Irrational Numbers: Definition, Types, Properties & Examples (2024)

Types of Irrational Numbers

There are 2 types of irrational numbers:

1. Algebraic Irrational Numbers: An algebraic irrational number is a normal irrational number that is resulted from mathematical operations. Algebraic ones are those which have roots of the algebraic equation as the square root of 2.

2. Transcendental Irrational Numbers: A transcendental number is a number that is not algebraic, in the sense that it is not the solution of an algebraic equation with rational number coefficients. Transcendental numbers are irrational, but not all irrational numbers are transcendental. In 1844 Joseph Liouville discovered the existence of transcendental numbers. Transcendental numbers are usually the most famous – \(\pi\), e, etc.

Properties of Irrational Numbers

Irrational numbers show distinct properties which are given below:

• An irrational number is always a real number.
• An irrational number cannot be expressed as a fraction.
• An irrational number is non-repeating and non-terminating as the decimal part never ends and never repeats itself.
• The value of the square root of any prime number is an irrational number.
• The sum of a rational number and an irrational number is irrational. The product of a rational number and an irrational number is irrational. This means that any operation between a rational and an irrational number, be it addition, subtraction, multiplication or division will always result in an irrational number only.
• If r is one irrational number and s is another irrational number, then r + s and r – s may or may not be irrational numbers and rs and r/s are may or may not be irrational numbers. This means that any operation between two irrational numbers, be it addition, subtraction, multiplication or divisions will not always result in an irrational number.
• If a and b are two distinct positive irrational numbers, then \(\sqrt{ab}\) is an irrational number lying between a and b.
• For any two irrational numbers, their least common multiple (LCM) may or may not exist.
• Irrational number is simplifications of Surds. When we can’t simplify a number to remove a square root or cube root etc. then it is a surd. For example, \(\sqrt{2}\) (square root of 2) can’t be simplified further so it is a surd.

There are 5properties of natural numbers: Closure Property, Commutative Property, Associative Property, Identity Property and Distributive Property.

List of Irrational Numbers

Here’s a list of important irrational numbers that are commonly used:

Square Root of Primes: \(\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7}, \sqrt{11}, \sqrt{13}, \sqrt{17}, \sqrt{19}…\)

The first irrational to be discovered was \(\sqrt{2}\). The Pythagoreans- and ancient Greek philosophical university and religious brotherhood- stumbled upon \(\sqrt{2}\) as the length of a diagonal of a square with side lengths 1 in the sixth century B.C. It was Hippasus who was one of the students of Pythagoras who discovered irrational numbers. However, the other followers of Pythagoras could not accept the existence of irrational numbers, and it is said that Hippasus was drowned at sea as a punishment from the gods!

Logarithms of primes with prime base: \(log_23, log_25, log_27, log_35, log_37…\)

The logarithm of a prime number with a prime base, like log_35 or log_72, is irrational.

\(\begin{matrix}
\text{ Assume } log_35 = {x\over{y}}, \text{ x and y are integers }, y ≠ 0.\\
3{x\over{y}} = 5 ( 3 < 5 \text{ therefore } {x\over{y}} > 1)\\
(3{x\over{y}})y = 5y\\
3x = 5y
\end{matrix}\)

᠎3 and 5 are prime numbers. x and y are integers. So the above equation is not balanced. Our assumption has led us to a contradiction. Therefore the assumption \(log_35 = {x\over{y}}\) = rational number is false. So log35 is an irrational number.

Sum of Rational and Irrational: \(3 + \sqrt{2}, 4 + \sqrt{7}…\)

Adding a rational number to an irrational number is an easy way to create a new irrational number. See the lists of numbers created using this method:

• \(1 + \sqrt{2}, 2 + \sqrt{2}, 3 + \sqrt{2}, …..\)
• 1 + π, 2 + π, 2 + π, ….
• \(1 + log_35, 2 + log_35, 3 + log_35, …\)

Learn various concepts of Binomial Theorem

Product of Rational & Irrational Numbers

What works for the sum of a rational and an irrational number, works for their product also. This provides yet another method to create examples of an irrational number. See the lists of such numbers below:

• \(2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}, …\)
• 2π, 3π, 4π, …
• \(2log_35, 3log_35, 4log_35, …\)

Infinite Continued Fraction

This is one of the ways to represent irrational numbers. It takes the form:

\(a_0+{1\over{a_1+{b_0\over{a_2+{b_1\over{…}}}}}}\)

For irrational numbers, we can limit \(a_i,b_i\) to be integers.

