CLASSIFYING RATIONAL AND IRRATIONAL NUMBERS

A rational number is a type of real number, which is in the form of p/q where q is not equal to zero.

Any fraction with non-zero denominators is a rational number.

Integers are rational numbers ?

Yes

For example,

-5, -4, -3, ....., 0, 1, 2, 3,..........

Every numbers can be written as a fraction, so they are rational.

Whole numbers are rational numbers ?

Yes

Is 0 a rational number ?

Yes

0 can be considered as fraction with any denominator.

Classify each number as RATIONAL (Q) or IRRATIONAL (I)

Problem 1 :

√47

Solution :

√47 is an irrational, because 47 is not a perfect square.

√47 = 6.8556546….. (irrational)

6.8556546….. is a non terminating and non repeating decimal.

So, √47 is irrational number.

Problem 2 :

11/9

Solution :

11/9

11/9 is in the form of a/b

11/9 is a rational number.

Problem 3 :

19/4

Solution :

19/4

19/4 is in the form of a/b

19/4 is a rational number.

Problem 4 :

√96

Solution :

√96 is an irrational, because 96 is not a perfect square.

√96 = 9.7979589711….. (irrational)

√96 is irrational.

Problem 5 :

19/14

Solution :

19/14

19/14 is in the form of a/b

19/14 is a rational number.

Problem 6 :

15/4

Solution :

15/4

15/4 is in the form of a/b

15/4 is a rational number.

Problem 7 :

√84

Solution :

√84 is an irrational, because 96 is not a perfect square.

√84 = 9. 161513….. (irrational)

√84 is irrational number.

Problem 8 :

-9

Solution :

-9

-9/1 is in the form of a/b

-9 is a rational number.

Problem 9 :

√72

Solution :

√72 is an irrational, because 96 is not a perfect square.

√72 = 8.48528137….. (irrational)

√72  is irrational number.

Problem 10 :

0

Solution :

0

0/1 is in the form of a/b

0 is a rational number.

Problem 11 :

8/9

Solution :

8/9

8/9 is in the form of a/b

8/9 is a rational number.

Problem 12 :

Which statement is not always true?

1) The product of two irrational numbers is irrational.

2) The product of two rational numbers is rational.

3) The sum of two rational numbers is rational. 

4) The sum of a rational number and an irrational number is irrational.

Solution :

The product of two irrational numbers is irrational.

For example, √2 is an irrational number. √2(√2) = √4 ==> 2

2 is a rational number not irrational. So, it is not true.

So, option a is not true always.

Problem 13 :

Determine if the product of 3√2 and 8 √18 is rational or irrational ? Explain your answer.

Solution :

The product of 3√2 and 8 √18

= 3√2 (8 √18)

= 24(√2√18)

= 24√2√(2x3x3)

= 24 x 3√2√2

= 72(2)

= 144

The product of two irrational number is a rational number.

Problem 14 :

Which of the following numbers is irrational?

a) 0.252525…     b) 0.875       c) 0.3754152…     d) -0.121212… 

Solution :

A number which cannot be written in the form of p/q is known an rational number.

Option a :

Let x = 0.252525......... ------(1)

Since two digits is repeating, then we have to multiply by 100.

100x = 25.2525........------(2)

(2) - (1)

100x - x = 25.2525....... - 0.252525.......

99x = 25

x = 25/99

So, it is rational number,

Option b :

x = 0.875

Since we have three digits after the decimal, we have to multiply both sides by 1000.

1000x = 875

x = 875/1000

So, it is rational number.

Option c :

0.3754152… 

The given number is not repeating decimal, after the decimal we donot know how many digits are repeating. Then it cannot be converted it as fraction. Then it is irrational number.

Problem 15 :

The product of any two irrational numbers is:

a) always an irrational number           b) always a rational number

c) always an integer             d) sometimes rational, sometimes irrational

Solution :

Let us consider the following examples,

√2 x √2 = √(2 x 2)

= 2 (rational)

√2 x √7 = √14 (irrational)

So, product of two rational numbers is sometimes rational sometimes irrational.

Problem 16 :

Between two rational numbers:

a) there is no rational number          b) there is exactly one rational number

c) there are infinitely many rational numbers

d) there are only rational numbers and no irrational numbers

Solution :

Let us consider two rational numbers, 1/5 and 2/5

Converting into decimal, we get

1/5 = 0.2

2/5 = 0.4

0.25, 0.29, 0.31, ...............

From this, we understand that there are infinitely many rational numbers.

Problem 17 :

If a = -2 , b = -1, then find 𝑎^−𝑏 − 𝑏^𝑎.

Solution :

𝑎^−𝑏 − 𝑏^𝑎 = 1/𝑎^𝑏  - 𝑏^𝑎

applying the value of a and b, we get

= 1/(-2)^-1  - (-1)^-2

= 1/(1/2) - 1/(-1)²

= 2 - 1

= 1

Problem 18 :

Find the three rational numbers between:

(i) -1 and -2

(ii) 0.1 and 0.11

Solution :

(i) -1 and -2

Let a = -1 and b = -2

c = (a + b)/2

-3/2, -7/2 and -11/2 are three rational numbers in between them.

(ii) 0.1 and 0.11

Let a = 0.1 and b = 0.11

c = (a + b)/2

So, 0.105, 0.1075 and 0.10875 are rational numbers in between 0.1 and 0.11.