Irrational Numbers

Irrational numbers are the real numbers that cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio, such as p/q, where p and q are integers, q≠0. It is a contradiction of rational numbers

Irrational numbers are expressed usually in the form of R\Q, where the backward slash symbol denotes ‘set minus’. it can also be expressed as R – Q, which states the difference of set of real numbers and set of rational numbers.

The calculations based on these numbers are a bit complicated. For example, √5, √11, √21, etc., are irrational. If such numbers are used in arithmetic operations, then first we need to evaluate the values under root. These values could be sometimes recurring also. Now let us find out its definition, lists of irrational numbers, how to find them, etc., in this article.

Table of Contents:

Irrational Numbers Definition

An irrational number is a real number that cannot be expressed as a ratio of integers, for example, √ 2 is an irrational number. Again, the decimal expansion of an irrational number is neither terminating nor recurring

How do you know a number is irrational? 

The real numbers which cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0 are known as irrational numbers. For Example  √ 2 and √ 3 etc. are irrational. Whereas any number which can be represented in the form of p/q, such that, p and q are integers and q ≠ 0 is known as a rational number.

Is Pi an irrational number?

Pi (π) is an irrational number because it is non-terminating. The approximate value of pi is 22/7.

Symbol

Generally, the symbol used to represent the irrational symbol is “P”.  Since the irrational numbers are defined negatively, the set of real numbers (R) that are not the rational number (Q), is called an irrational number. The symbol P is often used because of the association with the real and rational number. (i.e) because of the alphabetic sequence P, Q, R. But mostly, it is represented using the set difference of the real minus rationals, in a way R- Q or R\Q.

Properties

The following are the properties of irrational numbers:

  • The addition of an irrational number and a rational number gives an irrational number.  For example, let us assume that x is an irrational number, y is a rational number, and the addition of both the numbers x +y gives a rational number z.
  • Multiplication of any irrational number with any nonzero rational number results in an irrational number. Let us assume that if xy=z is rational, then x =z/y is rational, contradicting the assumption that x is irrational. Thus, the product xy must be irrational.
  • The least common multiple (LCM) of any two irrational numbers may or may not exist.
  • 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.
  • The set of irrational numbers is not closed under multiplication process, unlike the set of rational numbers.

List of Irrational Numbers

The famous irrational numbers consist of Pi, Euler’s number, Golden ratio. Many square roots and cube roots numbers are also irrational, but not all of them. For example, √3 is an irrational number but √4 is is a rational number. Because 4 is a perfect square, such as 4 = 2 x 2 and √4 = 2, which is a rational number.

Pi, π  3.14159265358979…
Euler’s Number, e 2.71828182845904…
Golden ratio, φ 1.61803398874989….

Irrational Number Proof

The following theorem is used to prove the above statement

Theorem: 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.

Proof: Using the Fundamental Theorem of Arithmetic, the positive integer can be expressed in the form of the product of its primes as:

a = p1 × p× p3………..  × pn …..(1)

Where, p1, p2p3, ……, pn represent all the prime factors of a.

Squaring both the sides of equation (1),

a2 = ( p1 × p× p3………..  × pn) ( p1 × p2 × p3………..  × pn)

⇒a2 = (p1)2 × (p2)2 × (p3 )2………..× (pn)2

According to the Fundamental Theorem of Arithmetic, the prime factorization of a natural number is unique, except for the order of its factors.

The only prime factors of a2 are p1, p2, p3……….., pn. If p is a prime number and a factor of a2, then p is one of  p1, p2 , p3……….., pn. So, p will also be a factor of a.

Hence, if a2  is divisible by p, then p also divides a.

Now, using this theorem, we can prove that 2 is irrational.

How to Find an Irrational Number?

Let us find the irrational numbers between 2 and 3.
We know, square root of 4 is 2; √4 =2
and the square root of 9 is 3; √9 = 3
Therefore, the number of irrational numbers between 2 and 3 are 5, 6, 7, and 8, as these are not perfect squares and cannot be simplified further. Similarly, you can also find the irrational numbers, between any other two perfect square numbers.

Another case:

Let us assume a case of 2. Now, how can we find if 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-prime numbers are those numbers whose common factor is 1).

Squaring both the sides of equation (1), we have

2 = p2/q2

⇒ p2 = 2 q 2    ………. (2)

From the theorem stated above, if 2 is a prime factor of p2, then 2 is also a prime factor of p.

So, = 2 × c, where c is an integer.

Substituting this value of p in equation (3), we have

(2c)2 = 2 q 2

⇒ q2 = 2c 2 

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.

Since according to 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, root 2 is irrational.

Similarly, we can justify the statement discussed in the beginning that if p is a prime number, then  p  is an irrational number. Similarly, it can be proved that for any prime number p, p is irrational.

Problems and Solutions

Question 1: Which of the following are Rational Numbers or Irrational Numbers?

2, -.45678…, 6.5,  3,  2

Solution: Rational Numbers – 2, 6.5 as these have terminating decimals.

Irrational Numbers – -.45678…,  3,  2 as these have a non-terminating non-repeating decimal expansion.

Question 2: Check if below numbers are rational or irrational. 

2, 5/11, -5.12, 0.31 

Solution: Since the decimal expansion of a rational number either terminates or repeats. So, 2, 5/11, -5.12, 0.31 are all rational numbers.

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Frequently Asked Questions – FAQs

What is an irrational number? Give an example.

An irrational number is a type of real number which cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio. If N is irrational, then N is not equal to p/q where p and q are integers and q is not equal to 0.
Example: √2, √3, √5, √11, √21, π(Pi) are all irrational.

Are integers irrational numbers?

Integers are rational numbers but not irrational. All the integers whether they are positive or negative or zero can be written in the form of p/q.
Example: 2, 3 and 5 are rational numbers because we can represent them as 2/1, 3/1 and 5/1.

Is an irrational number a real number?

Yes, an irrational number is a real number and not a complex number, because it is possible to represent these numbers in the number line.

What are the five examples of irrational numbers?

There are many irrational numbers that cannot be written in simplified form. Some of the examples are:
√8, √11, √50, Euler’s Number e = 2.718281, Golden ratio, φ= 1.618034.

What are the main irrational numbers?

The most common irrational numbers are:
Pi (π) = 22/7 = 3.14159265358979…
Euler’s Number, e = 2.71828182845904…
Golden ratio, φ = 1.61803398874989….
Root, √ = √2, √3, √5, √7, √8, any number under root which cannot be further simplified.

2 Comments

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