Name: Applications of Euclid's Division Lemma
Uploaded: 2019-07-19
Duration: 18 min 17 s
Description: Can you prove that only one out of three consecutive positive integers is divisible by 3? Watch how, along with more examples of the applications of Euclid's...

Question

Prove that $\sqrt{5}$ is a irrational number and hence show that $3 + \sqrt{5}$ is an irrational number.

Accepted Answer

Let $\sqrt{5}$ be a rational number, then $\sqrt{5} = \frac{p}{q}, where p, q$ are integers, $q \neq 0 and p, q$ have no common factors.

$\Rightarrow 5 = \frac{p^{2}}{q^{2}} \Rightarrow p^{2} = 5 q^{2} . . . . (i)$

As $5$ divides $5 q^{2}$ , so $5$ divides $p^{2}$ but $5$ is prime.

$\Rightarrow 5$ divides $p$

Let $p = 5 m, where m$ is an integer.

Substituting this value of $p$ in $(i)$ , we get

${(5 m)}^{2} = 5 q^{2} \Rightarrow 25 m^{2} = 5 q^{2} \Rightarrow 5 m^{2} = q^{2}$

As $5$ divides $5 m^{2}$ , so $5$ divides $q^{2}$ but $5$ is prime.

$\Rightarrow 5$ divides $q$

Thus, $p and q$ have a common factor $5$ .This contradicts that $p a n d q$ have no common factors (except $1$ )

Hence, $\sqrt{5}$ is not a rational number. So, we conclude that $\sqrt{5}$ is an irrational number.

Now in $3 + \sqrt{5}$ , $3$ is rational and $\sqrt{5}$ is a irrational number.

Also we know the sum or difference of a rational and irrational number is always irrational.

Hence, $3 + \sqrt{5}$ is an irrational number.

Prove that $\sqrt{5}$ is a irrational number and hence show that $3 + \sqrt{5}$ is an irrational number.

Important Questions on Real Numbers

Learn and Prepare for any exam you want !

Prove that 5 is a irrational number and hence show that 3+5 is an irrational number.

Important Questions on Real Numbers

Learn and Prepare for any exam you want !

Prove that $\sqrt{5}$ is a irrational number and hence show that $3 + \sqrt{5}$ is an irrational number.