Let’s start with its meaning, the meaning of irrational is “no ratio”, which means the numbers can not be represented as a ratio of integers. Numbers that are real and can not be represented as an integers ratio. For instance, √3 is an irrational number. The decimal expression of these numbers never terminates. The calculation of these numbers is quite complicated. Usually, it can not be expressed as a fraction, like a/b, such that a,b both are integers and where b ≠ 0. We can say that Pi (22/7) is also an irrational number because it never terminates after the decimal.

## Some properties of irrational numbers-

- By adding rational and irrational numbers, the result will be an irrational number. For instance: a is rational, and b is irrational. If we add both numbers a+b =c, the result (c) will be irrational.
- Multiplication of any non-zero rational number with an irrational number, the result will be an irrational number. For instance: a is a rational number, and b is an irrational number. If we multiply both numbers ab=c, the result (c) will be an irrational number.
- The addition of two irrational numbers and the multiplication of two irrational numbers may be rational. For instance, √3√3 = 3 here, √3 is an irrational number and if we multiply it two times we get a result 3 which is a rational number.
- The LCM ( least common multiple) of two rational numbers may or may not exist.
- All irrational numbers are considered real numbers. For example, the square root of 5. It does not have a perfect square but always results in an irrational number.
- They consist of non-terminating decimals.
- On a number line, the representation of an irrational number is possible.

Let us see, some examples of an irrational number

There are many irrational numbers whose calculation is very complicated. They may not be written and simplified. Let’s take some examples of irrational numbers:

- Pi(∏) = 22/7 = 3.14159265…….. It consists of never-ending decimal, so pi is an irrational number.
- Any number in a square root which can not be further simplified is included as an irrational number. For example √3,√7,√5

Now, Do you know what are irrational numbers? Very simple, the numbers which are not rational, referred to as irrational. There is a lot of differences between rational numbers and irrational numbers which, are given below-

Ratinal numbers | Irrational numbers |

Can be expressed as a fraction. | Cannot be expressed as a fraction. |

They consist of terminating decimals. | They never consist terminating decimals. |

Rational numbers are the super-set of all-natural, whole, and integer numbers. | An irrational number is a separate set. |

Examples- 4/2, 0.12 | Examples- 22/7, √7 |

Now let us take some questions of irrational numbers and apply these properties to them.

Question – Show that 5√2 is irrational.

Solution- Let us assume, 5√2 is a rational number.

That means, there is a coprime a and b (b = 0) such that 5√2 = a/b.

By rearranging it √2 = a/5b.

Here 5, a, and b are integers. a/5b is a rational number and so, √2 is rational.

But this is a contradiction because √2 is irrational.

So, 5√2 is irrational.

Question – Is 120 an irrational number?

Solution – The number 120 is not irrational. The reason is, this can be expressed as the quotient of two integers: 120 = 120/1.

Question – Which of the following are rational numbers or irrational numbers and why?

3, -2.4567894568…., √7, -5.56

Solution-

Irrational numbers: -2.4567894568…., √7. Because these two numbers have non-terminating decimal expansion.

Rational numbers: 3, -5.56. Because these two numbers have terminating decimals.

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