Question

Show that 5√3 is irrational​

Answer 1

Answer:

Every natural number, every whole number, every integer and every fraction is a rational number.

Here 5√3 is not from above so it's not rational.

Or Let us assume the given number be rational and we will write the given number in p/q form

⇒

$5 - \sqrt{3 } = \frac{p}{q}$

⇒

$\sqrt{3} = \frac{5q - p}{q}$

We observe that LHS is irrational and RHS is rational, which is not possible.

This is contradiction.

Hence our assumption that given number is rational is false

⇒

$5 - \sqrt{3}$

is irrational

Answer 2

Step-by-step explanation:

Given :-

5-√3

To find:-

Show that 5 - √3 is an irrational number ?

Solution :-

Let us assume that 5-√3 is a rational number.

It must be in the form of p/q

Where p and q are integers and q≠0

Let 5-√3 = a/b

Where a and b are co-primes

=> -√3 = (a/b)-5

=> √3 = 5-(a/b)

=> √3 = (5b-a)/b

=> √3 is in the form of p/q

=> √3 is a rational number

But √3 is not a rational number

It is an irrational number.

This contradicts to our assumption that 5-√3 is a rational number.

So , 5-√3 is an irrational number.

Hence , Proved .

Answer:-

5-√3 is an irrational number.

Note :-

Let q is a rational number and s is an irrational number then

The difference of a rational and irrational number is an irrational number

So, 5 is a rational number

√3 is an irrational number

Their difference = 5-√3 is an irrational number.

Used Method:-

• Method of Contradiction (Indirect method)

Points to know :-

• The numbers in the form of p/q, where p and q are integers and q≠0 are called Rational numbers and they are denoted by Q.

• The numbers can not be written in the form of p/q, where p and q are integers and q≠0 are called Irrational numbers and they are denoted by Q' or S.

• Let q is a rational number and s is an irrational number then
• q+s is also an irrational number.

• q-s is also an irrational number

• q×s is also an irrational number.

• q/s is also an irrational number.