TO CHECK THE GIVEN NUMBER IS RATIONAL NUMBERS

Problem 1 :

Which of these are rational?

a. 8     b. -8     c. 2 1/3     d. -3 1/4

e. √3    f. √400     g. 9.176     h. π - π

Solution :

A number which can be written in the form of p/q is called rational number.

a) 8

8 is a integer and it can be written as 8/1. This can be written in the form of p/q. So, it is rational.

b) -8

This can be represented as -8/1. So, it is rational.

c) 2 1/3

This mixed fraction can be converted as improper fraction as 7/3. So, it is rational.

d) -3 1/4 = - 13/4

It is rational.

e) √3

√3 = 1.732 ..........

It cannot be converted as fraction, so it is not a rational number.

f) √400

√400 = 20

It can be represented as 20/1, then it is rational.

g) 9.176

Since it is recurring decimal, it can be converted as fraction. Then, it is rational.

h) π - π

= 0

Rational number:

8, -8, 2 1/3, -3 1/4, √400, π - π

Problem 2 :

Show that these numbers are rational:

a. 0.7.....       b. 0.41.....       c. 0.324.......

Solution :

Rational number can be written as a fraction, integers, terminating decimals and repeating decimals.

a.

Let x = 0.7777..... ----(1)

10x = 7.7777...... ------(2)

(2) - (1)

10x - x = 7.7777..... - 0.7777......

9x = 7

x = 7/9

0.7777...... = 7/9

This can be represented as fraction. So, it is rational.

b.

0.41 = 0.414141...

Let x = 0.414141... ----(1)

100x = 41.4141....... ------(2)

(2) - (1)

100x - x = 41.4141...... - 0.414141.......

99x = 41

x = 41/99

This can be represented as fraction. So, it is rational.

c.

0.324 = 0.324324....

Let x = 0.324324....... ----(1)

1000x = 324.324324...... ------(2)

(2) - (1)

1000x - x = 324.324324......- 0.324324.......

999x = 324

x = 36/111

This can be represented as fraction. So, it is rational.

Problem 3 :

a. Why is 0.527 a rational number?

b. 0.9 is a rational number. In fact, 0.9 € Z. Give evidence to support this statement.

Solution :

a.

Yes, 0.527 is a rational Number. Since it is ending decimal, it can be represented as fraction.

b.

Yes, it is a rational number because it has terminating decimal representation.

Also, it can be expressed in the form of p/q where p and q are Integers and q not equal to zero.

9/10

Problem 4 :

Explain why these statements are false:

a. The sum of two irrationals is irrational.

b. The product of two irrationals is irrational.

Solution :

a. 

Consider two irrational numbers (3 + √2) and (3 - √2)

Sum of two irrational numbers

= (3 + √2) + (3 - √2)

= 3 + √2 + 3 - √2

= 6 which is rational number

So, it is false statement.

b. 

Consider two irrational numbers (3 + √2) and (3 - √2)

Product of two irrational numbers

= (3 + √2)(3 - √2)

= (3)² - (√2)² 

= 9 - 2 

= 7 which is rational number

So, it is false statement.

Problem 5 :

a. Explain why 1.3 is a rational number.

b. True or false. √4000 € Q ?

Solution :

Here the decimal 1.3 can be converted as fraction which is 13/10. So, it is a rational number.

b.

= √4000

= √(2 x 2 x 2 x 2 x 2 x 5 x 5 x 5)

= (2 x 2 x 5)√(2 x 5)

= 20√10

The product of rational and irrational is irrational. So, it is irrational.

Problem 6 :

a. True or false: 1/√7 € Q ?

b. Show that 0.41 is a rational number.

Solution :

1/√7

√7 is a irrational number. Dividing 1 by an irrational number, we will get irrational number. Then, it is true.

b. 

Let x = 0.41

It is a terminating decimal, by converting it as fraction, we get 41/100. Which is rational number.

Problem 7 :

Is the sum of 3√2 and 4√2 rational or irrational 

Solution :

Sum of 3√2 and 4√2 is,

= 3√2 + 4√2

= 7 √2

Multiplying rational number by irrational number will be a irrational number.

Problem 8 :

Jacob is working on his math homework. He decides that the sum of the expression 1/3 + 6√5/7 must be rational because it is fraction. Is Jacob correct ?

Solution :

= 1/3 + 6√5/7

= [7 + 3(6)√5]/21

= (7 + 18√5)/21

It must be an irrational number, because adding, subtracting, multiplying or dividing by irrational by rational number will product rational number as result.

Problem 9 :

State whether 7 - √2 is rational or irrational

Solution :

= 7 - √2

Subtracting rational and irrational will be a irrational. Then the given number is irrational.

Problem 10 :

Is the product of √16 and 4/7 rational or irrational ?

Solution :

= √16 x (4/7)

= √(4 x 4) x (4/7)

= 4 x 4/7

= 16/7

Since the given number can be represented in the form of p/q, it must be a rational number,.