CLASSIFYING REAL NUMBERS AS RATIONAL OR IRRATIONAL

What are real numbers ?

The numbers which are from -infinity to +infinity is known as real numbers. More simply, the numbers on the number line is real numbers.

What is rational number ?

Rational numbers are any number that can be written as a fraction.

What is irrational number ?

Irrational numbers are numbers that cannot be written as a fraction and include never-ending decimal numbers,

Problem 1 :

Which irrational numbers is between 4 and 5.

a. √12    b.√20    c. √34     d. √80

Solution :

Option a :

√12 lies between

√9 < √12 < √16

Its approximate value will be between 3 to 4.

Option b :

√20 lies between

√16 < √20 < √25

Its approximate value will be 4 to 5.

So, option b is correct.

Problem 2 :

Which number is an integer ?

a. -11/5    b.-7    c. √15     d. 1/2

Solution :

Option a :

-11/5 is decimal

Option b :

-7 is negative integer

Problem 3 :

Which number is irrational ?

a. 9.2727....    b.√2    c. 5√9     d. -37/71

Solution :

Option a :

Converting 9.2727....... as fraction

Let x = 9.2727........ ----(1)

Since 2 digits is repeating, we multiply both sides by 100.

100x = 927.27..... ----(2)

(2) - (1)

100x - x = 927.27.....  -  9.2727..... 

99x = 918

x = 918/99

x= 306/33

x = 102/11

Option b :

√2 = irrational number

Problem 4 :

Which of the following is a rational number but is NOT an integer?

a. 8     b. 24      c. 40/4     d. 6.2

Solution :

a) 8 is integer and we can consider it as 8/1. So it is rational.

b) 24 is integer and we can consider it as 24/1. So it is rational.

c) 40/4 is a rational.

d) 6.2 is not integer it is decimal.

Problem 5 :

Barbara was asked to create a set of numbers that contained only integers. Which of the sets does NOT contain only integers?

a. {11, 6, -3, -4, 600, 24/12}

b. {9, 100, -4, 12, -6, 20/5}

c. {5, 3, -8, -14, 3.5, 24/12}

d. {22, -12.0, 9, -14, 28, 4}

Solution :

In option c, we have quantity 3.5 which is decimal. So, option c does not contain only integers.

Problem 6 :

Rational numbers are a dense set. This means that between any two rational numbers on a number line there is another rational number. Which rational number is between 2.46 and 2.47 on a number line?

a. 2.48   b. G 2.4   c. 2.53    d. 2.468

Solution :

There are infinite numbers between 2.46 and 2.47,

2.461, 2.462, 2.463, .........

So, 2.468 is one of the numbers between 2.46 and 2.47.

Problem 7 :

Which statement is NOT true about rational numbers?

a. The sum of two rational numbers is also a rational number.

b. 0 is not a rational number.

c. The product of two rational numbers is also a rational number.

d. The opposite of a rational number is also a rational number.

Solution :

The opposite of rational number is a irrational number.

Say true or false

Problem 8 :

i) √40 has an infinite non repeating decimal expansion.

ii)  The number 0.5656...... is a rational number.

iii)  -200 and 500 are integers

iv)  All numbers with infinite decimal expansions are irrational.

v)  The numbers -8, -3, 5, 17 are all whole numbers.

Solution :

i) √40 has an infinite non repeating decimal expansion.

True

ii)  The number 0.5656...... is a rational number.

To prove this, let us consider x = 0.5656......  -----(1)

Since two digits are repeating, multiply both sides by 100.

100x = 56. 56........  -----(2)

(2) - (1)

100x - x = 56.56......... - 0.5656.......

99x = 56

x = 56/99

So, it is true.

iii)  -200 and 500 are integers

-200 is negative integer, 500 is positive integer. So, it is true.

iv)  All numbers with infinite decimal expansions are irrational.

False

The decimal form of irrational numbers will contain only infinite non repeating decimals. If it contains repeating decimals, it can be converted into fraction. So, it is rational.

v)  The numbers -8, -3, 5, 17 are all whole numbers.

True.

Problem 9 :

Which number is irrational

a. 9.2727.....      b.√2         c. 5√9       d.-37/71

Solution :

√2 is irrational.

Problem 10 :

Any number with a finite decimal expansion must be 

a. rational    b. irrational

Solution :

The number which has finite decimal expansion is rational.

Problem 11 :

All integers are 

a.  Whole      b. rational    c. irrational

Solution :

All integers are rational.

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