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

Proof

We prove this by contradiction.
Let be a rational number and p an irrational number.

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

Multiplying both sides by :

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

So our assumption ( product of a rational number with an irrational number p is a rational number ) is false. Therefore the result of this product is an irrational number.

✩ Known Irrationals

  1. Square Root of Prime (): √2, √3, √5, √7, √11, √13, √17, √19 …
  2. Special Numbers: Pi ( π ) , Euler’s number ( e ), Golden Ratio
  3. Logarithms of primes with prime base: log23, log35

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 is rational
  • √3 is irrational
  • Product is irrational

Example 2

  • is rational
  • √5 is irrational
  • Product is irrational

Example 3

is rational, is irrational

Their product is irrational.

Example 4

Following products are irrational because π is an irrational number.

  • 2 × π = 2π

✩ Irrational Result of Operations

Following operations between rational and irrational numbers result in an irrational number. Whatever the order of operations, the outcome is always an irrational number.

  1. Rational + Irrational: [ 3 + √2 ], [ 4 + √7 ], …
  2. Rational Irrational: [ 5 – √2 ], [ √3 – 6 ], …
  3. Non zero rational × Irrational: [ 4 × π = 4π ], [ 6 × √3 = 6√3 ], …
  4. Non zero rational ÷ Irrational: [ 2 ÷ √2 ], [ π ÷ 2 ], …

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

= 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

is rational or irrational?

Answer

Is 31 a prime number?

2 Question

is rational or irrational?

Answer

Can you multiply the numbers?

3 Question

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

Answer

Can you apply the formula (a − b)(a + b) = a2 − b2?

4 Question

Prove that the area of circle with radius is a rational number.

Answer

Can you find the area using the formula Area = πr2?

5 Question

Given , where x, y are rational numbers. Find x, y.

Answer

Can you rationalize the denominator?

Answers

1 Answer

31 is a prime number.

Square root of prime is irrational, so its product with 2 is also irrational.

2 Answer

On multiplication we get:

4√5 × √5 = 4 × 5 = 20

This is a rational number.

3 Answer

Applying the formula for difference of squares, we get:

(4 − √7)(4 + √7) = 42 − (√7)2

= 16 − 7 = 9

The result is a rational number.

4 Answer

The area of a circle = πr2. Applying this formula:

Area

The result is a rational number.

5 Answer

Let us rationalize the denominator by multiplying the left side of the equation with :

−9 − 2√5 = x + y√5

⇒ x = − 9, y = − 2


Related

Examples of Irrational Numbers (With Lists) ➤

Square root of a prime number is Irrational ➤

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