## Chapter 1 Number Systems Exercise 1.5

Question 1: Classify the following as rational or irrational.

i) 2 - √5

ii) (3 + √23) - √23

iii) 2√7/7√7

iv) 1/√2

v) 2π

Answer:

i) Irrational; The difference of a rational and an irrational number is irrational.

ii) Rational; (3 + √23) - √23 = 3 + √23 - √23 = 3, which is a rational number

iii) Rational; 2√7/7√7 = 2/7, which is a rational number.

iv) Irrational; The division of a rational and an irrational number is irrational.

v) Irrational; The product of a rational and an irrational number is irrational.

Question 2: Simplify each of the following expressions:

i) (3 + √3)(2 + √2)

ii) (3 + √3)(3 - √3)

iii) (√5 + √2)²

iv) (√5 - √2)(√5 + √2)

Answer:

i)

= (3 + √3)(2 + √2)

= (3 × 2) + (3 × √2) + (√3 × 2) + (√3 × √6)

= 6 + 3√2 + 2√3 + √18

ii)

= (3 + √3)(3 - √3)

by (a + b)(a - b) = a² - b²

a = 3

b = √3

= (3)² - (√3)²

= 9 - 3

= 6

iii)

= (√5 + √2)²

by (a + b)² = a² + 2ab + b²

a = √5

b = √2

= (√5)² + 2(√5)(√2) + (√2)²

= 5 + 2√10 + 2

= 7 + 2√10

iv)

= (√5 - √2)(√5 + √2)

by (a + b)(a - b) = a² - b²

a = √5

b = √2

= (√5)² - (√2)²

= 5 - 2

= 3

Question 3: Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, π = c/d. This seems to contradict the fact that π is irrational. How will you resolve this contradiction?

Answer: The ratio of circumference to the diameter of a circle is constant represented by π whose approximate value is 3.14… or 22/7.

Question 4: Represent √9.3 on the number line.

Answer:

Question 5: Rationalise the denominators of the following:

i) 1/√7

ii) 1/√7 - √6

iii) 1/√5 + √2

iv) 1/√7 - 2

Answer:

i)

= 1/√7

= 1/√7 × √7/√7

= √7/7

ii)

= 1/√7 - √6

= 1/√7 - √6 × √7 + √6/√7 + √6

= √7 + √6/(√7)² - (√6)²

= √7 + √6/1

= √7 + √6

iii)

= 1/√5 + √2

= 1/√5 + √2 × √5 - √2/√5 - √2

= √5 - √2/(√5)² - (√2)²

= √5 - √2/5 - 2

= √5 - √2/3

iv)

= 1/√7 - 2

= 1/√7 - 2 × √7 + 2/√7 + 2

= √7 + 2/(√7)² - (2)²

= √7 + 2/3

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