Rational Numbers: The real numbers which can be represented in the form of the ratio of two integers, say P/Q, where Q is not equal to zero are called rational numbers.

Irrational Numbers: The real numbers which cannot be expressed in the form of the ratio of two integers are called irrational numbers.

Rational NumbersNumbers that can be expressed as a ratio of two number (p/q form) are termed as a rational number.

Rational Number includes numbers, which are finite or are recurring in nature.

Rational Numbers includes perfect squares such as 4, 9, 16, 25, and so on.

Both the numerator and denominator are whole numbers, in which the denominator is not equal to zero.

Example: 3/2 = 1.5, 3.6767.

Irrational NumbersNumbers that cannot be expressed as a ratio of two numbers are termed as an irrational number.

These consist of numbers, which are non-terminating and non-repeating in nature.

Irrational Numbers includes surds such as √2, √3, √5, √7 and so on.

Irrational numbers cannot be written in fractional form.

Example: √5, √11

What are Irrational Numbers?

Irrational numbers are the numbers that cannot be expressed as a ratio of two integers or in a fraction form. These numbers cannot be represented in the form of p/q where p and q are integers and q ≠ 0. They are the opposite of rational numbers.

Examples of Irrational Numbers

The most common examples of irrational numbers are:

- π (pi) = 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679...
- √2 (square root of 2) = 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727...
- √3 (square root of 3) = 1.7320508075688772935274463415058723669428052538103806280558069794519330169088000370811461867572485756...
- √5 (square root of 5) = 2.2360679774997896964091736687312762354406183596115257242708972454105209256378048994144144085147968109...

Properties of Irrational Numbers

- They cannot be written as a ratio of two integers.
- They are non-terminating and non-repeating decimals.
- They cannot be expressed in fraction or decimal form.
- They are infinite in nature.
- They cannot be represented on a number line using whole numbers.

Comparison between Rational and Irrational Numbers

- Rational numbers can be expressed as a ratio of two integers, whereas irrational numbers cannot be expressed in such form.
- Rational numbers can be represented on a number line using whole numbers, whereas irrational numbers cannot be represented on a number line using whole numbers.
- Rational numbers are terminating or repeating decimals, whereas irrational numbers are non-terminating and non-repeating decimals.
- Rational numbers have a finite number of digits after the decimal point, whereas irrational numbers have an infinite number of digits after the decimal point.
- Rational numbers can be converted into decimals, whereas irrational numbers cannot be converted into decimals.

Summary

Irrational numbers are the numbers that cannot be expressed in the form of p/q where p and q are integers and q ≠ 0. They are non-terminating and non-repeating decimals and cannot be represented on a number line using whole numbers. They are infinite in nature and cannot be converted into decimals. In contrast, rational numbers can be expressed as a ratio of two integers, can be represented on a number line using whole numbers, and can be converted into decimals.