Real Numbers exercise 1.2.3

Belongs to the unit Real Numbers

1. Find 5 irrational numbers between 4 and 5.

Solution:
The numbers a = 4.101001000….
b = 4.02002000…..
c = 4.303003000……
d = 4.404004000……
e = 4.05005000…..
Are 5 irrational numbers between 4 and 5 since 4 < a < b < c < d < e < 5
2. Find 6 rational numbers between √𝟐 and √𝟑

Solution:

Let √2,

9.png
Therefore, √𝟐 = 1.414213

Let √3,

10.png

√3 = 1.7320508

We consider r1 = 1.422222……..

2 = 1.433333…….

r3 = 1.44444……..

r4 = 1.5555……..

r5 = 1.666666……

r6 = 1.7111111……

These are rational numbers between 2 and 3

√2< r1< r2 < r3 < r4 < r5 < r6 < √3

 

3. Write 5 irrational numbers between √𝟑 and √𝟓

Solution:

10

Therefore, √3 = 1.7320508

Then, √5,

11.png

Therefore, √5 = 2.236067

Consider a = 1.7404004000……
b = 1.7505005000……
c = 1.8202002000……
d = 1.9303003000……
e = 2.101001000… are 5 irrational numbers between √3 and √5

4. Prove that √2 + √5 is irrational.

Solution:

If possible, let √2 + √5 be rational
Then (√2 + √5) is rational
═> (√2 + √5) is rational
═> 2 + 5 + 2√2 .√5
═> 7 + 2 √10 is rational
But (7 + 2 √10 ) being the sum of a rational and an irrational. These we arrive at a contradiction to the assumption that ( √2 + √5 ) is rational.
Hence is ( √2 + √5 ) is irrational

5. Give an example of an irrational number such that its 8-th power is a rational number.

Solution:

√28 = 21/8 is an irrational number.

The 8th power is (21/8 )8 = 21/8 x 8 = 21 = 2

2 is a rational number.

6. Why is 0.111222333444……., where each number appears 3 times in a row. Irrational?

Solution: 
0.111222333444….., where each number appears 3 times in a row is a non- terminating, non-recurring decimal expansion. Hence it is irrational.
7. Find 3 rational numbers between 0.1122334455….. and 0.1234567…. here each number appears 2 times in the first expansion and each number appears only once in the second expansion.

Solution:

Consider r1 = 0.12222…..

r2 = 0.122333…..

r3 = 0.122444……

r4 = 0.122555……

r5 = 0.122666……

This is 5 rational numbers between 0.1122334455….. and 0.1234567……

8. Find the 3 irrational numbers between 2/3 and ¾ using their decimal expansion.

Solution:

12.png

Consider a = 0.6707007000….

b = 0.6808008000….

c = 0.6909009000…

d = 0.707007000….

e = 0.7202002000….

These are 5 irrational numbers and 2/3 < a < b < c < d < e < 3/4