Solution:

  1. √5 and 5-√5
  2. √3+2 and 3-√3

a) π is an irrational number. ( True)

b) -√3 is an irrational number. (True)

c) Irrational numbers cannot be represented by points on the number line. (False)

d) All real number are rational ( False)

e) Every real number is not a rational number. (True)

Solution:

√2 = 1.41421356.......

Solution:

1) 0.75

2) -100

3) \(\frac{7}{20}\)

4) 0

Solution:

-6/25 ÷ 3/5

= -6/25 × 5/3

= {(-6) × 5}/(25 × 3)

= -30/75

= -2/5

Solution:

11/24 ÷ (-5)/8

= 11/24 × 8/(-5)

= (11 × 8)/{24 × (-5)}

= 88/-120

= -11/15

Solution:

(-25/9) × (-18/15)

= (-25) × (-18)/9 × 15

= 450/135

= 10/3

Solution:

(-11)/3 is not a positive rational. Since both the numerator and denominator are of the opposite sign.

Solution:

25/(-27) is not a positive rational. Since both the numerator and denominator are of the opposite sign.

Solution:

\(\frac{1}{2}\) and \(\frac{3}{5}\)

0.50 0.60

0.51 0.61

0.52 0.62

0.54 0.64

0.56 0.66

0.58 0.68

Solution:

1) \(\sqrt{3 }\) and 3 -\(\sqrt{3}\)

2) \(\sqrt{7 }\)+ 2 and 7 -\(\sqrt{7}\)

Solution:

1) \(\sqrt{5}\) and -\(\sqrt{5}\)

2) \(\sqrt{7 }\) and -\(\sqrt{7}\)

Solution:

( 2+\(\sqrt{3}\)) + ( 2 - \(\sqrt{3}\)) = 4, which is rational.

The sum of two irrational numbers is not always an irrational number.

Solution:

( 2 + \(\sqrt{3}\)) \(\times\) ( 2 - \(\sqrt{3}\))

= 22 + (\(\sqrt{3}\))2

= 4 - 3

= 1, which is rational.

So, the product of two irrational numbers is not always irrational numbers.

Solution:

a) \(\frac{4}{7}\) (Rational)

b) -\(\frac{2}{5}\) (Rational)

c) \(\sqrt{4}\) (Rational)

d) \(\sqrt[3]{8}\) (Rational)

e) \(\sqrt{3}\) (Irrational)

f) 2 - \(\sqrt{3}\) (Irrational)

g) - \(\sqrt{7}\) (Irrational)

h) \(\sqrt{25}\) (Rational)

i) 0.75 (Rational)

Solution:

9/16 ÷ 5/8

= 9/16 × 8/5

= (9 × 8)/(16 × 5)

= 72/80

= 9/10

Solution:

Let the other number be x.

Then, x × (-4)/9 = -28/27

or, x = (-28)/27 ÷ (-4)/9

or, x = (-28)/27 × 9/-4

or, x = {(-28) × 9}/{27 × (-4)}

or, x = -(28 × 9)/-(27 × 4)

or,x =(287×91)/(273×41)

\(\therefore\)x = 7/3

Hence, the other number is 7/3.

Solution:

6/11 × (-55)/36

= 6 × -55/11 × 36

= -330/396

= -5/6

Solution:

The additive inverse of 2/5 is -2/5

4/7 - 2/5

= 4/7 + (-2/5)

= 4 × 5/7 × 5 + (-2) × 7/5 × 7

= 20/35 + -14/35

= 20 + (-14)/35

= 6/35