Theorems related to irrational numbers | Real Numbers | Assessment | English | Grade 10

Which of the following are rational numbers? 3, 9/7, √3, 1.010101.., 8√5

• 1.010101…, 3, 8√5

• 3, 9/7,√3

• 3 ,9/7, 1.010101…

• √3, 3,9/7

Answer explanation

Terminating decimals, Recurring decimals and fractions with denominator of the 2ᵐ x 5ⁿ form are rational numbers

The decimal expansion of an irrational number is ____

• Terminating and repeating

• Non terminating and repeating

• Terminating and non repeating

• Non terminating and non repeating

Answer explanation

An irrational number has a non-terminating and non-repeating decimal expansion

The sum of an irrational number and a rational number gives an irrational number.' This statement is _______

• Always true

• Sometimes true

• Never True

Answer explanation

To see why this is true, suppose x is irrational, y is rational, and the sum x+y is a rational number z. Then we have x = z-y, and since the difference of two rational numbers is rational, this implies x is rational. This is a contradiction since x is irrational. Therefore, the sum x+y must be irrational.

Which statement is true for √5/7?

• It is a Rational Number

• It is an Irrational Number

Answer explanation

Let √5/7 is a rational number. Therefore, we can find two integers a, b (b ≠ 0) such that √5/7 = a/b Let a and b have a common factor other than 1. Then we can divide them by the common factor, and assume that a and b are co-prime. a = √5/7 b a² = 5/49 b² Therefore, a² is divisible by 5/49 and it can be said that a is divisible by 5/49. Let a = 5/49k, where k is an integer (5/49k)² = 5/49b² b² = 5/49 k². This means that b² is divisible by 5/49 and hence, b is divisible by 5/49. This implies that a and b have 5/49 as a common factor. And this is a contradiction to the fact that a and b are co-prime. Hence, cannot be expressed as p/ q or it can be said that √5/7 is irrational.

Which statement is true for 7 + 2√5? Prove it.

• It is an Irrational Number

• It is a Rational Number

Answer explanation

It is irrational. PROOF: Let us assume 7 + 2√ 5 is rational.Therefore, we can find two integers a, b (b ≠ 0) such that 7 + 2√ 5 = a/b 2√ 5 = a/b -7 √ 5 = 1/2 {a/b -7} Since a and b are integers, 1/2 {a/b -7} will also be rational and therefore, √5 is rational. This contradicts the fact that √5 is irrational. Hence, our assumption that 7 + 2 √5 is rational is false. Therefore, 7 + 2 √5 is irrational