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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:
In a right triangle with a base length of 1 unit, the hypotenuse is √2, which is irrational. (√2 = 1⋅414213⋅⋅⋅⋅)
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
${\dfrac{5\sqrt{2+4}}{\sqrt{3}}}$
${\begin{aligned}\dfrac{5\sqrt{2+4}}{\sqrt{3}}\\ \simeq \dfrac{5\sqrt{6}}{\sqrt{3}}\end{aligned}}$
= ${ 5\sqrt{2} }$
Irrational numbers are universally represented by the alphabet ‘P’. The universal symbols for rational numbers is ‘Q’, real numbers is ‘R’.
Let us solve some examples to understand the concept better.
π × π
Let us see if π × π = rational or irrational
π × π
= π2
= irrational, ∵ π is irrational
√7 × √7
√7 × √7
= 7
= rational
Show that √7 + √5 is irrational.
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
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.