Prove that 3 + 2√5 is irrational?

By BYJU'S Exam Prep

Updated on: October 17th, 2023


Let us assume that the given number 3 + 2√5 is a rational number.  It can therefore be expressed in the form a/b. 3 + 2√5 = a/b. Here, b is not equal to zero and a and b are coprime numbers. So 3 + 2√5 = a/b.

we get 2√5 = a/b – 3

2√5 = (a – 3b)/b

√5 = (a – 3b)/2b.

The above result 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.

Irrational Number Characteristics

An irrational number results from adding an irrational number to a rational number. As an illustration, suppose that x is an irrational number, y is a rational number, and the sum of both is a rational number, z. Below mentioned points elaborate on the characteristics of Irrational Numbers.

  • Any irrational number multiplied by any non-zero rational number yields an irrational number. Let’s assume that, contrary to the presumption that x is irrational, x = z/y is rational if xy=z is rational. Consequently, the xy product must be illogical.
  • There could or might not be a least common multiple (LCM) between any two irrational numbers.
  • Two irrational numbers can be rationally added together or multiplied; for instance, √2. √2 = 2. In this case, the number 2 is irrational. Once it has been multiplied twice, the result is a rational number. (i.e.) 2. 
  • In contrast to the set of rational numbers, the set of irrational numbers is not closed under the multiplication operation.


Prove that 3 + 2√5 is irrational

It is proved that 3 + 2√5 is an irrational Number. To prove this number is an Irrational number, you have to suppose that it is a rational number. The rest calculation is explained above.

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