Question

Insert an irrational number between the following:

`-1/3` and `-4/5`

Answer 1

Given: Insert an irrational number between `-1/3` and `-4/5`.

Step-wise calculation:

1. Convert the fractions to decimals for comparison:

`-1/3 ≈ -0.3333`,

`-4/5 = -0.8`

So the number must lie between –0.8 and –0.3333.

2. Consider the number `-sqrt(0.2)`: 

`sqrt(0.2)`

= `sqrt(1/5)`

= `1/sqrt(5) ≈ 0.4472`

Therefore, `-sqrt(0.2) ≈ -0.4472`

3. Check that –0.4472 lies between –0.8 and –0.3333: 

–0.8 < –0.4472 < –0.3333

4. `sqrt(0.2)` is irrational because 0.2 is not a perfect square rational number and the square root of a non-perfect square rational number is irrational.

Thus, `-sqrt(0.2)` is an irrational number between `-4/5` and `-1/3`.