Question

Insert an irrational number between the following:

`1/3` and `1/2`

Answer 1

Given: Insert an irrational number between `1/3` and `1/2`.

Step wise calculation:

1. First, calculate the decimal equivalents:

`1/3 = 0.3333`

`1/2 = 0.5`

2. We need an irrational number between 0.3333... and 0.5.

3. Recall that irrational numbers have non-terminating, non-repeating decimal expansions.

4. One way is to find an irrational number x such that 0.3333... < x < 0.5

5. Take for example, `x = sqrt(0.17)`

Check if it’s between `1/3` and `1/2`:

0.17 = 0.17

`sqrt(0.17) ≈ 0.4123`

Since 0.3333 < 0.4123 < 0.5, 

`x = sqrt(0.17)` lies between `1/3` and `1/2`.

6. `sqrt(0.17)` is irrational because 0.17 is not a perfect square of a rational number.