Question

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Question- Prove that 3+2√5 is irrational.​

Answer 1

Step-by-step explanation:

Given:-

3+2√5

To find:-

Prove that 3+2√5 is irrational..

Solution:-

Given that 3+ 2√5

Let us assume that

3+2√5 is an irrational number

It must be in the form of p/q

Let 3+2√5 = a/b

Where a and b are co - primes

=> 3+2√5 = a/b

=> 2√5 = (a/b)-3

=> 2√5 = (a-3b)/3

=> √5 = [(a-3b)/3]/2

=> √5 = (a-3b)/(3×2)

=> √5 = (a-3b)/6

=> √5 is in the form of p/q

By the definition of rational numbers

=>√5 is a rational number.

But √5 is not a rational number

We get a contradiction to our assumption that is

3+2√5 is a rational number.

So, 3+2√5 is not a rational number

3+2√5 is an irrational number.

Hence, Proved

Used Method:-

Indirect method or Method of Contradiction

• The numbers in the form of p/q are rational numbers ,where p and q are integers and q≠0
• The numbers are not in the form of p/q are irrational numbers .
• The natural number n ,then √n is an irrational number where n is prime number.

Answer 2

Answer:

Prove that 3 + 2√5 is irrational

Given: 3 + 2√5

To prove: 3 + 2√5 is an irrational number.

Proof:

Let us assume that 3 + 2√5 is a rational number.

So, it can be written in the form a/b

3 + 2√5 = a/b

Here a and b are coprime numbers and b ≠ 0

Solving 3 + 2√5 = a/b we get,

=>2√5 = a/b – 3

=>2√5 = (a-3b)/b

=>√5 = (a-3b)/2b

This shows (a-3b)/2b is a rational number. But we know that √5 is an irrational number.

So, it contradicts our assumption. Our assumption of 3 + 2√5 is a rational number is incorrect.

3 + 2√5 is an irrational number

Hence proved

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