Irrational Numbers

Irrational Numbers

Irrational Numbers- Introduction: 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 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 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.

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.

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

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. Also, the value of π is 3.14159 26535 89793 23846 264…

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

Since irrational numbers are the subsets of the 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 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 the 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 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 have a look at the values of famous irrational numbers.

Are Irrational Numbers Real Numbers?

In Mathematics, all the irrational numbers are considered as real numbers, which should not be rational numbers. It means that irrational numbers cannot be expressed as the ratio of two numbers. The irrational numbers can be expressed in the form of non-terminating fractions and in different ways. For example, the square roots which are not perfect squares will always result in an irrational number.

Sum and Product of Two Irrational Numbers

Now, let us discuss the sum and the product of the 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 is sometimes 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 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

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, 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, 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 numbers 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.