# Irrational numbers

A number which cannot be expressed as a ratio of two quantities is defined an irrational number.

The meaning of irrational is opposite (inverse) to rational. It means, irrational numbers are formed based on opposite principle of forming rational numbers. Rational numbers are known as the numbers which can be expressed a fraction of two integers. However, there are some numbers which cannot be expressed as a ratio of two integers. Such numbers are called irrational numbers.

Rational numbers is denoted by a symbol $Q$. The set of irrational numbers is denoted by a symbol ${Q}^{‘}$. The inverse symbol over $Q$ represents the inverse of the rational numbers.

## Examples

Surds, some decimal numbers, transcendental numbers and etc. are best examples for irrational numbers. The following examples understand you irrational numbers clearly.

Surds

Surds represent arithmetical quantities but they cannot be expressed as a ratio of two integers because they cannot be written in fractional form.

$\sqrt{5}=2.2360679774997896964091736687313$

The approximate value of the square of $5$ is given here and it cannot be written as a ratio of two integers because it’s not possible algebraically. Therefore, surds are best examples for irrational numbers.

Decimals

Decimals can be expressed as a ratio of two integers but it is not possible to express in rational form in the case of some decimal numbers.

$1.4142135623730950488016887242097$

It is an example decimal which cannot be written in fractional form. Such decimal numbers are known irrational numbers.

Transcendental Numbers

Transcendental numbers are mathematical constants which cannot be expressed as ratio of two integers.

Natural Logarithmic base $\left(e\right)=2.7182818284590452353602874713527$

Pi $\left(\pi \right)=3.1415926535897932384626433832795$

Natural Logarithmic base $\left(e\right),$ Pi $\left(\pi \right)$ and etc. are best example for irrational numbers.

### Representation

The numbers which belong to irrational numbers group is written in a set form mathematically. The set of irrational numbers is denoted by the ${Q}^{‘}$ and the set along with irrational numbers is written in mathematical language as follows.

${Q}^{‘}=\left\{\dots .,-3.1428571428571,\frac{1}{2–\sqrt[7]{5}},\sqrt{2},\sqrt{3},\sqrt{\frac{71}{2}},\dots .\right\}$

Irrational numbers are collection of infinite numbers. Thence, the set of irrational numbers is also known as an infinite set.

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