Pi (π):Number Pi originated from geometry. It is the ratio of the circumference and the diameter of a circle. It remains constant, independent of the size of the circle. If the circumference of a circle is rational, the radius is irrational. Hence, pi is an irrational number. In India, some interesting values of \(\pi\) began to emerge. In 499, Aryabatha published \(\pi = 3.1416…\); Born in 598, Brahmagupta published \(\pi = \sqrt{10} = 3.1622…\); and Bhaskara, born 1114, said that \(\pi = 3.14156…\) China beat them all with Liu Hui (3.141024 to 3.142764) and Tsu Chung- Chih \((\frac{355}{113} = 3.1415929)\)

Number e:The Number e is the sum of Infinite Quotients. The number e is a recent discovery by Jacob Bernoulli. He was trying to compute a continuously compounded interest growth in the 17th century. In short, he was evaluating \((1+1/n)^n\), as n grows to infinity. Later Euler calculated this number. Euler used the following formula, an endless summation, to calculate the value of e up to 18 digits.

\(e = 1 + {1\over{1!}} + {1\over{2!}} + {1\over{3!}} + {1\over{4!}} + {1\over{5!}} ….\)

Euler also found that e could be represented as a continuous, infinite fraction and proved that it is an irrational number.

Are Irrational Numbers Real Numbers?

All rational numbers are real numbers. An irrational number is a real number that cannot be expressed as the ratio of two integers.

Learn about Rational Numbers Between Two Numbers and Complex Numbers

Sum of Two Irrational Numbers

If r is one irrational number and s is another irrational number, then r + s and r – s may or may not be irrational numbers. This means that any operation between two irrational numbers, be it addition, subtraction, multiplication or divisions will not always result in an irrational number.

Irrational number + Irrational number = Irrational / Rational number.

Example: Consider two irrational numbers \(2 + \sqrt{3} and 5 + \sqrt{3}\). Adding the number will result in, \(2 + \sqrt{3} + 5 + \sqrt{3} = 7 + 2\sqrt{3}\). This is again the irrational number. Now consider another two irrational numbers \(2 + \sqrt{3} \text{ and } 2 – \sqrt{3}\). Adding the number will result in, \(2 + \sqrt{3} + 2 – \sqrt{3} = 4\). This is a rational number. Similarly, Irrational number − Irrational number = Irrational / Rational number.

Product of Two Irrational Numbers

If r is one irrational number and s is another irrational number, then rs and r/s are may or may not be irrational numbers. This means that any operation between two irrational numbers, be it addition, subtraction, multiplication or divisions will not always result in an irrational number. Irrational number × Irrational number = Irrational / Rational number.

Consider two irrational numbers \(2\sqrt{3} \text{ and } \sqrt{3}\). Multiplying the number will result in, \(2\sqrt{3} × \sqrt{3} = 2 × 3 = 6\). It is a rational number. Now consider another two irrational numbers \(2\sqrt{3} \text{ and } \sqrt{2}\). Multiplying the numbers will result in, \(2\sqrt{3} × \sqrt{2} = 2\sqrt{6}\). It is an irrational number. Similarly, Irrational number ÷ Irrational number = Irrational / Rational number.

How to Find Irrational Number Between Two Rational Numbers

Assume that we have two rational numbers a and b, then the irrational number between the two will be \(\sqrt{ab}\).

Let’s take an example of numbers 3 and 4. There can be an infinite number of irrational numbers between these numbers. The numbers between the squares of 3 and 4, i.e., between 9 and 16 are 10, 11, …14, 15. The square root of any of these numbers is always an irrational number. Hence, we take the square root of the product of the two vectors \(\sqrt{3\times4} = \sqrt{12}\)

How to Find Irrational Number Between Two Irrational Numbers

Assume that we have two irrational numbers a and b, then the irrational number between the two will be the square of both the numbers and take the square root of their average. If the square root is irrational, then we get the number we want. If we do not have the number you are looking for, we can repeat the procedure using one of the original numbers and the newly generated number.

Find the rational numbers between \(\sqrt{2} \text{ and } \sqrt{3}\)

The difference between \(\sqrt{2} \text{ and } \sqrt{3}\). 6 is between \(4\sqrt{2}\text{ and } 4\sqrt{3} \text{ and } \frac{3}{2} \text{ is between } \sqrt{2} \text{ and } \sqrt{3}\).

How to Find Irrational Numbers Between Decimals

The Irrational numbers have non-recurring and non-Terminating decimals. It doesn’t have the specific pattern of repeating the numbers after the decimal point, and these numbers are neverending. So we can write the numbers which are not repeated in any manner.

Decimal Expansion of Irrational Numbers

For all rationals of the form p/q (q ≠ 0).

On a division of p by q, two main things happen. Either the remainder becomes zero or never becomes zero and we get a repeating string of remainders.

Case I: The remainder becomes zero In this case, the decimal expansion terminates or ends after a finite number of steps. We call the decimal expansion of such numbers terminating.

e.g., 78, 12, 34, etc.

Case II: The remainder never becomes zero. In this case, we have a repeating block of digits in the quotient, this expansion is called non-terminating recurring.

e.g., 23 = 0.6666…..

22/7 = 3.142857142857……..

The repeated digits are written as \(23 = \bar{0.6}\)

\({22\over{7}} = \bar{3.142857}\)

Learn about Mean Deviation

Difference between Rational Numbers and Irrational Numbers

The difference between Rational numbers and Irrational numbers is as given below: