Prove That √7 is an Irrational Number

By BYJU'S Exam Prep

Updated on: September 25th, 2023

Let us consider that √7 is a rational number. So it can be expressed in the form p/q, where p, q are co-prime integers and q is not equal to 0

√7 = p/q

In this case, q is not equal to zero and p and q are co-prime numbers.

√7 = p/q

On squaring on both sides

(√7)2 = (p/q)2

7 = (p/q)2

7 = p2/q2

On simplifying we get

7q2 = p2 ….. (1)

p2/7 = q2

So 7 divides p ad p and q are multiple of 7

p = 7m

p2 = 49 m2 …. (2)

From equations (1) and (2) we get

7q2 = 49m2

q2 = 7m2

q2 is a multiple of 7

q is a multiple of 7

So p, q have a common factor 7. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√7 is an irrational number.

Irrational Number

Real numbers that are irrational cannot be expressed using straightforward fractions. A ratio, such as p/q, where p and q are integers, q is not equal to 0, cannot be used to indicate an irrational number. It defies logic in terms of numbers. Ordinarily, irrational numbers are written as RQ, where the backward slash symbol stands for set minus. The difference between a set of real numbers and a set of rational numbers can alternatively be written as R – Q.

Therefore, it is proved that √7 is an irrational number.


Prove That √7 is an Irrational Number

It is proved that √7 is an irrational number. 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.

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