Prove that 3 + 2√5 is irrational

Solution:

Irrational numbers are the subset of real numbers that cannot be expressed in the form of a fraction, p/q where p and q are integers and q ≠ 0.

We have to prove that 3 + 2√5  is irrational. We will be solving this question with the help of the contradiction method.

Let's assume that 3 + 2√5 is rational. If 3 + 2√5 is rational that means it can be written in the form of a/b where a and b are integers that have no common factor other than 1 and b ≠ 0.

3 + 2√5 = a/b

b(3 + 2√5) = a 

3b + 2√5b = a

2√5b = a - 3b

√5 = (a - 3b)/2b

Since (a - 3b)/2b is a rational number, then √5 is also a rational number.

But, we know that √5 is irrational. 

Therefore, our assumption was wrong that 3 + 2√5 is rational. Hence, 3 + 2√5 is irrational.

☛ Check: NCERT Solutions for Class 10 Maths Chapter 1

Video Solution:

Prove that 3 + 2√5 is irrational

NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.3 Question 2

Summary:

We have proved that 3 + 2√5 is irrational using the contradiction method.

☛ Related Questions:

• There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time and go in the same direction. After how many minutes will they meet again at the starting point?
• Prove that √5 is an irrational number.
• Prove that the following are irrationals: (i) 1/√2 (ii) 7√5 (iii) 6 + √2
• Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion: (i) 13/3125 (ii) 17/8 (iii) 64/455 (iv) 15/1600 (v) 29/343 (vi) 23/2352 (vii) 129/225775 (viii) 6/15 (ix) 35/50 (x) 77/210