Product of rational & irrational numbers is irrational (Proof + Questions)

Proof

We prove this by contradiction.
Let $\frac{x}{y}$ be a rational number and p an irrational number.

Let us assume that product of these numbers is a rational number $\frac{a}{b}$. We are hoping to get a contradiction due to this assumption. The assumption results in the following equation:

$\frac{x}{y}\times p=\frac{a}{b}$

Multiplying both sides by $\frac{y}{x}$:

$p=\frac{a}{b}\times \frac{y}{x}=\frac{ay}{bx}$

The equation expresses $p$ as a product of two rational numbers. So it must be rational. This result contradicts the fact that $p$ is an irrational number.

So our assumption ( product of a rational number $\frac{x}{y}$ with an irrational number p is a rational number ) is false. Therefore the result of this product is an irrational number.

Examples

The outcome of multiplication in the following examples is an irrational number. Why?

One of the numbers in multiplication is a square root of prime number, which is irrational.

Example 1

$2\times \sqrt{3}=2\sqrt{3}$

• 2 is rational
• √3 is irrational
• Product is irrational

Example 2

$\frac{3}{5}\times \sqrt{5}=\frac{3\sqrt{5}}{5}$

• $\frac{3}{5}$ is rational
• √5 is irrational
• Product is irrational

Example 3

$(\frac{29}{21}+\frac{5}{3})\times \sqrt{11}$

$=(\frac{29+35}{21})\times \sqrt{11}$

$=\frac{64}{21}\sqrt{11}$

$\frac{64}{21}$ is rational, $\sqrt{11}$ is irrational

Their product is irrational.

Example 4

Following products are irrational because π is an irrational number.

• $\frac{1}{2}\times \pi =\frac{\pi }{2}$
• $\frac{3}{2}\times \pi =\frac{3\pi }{2}$
• 2 × π = 2π

Product of two irrational numbers

We know that product of two rational numbers is rational. The only remaining case is the product of two irrational numbers. In this case, the resulting number may be rational or irrational, depending on the multiplicand and multiplier.

Let us see some examples.

Example 5

√2 × √6

$=\sqrt{12}$

= 2√3

This product is an irrational number due to the presence of the square root of √3, which is an irrational number.

Example 6

√3 × √3

= √9

= 3

3 is a rational number. The product of two irrational numbers can be rational.

Example 7

(2 + √3) × (2 − √3)

The resulting number in each bracket above is a sum of a rational and an irrational number. So it is an irrational number. Let us see what happens to the product.

(2 + √3) × (2 − √3)

= 4 − 2√3 + 2√3 − (√3)²

= 4 − 3 = 1

The product is a rational number!

Questions

1 Question

$2\sqrt{31}$ is rational or irrational?

Answer

2 Question

$\sqrt{80}\times \sqrt{5}$ is rational or irrational?

Answer

3 Question

Find the product (4 − √7)(4 + √7). Is it rational?

Answer

4 Question

Prove that the area of circle with radius $\sqrt{\frac{3}{\pi }}$ is a rational number.

Answer

5 Question

Given $\frac{2+\sqrt{5}}{2-\sqrt{5}}=x+y\sqrt{5}$, where x, y are rational numbers. Find x, y.

Answer

Answers

1 Answer

2 Answer

3 Answer

4 Answer

5 Answer

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Related

Examples of Irrational Numbers (With Lists) ➤

Square root of a prime number is Irrational ➤

Sum of rational & irrational number is irrational (Proof) ➤