Subject: Compulsory Maths

The number can be in different form. Some numbers can be in a form of fraction, ratio, root and with the decimal. If the number is in the form of \(\frac{p}{q}\) (fraction) ,of two integer p and q where numerator p and q≠0 are called rational numbers.

5 \(\frac{2}{3}\), \(\frac{7}{4}\), \(\frac{3}{4}\), \(\frac{3}{5}\) etc are the examples of rational numbers.

Rational number can be:

- All natural number
- All whole number
- All integer
- All fraction

Numbers which cannot be expressed in a ratio (as a fraction of integer) or it can be expressed in decimal form is known as irrational numbers. It can neither be terminated nor repeated.

For example,

√7 = 2.64575131.............

√5 = 2.23620679....... etc are irrational numbers.

√2,√3,√5,√6,√7, etc. are the examples of irrational number where the numbers are a non-terminating and a non-repeating number.

- If we made an irrational number negative then it is always an irrational number.

For example, -√5 - If we add a rational number and an irrational number then a result is always an irrational number.

For example, 2 +√3 is irrational. - If we multiply a non-zero rational number with an irrational number then it is always an irrational number.

For example, 5√3 is an irrational number. - The sum of two irrational number is not always an irrational number.

For example, (2 +√3) + (2 -√3) = 4, which is irrational. - The product of two irrational number is not always an irrational number.

For example, ( 2 +√3) x (2 -√3) = 4 -3 =1, which is rational.

- The number in the form \(\frac{p}{q}\), where p and q are integers and q≠0 are called rational number.
- A rational number is a number that can be written as a ratio.
- An irrational number is a real number that cannot be expressed as a ratio of integers.
- Irrational numbers cannot be represented as terminating or repeating decimals.

- It includes every relationship which established among the people.
- There can be more than one community in a society. Community smaller than society.
- It is a network of social relationships which cannot see or touched.
- common interests and common objectives are not necessary for society.

Give the example of four rational number which aren't whole number.

Solution:

- 0.5
- 0
- -100
- \(\frac{3}{5}\)

Give an example of irrational number whose sum is rational.

Solution:

- √5 and 5-√5
- √3+2 and 3-√3

Give an example of two irrational number whose product is rational.

Solution:

- √3 and -√3
- √5 and -√5

Write five rational number between \(\frac{1}{2}\) and \(\frac{3}{5}\)

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

Identify the following statements as true or false.

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)

Find the irrational number of √2.

Solution:

√2 = 1.41421356.......

Give an example of four rational number, which isn't whole number.

Solution:

1) 0.75

2) -100

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

4) 0

Give an example of the irrational number, whose sum is rational.

Solution:

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

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

Give an example of two irrational number whose product is rational.

Solution:

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

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

Find the sum of two irrational numbers which is not always an irrational number?

Solution:

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

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

Find the product of two irrational numbers which is not always irrational numbers?

Solution:

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

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

= 4 - 3

= 1, which is rational.

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

Which of the following are 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)

Divide:

9/16 by 5/8

Solution:

9/16 ÷ 5/8

= 9/16 × 8/5

= (9 × 8)/(16 × 5)

= 72/80

= 9/10

Divide:

-6/25 by 3/5

Solution:

-6/25 ÷ 3/5

= -6/25 × 5/3

= {(-6) × 5}/(25 × 3)

= -30/75

= -2/5

Divide:

11/24 by -5/8

Solution:

11/24 ÷ (-5)/8

= 11/24 × 8/(-5)

= (11 × 8)/{24 × (-5)}

= 88/-120

= -11/15

The product of two numbers is -28/27. If one of the numbers is -4/9, find the other.

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.

Find the product of two rational numbers?

(-25/9) by (-18/15)

Solution:

(-25/9) × (-18/15)

= (-25) × (-18)/9 × 15

= 450/135

= 10/3

Find the product of two rational numbers.

6/11 × (-55)/36

Solution:

6/11 × (-55)/36

= 6 × -55/11 × 36

= -330/396

= -5/6

Subtract:

2/5 from 4/7

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

Determine whether the following rational numbers are positives or not.

(-11)/3

Solution:

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

Determine whether the following rational number is positives or negetive.

25/(-27)

Solution:

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

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