Prove that √7 is an Irrational Number

Answer: A real figure that cannot be stated as a ratio of numbers is indeed an irrational number; for example,  √ 2 seems to be an irrational number. Again, an irrational number’s fractional expansion is not terminated or repeated.

Proof of √7 as an irrational number

Let us suppose √7 seems to be a rational number.

√7 =p/q, for example. If p/q have a common component, we divide by it to get√7 = a/b, where a & b are co-prime numbers. That is a and b share no factor. √7  is a co-prime number (a/b). √7 =a/b a=7b squared a²=7b².

1 a² is divisible by 7 a=7c by substitution numbers in 1 (7c)²=7b² 49c²=7b² 7c²=b² b²=7c² b² is divided by 7 a and b have at least a single factor 7. This is because being a and b share no factor. This is due to our incorrect assumption. The number √7 is irrational.