Prove that Root 2 + Root 5 is Irrational

By BYJU'S Exam Prep

Updated on: September 25th, 2023

To prove √2 + √5 is irrational, we will be assuming it to be rational first. By solving this equation, we will be proving that √2 + √5 is irrational indeed. An irrational number refers to a real number that cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0. “P” is the symbol that is used to represent the irrational number in mathematics.

Table of content

1. Steps to Prove √2 + √5 is Irrational
2. Irrational Number
3. Symbol of Irrational Number
4. Prove that Root 2 + root 5 is Irrational

Steps to Prove √2 + √5 is Irrational

To prove that root 2 + root 5 is irrational, we need to first assume that √2 + √5 is rational in nature.

√2 + √5 = a/b

Squaring on both sides, we get

(√2 + √5)² = (a/b)²

On simplifying we get

7 + 2√10 = a²/b²

√10 = ½ (a²/b² – 7) ….. (1)

From (1) RHS is a rational number but LHS is irrational

So our assumption is wrong. Thus, root √2 + √5 is irrational.

Irrational Number

A real number that cannot be stated as a ratio of integers is said to be irrational; an example of this is the number √2. Any irrational number, such as p/q, where p and q are integers, q≠0, cannot be expressed as a ratio. Once more, an irrational number’s decimal expansion is neither end nor recurrent.

Symbol of Irrational Number

The irrational symbol is typically represented by the letter P. The group of real numbers (R) that are not rational numbers (Q) are referred to as irrational numbers because they are defined negatively. Because P comes after Q and R in the alphabet, it is frequently used in conjunction with real and rational numbers. However, it is typically expressed as the set difference of the real minus rationals, denoted as R-Q or R\Q.

Hence, it is proved that √2 + √5 is irrational

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