Irrational Numbers

Irrational numbers are real numbers that cannot be represented as simple fractions. An irrational number cannot be expressed as a ratio, such as p/q, where p and q are integers, q≠0. It is a contradiction of rational numbers. Irrational numbers are usually expressed as R\Q, where the backward slash symbol denotes ‘set minus’. It can also be expressed as R – Q, which states the difference between a set of real numbers and a 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 the root. Now let us find out its definition, lists of irrational numbers, how to find them, etc., in this article.

What are Irrational Numbers?

An irrational number is a real number that cannot be expressed as a ratio of integers; for example, √2 is an irrational number.  We cannot express any irrational number in the form of a ratio, such as p/q, where p and q are integers, q≠0. Again, the decimal expansion of an irrational number is neither terminating nor recurring. 

Irrational Meaning: The meaning of irrational is not having a ratio or no ratio can be written for that number. In other words, we can say that irrational numbers cannot be represented as the ratio of two integers.

What are the examples of Irrational Numbers?

The common examples of irrational numbers are pi(π=3⋅14159265…), √2, √3, √5, Euler’s number (e = 2⋅718281…..), 2.010010001….,etc.

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?

Yes, Pi (π) is an irrational number because it is neither terminating nor repeating decimal. Also, Pi is not equal to 22/7 as 22/7 is a rational number while pi is an irrational number. The value of π is 3.141592653589………..

Note- Rational numbers (Q) and Irrational numbers (P or Q’ ) are always alternate with each other.

Therefore, 22/7 ≠  π  but they are alternate or next to each other.

Irrational Number Symbol

Generally, the symbol used to represent the irrational symbol is “P”.  Since 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 of Irrational numbers

Since irrational numbers are the subsets of real numbers, irrational numbers will obey all the properties of the real number system. 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 an irrational 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 the multiplication process, unlike the set of rational numbers.

List of Irrational Numbers

The famous irrational numbers consist of Pi, Euler’s number, and Golden ratio. Many square roots and cube root numbers are also irrational, but not all of them. For example, √3 is an irrational number, but √4 is a rational number. Because 4 is a perfect square, such as 4 = 2 x 2 and √4 = 2, which is a rational number.  It should be noted that there are infinite irrational numbers between any two real numbers. For example, say 1 and 2, there are infinitely many irrational numbers between 1 and 2. Now, let us look at famous irrational numbers’ values.

Note – √prime number always gives an irrational number.

Are Irrational Numbers Real Numbers?

In Mathematics, all irrational numbers are considered real numbers, which should not be rational numbers. It means irrational numbers cannot be expressed as the ratio of two numbers. For example, the square roots that are not perfect will always result in an irrational number. 

Sum and Product of Two Irrational Numbers

Now, let us discuss the sum and the product of irrational numbers.

Product of Two Irrational Numbers

Statement: The product of two irrational numbers is sometimes rational or irrational

For example, √2 is an irrational number, but when √2  is multiplied by √2, we get the result 2, which is a rational number.

(i.e.,) √2 x √2 = 2 

We know that π is also an irrational number, but if π is multiplied by π, the result is π², which is also an irrational number.

(i.e..) π x π = π²

It should be noted that while multiplying the two irrational numbers, it may result in an irrational number or a rational number. 

Sum of Two Irrational Numbers

Statement: The sum of two irrational numbers may be rational or irrational.

Like the product of two irrational numbers, the sum of two irrational numbers will also result in a rational or irrational number.

For example, if we add two irrational numbers, say 3√2+ 4√3, a sum is an irrational number.

But, let us consider another example, (3+4√2) + (-4√2 ), the sum is 3, which is a rational number.

So, we should be very careful while adding and multiplying two irrational numbers, because it might result in an irrational number or a rational number.

Irrational Number Theorem and 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 × p₂ × p_3………..  × p_{n …..(1)}

Where, p_1, p_2, p₃, _……, p_n represent all the prime factors of a.

Squaring both the sides of equation (1),

a² = ( p₁ × p₂ × p_3………..  × p_{n) (} p₁ × p₂ × p₃………..  × p_n)

⇒a² = (p₁)² × (p₂)² × (^p₃ )^{2_………..}× (p_n)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, p_2, p_{3………..,} p_n. If p is a prime number and a factor of a², then p is one of  p₁, p_{2 ,} p_{3………..,} p_n. 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 the 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 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 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, p = 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.

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, 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.

Irrational Numbers Solved Examples

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 the 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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