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Last modified on August 3rd, 2023

Irrational numbers are real numbers that cannot be written as a simple fraction or ratio. In simple words, the irrational numbers are those numbers those are not rational. Hippasus, a Greek philosopher and a Pythagorean, discovered the first evidence of irrational numbers 5th century BC. However, his theory was not accepted.

Irrational numbers can’t be written as p/q form (ratio), where the denominator, q is not zero (q ≠ 0).

Few examples of irrational numbers are given below:

- π (pi), the ratio of a circle’s circumference to its diameter, is an irrational number. It has a decimal value of 3.1415926535⋅⋅⋅⋅ which doesn’t stop at any point.
- √x is irrational for any integer x, where x is not a perfect square.

In a right triangle with a base length of 1 unit, the hypotenuse is √2, which is irrational. (√2 = 1⋅414213⋅⋅⋅⋅)

- Euler’s number, e = 2⋅718281⋅⋅⋅⋅
- Golden ratio, φ = 1.6180339⋅⋅⋅⋅

Given below are some popular irrational numbers.

Number | value |
---|---|

Pi (π) | 3.1415926535⋅⋅⋅⋅ |

Euler’s Number (e) | 2⋅718281⋅⋅⋅⋅ |

Golden ratio, φ | 1.6180339⋅⋅⋅⋅ |

Let us solve an example

**${\dfrac{5\sqrt{2+4}}{\sqrt{3}}}$**

Solution:

${\begin{aligned}\dfrac{5\sqrt{2+4}}{\sqrt{3}}\\ \simeq \dfrac{5\sqrt{6}}{\sqrt{3}}\end{aligned}}$

= ${ 5\sqrt{2} }$

Irrational numbers are universally represented by the alphabet ‘P’. The universal symbols for rational numbers is ‘Q’, real numbers is ‘R’.

- Are real numbers only
- Decimal expansion is non-terminating (continues endlessly)
- Addition of a rational and irrational number gives an irrational number as the sum; a + b = irrational number, here a = rational number, b = irrational number
- Multiplication of a rational and irrational number will always give an irrational number; here, a × b = irrational number, if a = rational number, b = irrational number
- Sum or product of 2 irrational numbers may give an irrational number; √2 × √2 = 2.
- The Least common factor (LCM) for 2 irrational numbers may or may not exist

Let us solve some examples to understand the concept better.

**π × π**

Solution:

Let us see if π × π = rational or irrational

π × π

= π^{2}

= irrational, ∵ π is irrational

**√7 × √7**

Solution:

√7 × √7

= 7

= rational

**Show that √7 + √5 is irrational.**

Solution:

Let us assume that √7 + √5 is rational

If √7 + √5 is rational,

Then, ${\dfrac{7-5}{\sqrt{7}+\sqrt{5}}=\sqrt{7}-\sqrt{5} }$ , implying √7 – √5 is also rational

(√7 + √5) – (√7 – √5)

= 2√5, (implies 2√5 is rational)

Hence, ${2\sqrt{5}\times \dfrac{1}{2}}$

= √5 is also rational. But this is a contradiction

Hence, we get the proof of irrational numbers by contradiction

We already have learnt that irrational numbers are real numbers which cannot be represented in the form of p/q, where p and q are integers, and q ≠ 0, and also can’t be simplified to a closed decimal value.

Taking √5 as an example,

It cannot be represented in the form of a fraction

It cannot be simplified either. Its value is 2.236067⋅⋅⋅⋅ and it is not a closed decimal value.

Thus, √5 is irrational.

Rational numbers are numbers that can be expressed in the form of a fraction (p/q) or ratio.

Given below the differences between rational and irrational numbers in a table.

Irrational Numbers | Rational Numbers |
---|---|

Cannot be expressed in the form of a fraction (p/q) | Can be expressed in the form of a fraction (p/q) |

The decimal expansion is non-terminating (continues endlessly) and non-recurring (with no group of digits repeating) at any point | The decimal expansion is terminating (ends at some point) or non-terminating recurring (repeating) |

Last modified on August 3rd, 2023

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Hey there! Wonderful little article, it explains things pretty clearly and introduces concepts that seem worth looking into, particularly the properties of irrational numbers. However, I find a lot of confusion with property #4, it seems to be incorrect. See this simple proof by contradiction: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:irrational-numbers/x2f8bb11595b61c86:sums-and-products-of-rational-and-irrational-numbers/v/proof-that-rational-times-irrational-is-irrational

There isn’t any example showcasing this property either in the examples section either, so I’m just assuming it was written in error. Would appreciate a response clearing up the confusion or a correction of the error if it is one, thank you!!

Thank you for your comment. We have edited that section.