 0.5, 2, 5......}.

In short it represents particular or fixed amount of quantity. It includes positive or negative numbers, natural number, whole numbers, fractional numbers, integers etc.

## Rational Number Properties

Let us consider rational number p,q and r . The law holds good if it is true for addition and multiplication.

Associative law for rational numbers :
Additive law : (p + q) + r = p + (q + r)

Multiplicative law : (p * q) * r = p * (q * r).

Commutative law for rational numbers :
Additive law : p + q = q + p

Multiplicative law : p * q = q * p

Identity for rational numbers:
0 is the additive identity for rational numbers

i.e., p + 0 = p and

1 is the multiplication identity for rational numbers

i.e., p * 1 = p.