# Free Number Systems 02 Practice Test - 9th Grade

Which of the following statements is correct?

A.

0 is a natural number

B.

-1 is a whole number

C.

6.5 is an integer

D.

1 is a whole number

#### SOLUTION

Solution : D

Natural numbers start from 1 and continue thereafter by adding 1 each time.

So 1, 2,3,4,5... are all natural numbers.

All the natural numbers and "0" together are referred to as whole numbers. Hence 1 is a whole number.

Natural numbers, their negatives and 0 constitute the set of Integers.

The numbers which have a non-terminating and non-repeating decimal expansion or cannot be represented in the form of pq (q is not equal to '0') are known asnumbers.

#### SOLUTION

Solution :

An irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction, where the denominator is different from zero. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat. 2, 3 are examples of irrational numbers.

Which of the following is a rational number?

A.

2

B.

π

C.

20

D.

4.8

#### SOLUTION

Solution : D

A rational number can be written in the form of pq where p and q are integers and q is not equal to zero. Rational numbers have their decimal expansions as either terminating or non-terminating but recurring.

π = 3.14159265358..., i.e., the digits after decimal point do not terminate and not recurring. Hence it is not rational

2 is approximated to 1.414213562373095..., i.e., it cannot be expressed as a recurring and non-terminating decimal and hence is not rational.

20 is not defined and for a rational number, the denominator should be non-zero.

4.8 on the other hand, satisfies all the properties of rational numbers.

Which of the following is a rational number between 14 and 13

A.

0

B.

18

C.

725

D.

15

#### SOLUTION

Solution : C

The decimal expansion of  14 is 0.25 and the decimal expansion of 13 is 0.333...

Now comparing with the decimal expansion 0.25 and 0.333... with the options:
a) Clearly, 0 is out of the range.
b) The decimal expansion of 18 is 0.125 which is also out of the range.
c) The decimal expansion of 725 is 0.28, so it lies between 14 and 13 .
d) The decimal expansion of 15 is 0.2, so it is out of the range.

1.666666...... is a rational number and can be expressed in pq form. Then p + q is ______.

A.

9

B.

6

C.

7

D.

8

#### SOLUTION

Solution : D

Let x = 1.6666......  ----- (i)
then, 10x = 16.666666 ----- (ii)

Subtracting (i) from (ii), we get
10x - x = (16.666666...) - (1.666666...)
9x = 15
Hence x = 159=15393=53
i.e., p = 5 and q = 3
So, p + q = 8.

Which of the following statements are correct?

A.

The product of two rational numbers is always a rational number

B.

The sum of two irrational numbers is always an irrational number

C.

Irrational numbers form part of the number line

D.

The sum or difference of a rational and an irrational numbers is irrational

#### SOLUTION

Solution : A, C, and D

Multiplication of two rational numbers is always a rational number.
This is because every rational number can be expressed as a pq form. Therefore, multiplying two rational numbers would result in multiplying two numbers in their pq forms, with the denominators not equal to 0, in each case. This would result in another number which is also in its pq form, with both the numerator and denominator being integers.

The sum of two irrational numbers need not always be irrational.
For example, consider 2+3 and 23, both of which are irrational numbers. Adding these gives us 4, which is a rational number.

Irrational numbers are real numbers too. Since we know that the number line constitutes all the real numbers, irrational numbers form part of it too.

The sum or difference of a rational and an irrational number is always irrational.

Rationalising 2+332 will give __________.

A.

526

B.

5+26

C.

7+26

D.

726

#### SOLUTION

Solution : B

We rationalise the denominator by dividing and multiplying the number by 2+3.

So, we have

2+332×3+23+2
=(3+2)23222
=(3)2+(2)2+2632

[Using the identity (a+b)2=a2+b2+2ab]

=5+26.

Simplify (256)12.

A.

16

B.

4

C.

116

D.

14

#### SOLUTION

Solution : C

We know that (am)n=amn .
(256)12=(28)12 (Since 256=2×2×2×2×2×2×2×2=28 )
=2(8×(12))
=24
=116

Every whole number is an/a ___________.

A.

integer

B.

irrational number

C. (a) and (b) above.
D. None of the above

#### SOLUTION

Solution : A

Whole numbers comprise the set {0,1,2,3...} whereas integers comprise the set {... -2,-1,0,1,2,...}.
Thus, we can see that whole numbers are a sub-set of integers.
Hence, every whole number is an integer.

Which of the following statements is incorrect?

A. Rational and irrational numbers together make up real numbers
B.

All irrational numbers are real numbers

C.

All real numbers are irrational

D.

Every point on the number line represents a unique real number

#### SOLUTION

Solution : C

Every irrational number is a real number but real numbers comprise of rational as well as irrational numbers. Moreover, any point on a number line represents a definite real number.