Name: Composite Number as Product of Primes
Uploaded: 2019-07-19
Duration: 6 min 23 s
Description: How do you write number 81? Number 81 can be written as 3x3x3x3, which cannot be divided further. What is this called? Explore more about it in this video.

Question

Prove that

\sqrt{7}

is an irrational number.

Accepted Answer

We need to prove, $\sqrt{7}$ is an irrational number.

Consider $\sqrt{7}$ is a rational number and it can be expressed in the form $\frac{p}{q}$ , where $p$ and $q$ are both integers and co-primes, i.e.,

There is no common factor in $p$ and $q$ except $1 (q \neq 0)$ .

$∴ \sqrt{7} = \frac{p}{q} \Rightarrow p = \sqrt{7} q$

$\Rightarrow p^{2} = 7 q^{2} \Rightarrow p^{2}$ is an even number $\Rightarrow p$ is an even number.

So, let $p = 7 m$ , where $m$ is an integer

$\Rightarrow p^{2} = {(7 m)}^{2} \Rightarrow 7 q^{2} = 49 m^{2} [∵ p^{2} = 7 q^{2}]$

$\Rightarrow q^{2} = 7 m^{2} \Rightarrow q^{2}$ is an even number $\Rightarrow q$ is an even number.

Thus, $p$ and $q$ are both even numbers. So, there exists $7$ as the common factor of both $p$ and $q$ .

But, we have let that there is no common factor in $p$ and $q$ , except $1$ .

This is a contradiction.

$∴ \sqrt{7}$ is not a rational number.

Hence, $\sqrt{7}$ is an irrational number.

Prove that $\sqrt{7}$ is an irrational number.

Important Questions on Real Numbers

Learn and Prepare for any exam you want !

Prove that 7 is an irrational number.

Important Questions on Real Numbers

Learn and Prepare for any exam you want !

Prove that $\sqrt{7}$ is an irrational number.