Prove that Root 2 + root 5 is Irrational

By K Balaji|Updated : November 7th, 2022

We need to prove √2 + √5 is irrational

Assume that √2 + √5 is rational

√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 rational number but LHS is irrational

So our assumption is wrong

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 ending 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 the rational number (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

Summary:-

Prove that Root 2 + root 5 is Irrational

It is proved that root 2 + root 5 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. Generally, the symbol used to represent the irrational symbol is “P”.

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