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 ${Q}^{\u2018}$. The inverse symbol over $Q$ represents the inverse of the rational numbers.

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.

The numbers which belong to irrational numbers group is written in a set form mathematically. The set of irrational numbers is denoted by the ${Q}^{\u2018}$ and the set along with irrational numbers is written in mathematical language as follows.

${Q}^{\u2018}=\{\dots .,-3.1428571428571,\frac{1}{2\u2013\sqrt[7]{5}},\sqrt{2},\sqrt{3},\sqrt{\frac{71}{2}},\dots .\}$

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

List of most recently solved mathematics problems.

June 18, 2018

Algebra Trigonometry

Find $x^3+\dfrac{1}{x^3}$ if value of $x+\dfrac{1}{x}$ equals to $2\cos{θ}$

Jun 13, 2018

Limit

Learn how to solve Limit of (1-cos6x)/(1-cos7x) as $x$ approaches $0$

Jun 09, 2018

Trigonometry

Find ΣtanAtanB, if A+B+C = 90°

May 30, 2018

Trigonometry

Find cos 40° + cos 80° + cos 160°

Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers.
Know more

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.