Irrational Numbers - Definition, Examples | Rational and Irrational Numbers (2024)

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 understanding 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 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.

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 right-angled 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 non-terminating and non-recurring 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.

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. Here are some tricks to identify irrational numbers.

• The numbers that are not perfect squares, perfect cubes, etc are irrational. For example √2, √3, √26, etc are irrational. But √25 (= 5), √0.04 (=0.2 = 2/10), etc are rational numbers.
• The numbers whose decimal value is non-terminating and non-repeating patterns are irrational. For example √2 = 1.4142135623730950488016887242097.... is irrational, whereas 1/7 = 0.14285714285714285714285714285714... is rational as we can observe that "142857" is keep getting repeated in the decimal portion.

Irrational Numbers Symbol

Before knowing the symbol of irrational numbers, let us 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. (R-Q) 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). Hence Irrational Numbers Symbol = Q'.

Set of Irrational Numbers

Set of irrational numbers can be obtained by writing all irrational numbers within brackets. But we know that there are infinite number of irrational numbers. So we cannot list the entire set of irrational numbers. But here are a few subsets of set of irrational numbers.

• All square roots which are not a perfect squares are irrational numbers. Example: {√2, √3, √5, √8}
• Euler's number, Golden ratio, and Pi are some of the famous irrational numbers. Example: {e, ∅, ㄫ}
• The square root of any prime number is an irrational number. Example: {√2, √3, √5, √7, √11, √13, ...}

The table illustrates the list of some of the irrational numbers.

Differences Between Rational and Irrational Numbers

Any number which is defined in the form of a fraction p/q or ratio is called a rational number. This may consists 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.

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 a² = b² + c²

AC²=AB²+BC² ⇒ AC²=1²+1² ⇒ 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 co-prime integers and \(q ≠ 0\) (Co-primes 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 p², 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 q² 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 co-primes 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.

☛Also Read

• Prove that Root 2 is Irrational
• Prove that Root 3 is Irrational
• Prove that Root 5 is Irrational
• Prove that Root 6 is Irrational
• Prove that Root 7 is Irrational
• Prove that Root 11 is Irrational

Rational and Irrational Numbers Worksheets

Rational and irrational numbers worksheets can provide a better understanding of why rational and irrational numbers are part of real numbers. Rational and irrational numbers worksheets include a variety of problems and examples based on operations and properties of rational and irrational numbers. It consists of creative and engaging fun activities where a child can explore end-to-end concepts of rational and irrational numbers in detail with practical illustrations.

Important Points on Irrational Numbers:

• 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 same rule works for quotient of two irrational numbers as well.
• The set of irrational numbers is not closed under the multiplication process, unlike the set of rational numbers.
• The sum and difference of any two irrational numbers is always irrational.

☛Related Articles:

Check out a few more interesting articles related to irrational numbers.

• Decimal Representation of Irrational Numbers
• Rational Numbers
• Rationalize the Denominator
• Is pi a rational or Irrational Number

FAQs on Irrational Numbers

What is the Definition of Irrational Numbers in Math?

Irrational numbers are a set of real numbers that cannot be expressed in the form of fractions or ratios made up of integers. Ex: π, √2, e, √5. Alternatively, an irrational number is a number whose decimal notation is non-terminating and non-recurring.

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 non-repeating digits to the right of the decimal.

Are Rational Numbers and Irrational Numbers Same?

No, rational and irrational numbers are not the same. All the numbers are represented in the form of p/q where p and q are integers and q does not equal to 0 is a rational number. Examples of rational numbers are 1/2, -3/4, 0.3, or 3/10. Whereas, we cannot express irrational numbers such as √2, ∛3, etc in the form of p/q.

What is the Difference Between Rational and Irrational Numbers?

Rational numbers are those that are terminating or non-terminating 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.

Why Rational Numbers and Irrational Numbers Are in the Set of Real Numbers?

The numbers which can be expressed in the form of decimals are considered real numbers. If we talk about rational and irrational numbers both forms of numbers can be represented in terms of decimals, hence both rational numbers and irrational numbers are in the set of real numbers.

Why Pi is an Irrational Number?

Pi is defined as the ratio of a circle's circumference to its diameter. The value of Pi is always constant. Pi (π) approximately equals 3.14159265359... and is a non-terminating non-repeating decimal number. Hence 'pi' is an irrational number.

How Many Irrational Numbers Lies Between Root 2 and Root 3?

We can have infinitely many irrational numbers between root 2 and root 3. A few examples of irrational numbers between root 2 and root 3 are 1.575775777..., 1.4243443..., 1.686970..., etc.

Are Irrational Numbers Non-Terminating and Non-Recurring?

Yes, irrational numbers are non-terminating and non-recurring. 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.
• Whereas non-terminating and non-recurring numbers are considered as the never-ending decimal expansion of irrational numbers.

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.