Irrational Numbers
Irrational numbers are those real numbers that cannot be represented in the form of a ratio. In other words, those real numbers that are not rational numbers are known as irrational numbers. Hippasus, a Pythagorean philosopher, discovered irrational numbers in the 5th century BC. Unfortunately, his theory was ridiculed and he was thrown into the sea. But irrational numbers exist, let's have a look at this page to get a better understand of the concept, and trust us, you won't be thrown into the sea. Rather, by knowing the concept, you will also know the irrational number list, the difference between irrational and rational numbers, and whether or not irrational numbers are real numbers.
What are Irrational Numbers?
Irrational numbers are the set of real numbers that cannot be expressed in the form of a fraction, p/q where p and q are integers. The denominator q is not equal to zero (q ≠ 0). Also, the decimal expansion of an irrational number is neither terminating nor repeating.
Irrational Numbers Definition: Irrational numbers are real numbers that cannot be represented as a simple fraction. These cannot be expressed in the form of ratio, such as p/q, where p and q are integers, q≠0. It is a contradiction of rational numbers.
Common Examples of Irrational Numbers
Given below are the few specific irrational numbers that are commonly used.
 ㄫ(pi) is an irrational number. π=3⋅14159265… The decimal value never stops at any point. Since the value of ㄫ is closer to the fraction 22/7, we take the value of pi as 22/7 or 3.14 (Note: 22/7 is a rational number.)
 √2 is an irrational number. Consider a rightangled isosceles triangle, with the two equal sides AB and BC of length 1 unit. By the Pythagoras theorem, the hypotenuse AC will be √2. √2=1⋅414213⋅⋅⋅⋅
 Euler's number e is an irrational number. e=2⋅718281⋅⋅⋅⋅
 Golden ratio, φ 1.61803398874989….
Properties of Irrational Numbers
Properties of irrational numbers help us to pick up irrational numbers out of a set of real numbers. Given below are some of the properties of irrational numbers:
 Irrational numbers consist of nonterminating and nonrecurring decimals.
 These are real numbers only.
 When an irrational and a rational number are added, the result or their sum is an irrational number only. For an irrational number x, and a rational number y, their result, x+y = an irrational number.
 When any irrational numbers multiplied by any nonzero rational number, their product is an irrational number. For an irrational number x and a rational number y, their product xy = irrational.
 For any two irrational numbers, their least common multiple (LCM) may or may not exist.
 Addition, subtraction, multiplication, and division of two irrational numbers may or may not be a rational number.
How to Identify an Irrational Number?
We know that the irrational numbers are real numbers only which cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0. For example, √ 5 and √ 3, etc. are irrational numbers. On the other hand, the numbers which can be represented in the form of p/q, such that, p and q are integers and q ≠ 0, are rational numbers.
Irrational Numbers Symbol
Before knowing the symbol of irrational numbers, we discuss the symbols used for other types of numbers.
 N  Natural numbers
 I  Imaginary Numbers
 R  Real Numbers
 Q  Rational Numbers.
Real numbers consist of both rational and irrational numbers. (RQ) defines that irrational numbers can be obtained by subtracting rational numbers (Q) from the real numbers (R). This can also be written as (R\Q). Symbol = Q'.
Set of Irrational Numbers
Set of irrational numbers can be obtained by writing few irrational numbers within brackets. The set of irrational numbers can be obtained by some properties.
 All square roots which are not a perfect square are irrational numbers. {√2, √3, √5, √8}
 Euler's number, Golden ratio, and Pi are some of the famous irrational numbers. {e, ∅, ㄫ}
 The square root of any prime number is an irrational number.
The table illustrates the list of some of the irrational numbers.
Irrational number  value 

π  3.14159265.... 
e  2.7182818..... 
√2  1.414213562... 
√3  1.73205080... 
√5  2.23606797.... 
√7  2.64575131.... 
√11  3.31662479... 
√13  3.605551275... 
√3/2  0.866025.... 
∛47  3.60882608 
Rational Numbers vs Irrational Numbers
Any number which is defined in the form of a fraction p/q or ratio is called a rational number. This may consist of the numerator (p) and denominator (q), where q is not equal to zero. A rational number can be a whole number or an integer.
 2/3 = 0.6666 = 0.67. Since the decimal value is recurring (repeating). So, we approximated it to 0.67
 √4 = 2 and 2, where both 2 and 2 are integers.
The table illustrates the difference between rational and irrational numbers.
Rational numbers  Irrational numbers 

It can be expressed in the form of a fraction or ratio.  It cannot be expressed in the form of a fraction or ratio. 
All perfect square numbers are rational  Any square root number which is not a perfect square is irrational 
The decimal value is either terminating or nonterminating repeating. Example: 0.33333, 0.656565.., 1.75 
The decimal value neither repeats nor terminates at any point. Example: π, √13, e 
Interesting Facts about Irrational Numbers
There are some cool and interesting facts about irrational numbers that make us deeply understand the why behind the what.
1. Accidental Invention of √2
The square root of 2 or √2 was the first invented irrational number when calculating the length of the isosceles triangle. He used the famous Pythagoras formula a^{2} = b^{2} + c^{2}
AC^{2}=AB^{2}+BC^{2} ⇒ AC^{2}=1^{2}+1^{2 }⇒ AC = √ 2
√2 lies between numbers 1 and 2 as the value is 1.41421... So he revealed that the length AC cannot be expressed in the form of fractions or integers.
2. The value of π
The value of π is approximately calculated to over 22 trillion digits without an end. A computer took about 105 days, with 24 hard drives, to calculate the value of pi.
3. Invention of Euler's Number e
The Euler's number is first introduced by Leonhard Euler, a Swiss mathematician in the year 1731. This 'e' is also called a Napier Number which is mostly used in logarithm and trigonometry.
Proof of an Irrational number:
Let's understand how to prove that a given non perfect square is irrational. Here is stepwise proof of the same.
To prove: √2 is an irrational number.
Suppose, √2 is a rational number. Then, by the definition of rational numbers, it can be written that,
√2=p/q ...(1) where p and q are coprime integers and \(q ≠ 0\) (Coprimes are those numbers whose common factor is 1).
Squaring both the sides of equation (1), we have
\(\begin{align}2 &= p^2/q^2\\⇒ p^2 &= 2 * q^2\qquad \dots(2)\end{align}\)
From the theorem, that states “Given p is a prime number and a2 is divisible by p, (where a is any positive integer), then it can be concluded that p also divides a”, if 2 is a prime factor of p^{2}, then 2 is also a prime factor of p.
So, p= 2 x c where c is an integer.
Substituting this value of p in equation (3), we have
\(\begin{align}(2c)^2 &= 2q^2\\⇒ q^2 &= 2c^2\end{align}\)
This implies that 2 is a prime factor of q^{2} also. Again from the theorem, it can be said that 2 is also a prime factor of q.
According to the initial assumption, p and q are coprimes but the result obtained above contradicts this assumption as p and q have 2 as a common prime factor other than 1. This contradiction arose due to the incorrect assumption that √2 is rational.
So, √2 is irrational.
History of irrational numbers
The first man to recognize the existence of irrational numbers might have died for his discovery. Hippassus of Metapontum was an ancient Greek philosopher of the Pythagorean school of thought. Supposedly, he tried to use his teacher's famous theorem: a^{2} + b^{2} = c^{2} to find the length of the diagonal of a unit square. This revealed that a square's sides are incommensurable with its diagonal and that this length cannot be expressed as the ratio of two integers. The other Pythagoreans believed dogmatically that only positive rational numbers could exist. They were so horrified by the idea of incommensurability that they threw Hippassus overboard on a sea voyage and vowed to keep the existence of irrational numbers an official secret of their sect. However, there are good reasons to believe Hippassus's demise is merely an apocryphal myth. Historical documents referencing the incident are both sparse and written 800 years after the time of Pythagoras and Hippassus. It wasn't until approximately 300 years after Hippassus's time that Euclid would give his proof for the irrationality of √2.
The Pythagoreans had likely manually measured the diagonal of a unit square. However, they would have regarded such a measurement as an approximation close to a precise rational number that gave the true length of the diagonal. Before Hippassus, Pythagoreans had no reason to suspect that there were real numbers that in principle, not merely in practice, could not be measured or counted to. Numbers were the spiritual basis of their philosophy and religion for the Pythagoreans. Cosmology, physics, ethics, and spirituality were predicated on the premise that "all is number." They believed that all thingsthe number of stars in the sky, the pitches of musical scales, and the qualities of virtuecould all be described by and apprehended through rational numbers.
Important Points
 The product of any two irrational numbers can be either rational or irrational. Example (a): Multiply √2 and π ⇒ 4.4428829... is an irrational number. Example (b): Multiply √2 and √2 ⇒ 2 is a rational number.
 The set of irrational numbers is not closed under the multiplication process, unlike the set of rational numbers.
 The addition or the multiplication of two irrational numbers may be rational; for example, √2 × √2 = 2. Here, √2 is an irrational number. If it is multiplied twice, then the final product obtained is a rational number, i.e 2.
Irrational Numbers Solved Examples

Example 1: John is playing "Roll a diceNumber game" with his friend. John takes a turn and rolls a dice. He gets 5. If he gets 5, he is supposed to collect all the irrational numbers from his friend. Help John to collect all the irrational numbers without missing even one. {e, 5, √9,√13, π, 2/8}
Solution:
5 is an integer. √9 is a perfect square. 2/8 has a recurring terminating decimal value. These numbers are rational numbers. The irrational numbers are e, √13, π. Therefore, John collected all the irrational numbers and those are e, √13 and π.

Example 2: Jade has a box with four irrational numbers. Jade wants only one irrational number which is closest to 3 and should not exceed 3. Help Jade to find out the right one. The irrational numbers in the box are √3, √6, √10, √5.
Solution:
First, we find the value of these irrational numbers. √3 = 1.732020.., √6 = 2.449489.., √10 = 3.162277.., √5 = 2.236067... Thus, √6 = 2.449489... comes closest to 3. Therefore, √6 is the closest number to 3.
Practice Questions on Irrational Numbers
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FAQs on Irrational Numbers
What are Irrational Numbers in Math?
Irrational numbers are a set of real numbers that cannot be expressed in the form of fractions or ratios. Ex: π, √2, e, √5
How can you Identify an Irrational Number?
For any number which is not rational is considered irrational. Irrational numbers can be written as decimals, but definitely not as fractions. Also, these numbers tend to have endless nonrepeating digits to the right of the decimal.
What are Rational Numbers?
All the numbers of the form of p/q where p and q are integers and q do not equal to 0 is a rational number. Examples of rational numbers are 1/2, 3/4, 0.3, or 3/10.
What is the Difference Between Rational and Irrational Numbers?
Rational numbers are those that are terminating or nonterminating repeating numbers, while irrational numbers are those that neither terminate nor repeat after a specific number of decimal places.
Is 2/3 an Irrational Number?
No, 2/3 is not an irrational number. 2/3 = 0.666666.... which is a recurring decimal. Therefore, 2/3 is a rational number.
What are Terminating Numbers?
Terminating numbers are those decimals that end after a specific number of decimal places. For example, 1.5, 3.4, 0.25, etc are terminating numbers. All terminating numbers are rational numbers as they can be written in the form of p/q easily.
Why are Irrational Numbers Called Surds?
A surd refers to an expression that includes a square root, cube root, or other root symbols. Surds are used to write irrational numbers precisely. All surds are considered to be irrational numbers but all irrational numbers can't be considered surds. Irrational numbers, which are not the roots of algebraic expressions, like π and e, are not surds.