Question

Find three irrational numbers between i) 2 and 2.5 ii) √2 and √3

Answer 1

Answer:

2.25 ,√6

Step-by-step explanation:

=1/2(2+2.5)

=1/2 X 4.5

= 2.25 answer 1st

2nd answer

= √2 X √3

= √6

Answer 2

Step-by-step explanation:

To find,

i) Three irrational numbers between 2 and 2.5.

For solving such types of questions, just remember the idea,

Suppose, if p and q are two different positive rational numbers, such that their product pq is not a perfect square, then it's root √pq, which is an irrational number lies between p and q.

∴ The irrational number between 2 and 2.5 is = √2 $\times$ 2.5, i.e, √5.

Thus, 2 < √5 < 2.5

But this seems lengthy enough for finding three irrational numbers as in above we've find only one.

Another method:

We know that,

2 = 2.000000.......

2.5 = 2.5000000......

So, the irrational number between 2 and 2.5 are,

1. 2.01734728472.....

2. 2.1047372756.....

3. 2.482649291......

ii) Three rational numbers between √2 and √3.

We know that,

√2 = 1.41421......

√3 = 1.732050.....

So, three rational numbers between √2 and √3 are,

1. 1.4257281758.....

2. 1.724656281057....

3. 1.635728185.....

So, you may follow the first method but it's lengthy. So, you may move towards the second method.

Explaination:

i) If you have asked for two rational numbers, then find their decimal expansion.

For eg: Find the irrational number between 2 and 3.5.

Here, we know that,

2 = 2.00000.....

3.5 = 3.50000.....

As we know, if the zeroes are beyond the decimal point it has no value.

So, may go randomly with any random numbers. But you have to write carefully the first three numbers i.e., one number which is before decimal point and other two should be in between 0 & 50 (depends on the decimal expansion).

So, the irrational number is,

2.0147381...

ii) If you have asked about two irrational numbers between given two irrational numbers then expand their decimal expansion and follow the same procedure in (i).