Question

show that√5+√3 is an irrational number

help me with this and please don't answer in this way( yes it is an irrational number)​

Answer 1

Answer:

If a number can't be written as p/q(q is not equal to zero) then this number is called irrational number. Another definition is if a number can't be rounded after decimal point then it is also. like as π=3.14159……., Pie=π can't be rounded after decimal point and it can't be written as p/q form.

Answer 2

Step-by-step explanation:

Given :-

√5+√3

To find :-

Show that √5+√3 is an irrational number.

Solution :-

Given number = √5+√3

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

=> √5= (a/b)-√3

On squaring both sides then

=>(√5)² = [(a/b)-√3]²

=> 5 = (a/b)²-2(a/b)(√3)+(√3)²

=> 5 = (a²/b²)-(2√3a/b)+3

=> 5-3 = (a²/b²)-(2√3a/b)

=> 2 = (a²/b²)-(2√3a/b)

=> 2 = (a²-2√3ab)/b²

=> 2b² = a²-2√3ab

=> 2b²-a² = -2√3ab

=> a²-2b² = 2√3 ab

=> (a²-2b²)/(ab) = 2√3

=> √3 = (a²-2b²)/(2ab)

=> √3 is in the form of p/q

=> √3 is an irrational number

But √3 is not a rational number.

This contradicts to our assumption.

=> √5+√3 is not a rational number.

√5+√3 is an irrational number.

Hence, Proved.

Answer :-

√5+√3 is an irrational number.

Used Method:-

Method of Contradiction ( Indirect method)

Note :-

The sum of two irrational numbers is always an irrational number.

√3 and √5 are irrational numbers then their sum √3+√5 is an irrational number.

Points to know:-

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

• q+s,q-s ,qs and q/s are irrational numbers.