9.2. Prove that V2+ V3 is an irrational number

1.	9.2. Prove that V2+ V3 is an irrational number
Answer» Step-by-step explanation: Given :- √2+√3 To Prove :- Prove that √2+√3 is an irrational number? Solution:- Given number =√2+√3 Let us ASSUME that √2+√3 is a RATIONAL number Then it must be in the form of p/q where p and q are integers and q≠0 =>√2+√3=a/b,where a and b are co-primes =>√3=(a/b)-√2 On squaring both sides then =>(√3)²=[(a/b)-√2]² =>3=(a/b)²-2(a/b)(√2)+(√2)² =>3=(a²/b²) - 2√2a/b+2 =>(a²/b²)-2√2a/b=3-2 =>(a²/b²)-2√2a/b=1 =>2√2a/b=(a²/b²)-1 =>2√2a/b=(a²-b²)/b² =>2√2=(a²-b²)(b)/ab² =>2√2=(a²-b²)/ab =>√2=(a2-b²)/(2AB) √2 is the form of a/b =>√2 is a rational number. But √2 is not a rational number . This is contradiction to our assumption. √2+√3 is not a rational number. √2+√3 is an irrational number. Hence, Proved Used method:- Method of Contradiction (Indirect method) Note :- SUM of any two irrational Numbers is also an irrational number. √2=1.414... √3=1.732... √2+√3=1.414...+1.732...=3.146...is an irrational number.

Answer»

Step-by-step explanation:

Given :-

√2+√3

To Prove :-

Prove that √2+√3 is an irrational number?

Solution:-

Given number =√2+√3

Let us ASSUME that

√2+√3 is a RATIONAL number

Then it must be in the form of p/q

where p and q are integers and q≠0

=>√2+√3=a/b,where a and b are co-primes

=>√3=(a/b)-√2

On squaring both sides then

=>(√3)²=[(a/b)-√2]²

=>3=(a/b)²-2(a/b)(√2)+(√2)²

=>3=(a²/b²) - 2√2a/b+2

=>(a²/b²)-2√2a/b=3-2

=>(a²/b²)-2√2a/b=1

=>2√2a/b=(a²/b²)-1

=>2√2a/b=(a²-b²)/b²

=>2√2=(a²-b²)(b)/ab²

=>2√2=(a²-b²)/ab

=>√2=(a2-b²)/(2AB)

√2 is the form of a/b

=>√2 is a rational number.

But √2 is not a rational number .

This is contradiction to our assumption.

√2+√3 is not a rational number.

√2+√3 is an irrational number.

Hence, Proved

Used method:-

Method of Contradiction (Indirect method)

Note :-

SUM of any two irrational Numbers is also an irrational number.

√2=1.414...

√3=1.732...

√2+√3=1.414...+1.732...=3.146...is an irrational number.