Question

Insert an irrational number between the following:

4 and 5

Answer 1

Given: `sqrt(20)` is an irrational number between 4 and 5.

Step-wise calculation:

1. Evaluate the squares of the boundary numbers: 4² = 16, 5² = 25.

2. Since 20 is between 16 and 25: 16 < 20 < 25.

Taking square roots preserves order because the function `f(x) = sqrt(x)` is increasing for x > 0.

So, `4 = sqrt(16) < sqrt(20) < sqrt(25) = 5`.

3. `sqrt(20)` is irrational because 20 is not a perfect square no integer squared equals 20. 

Hence, `sqrt(20)` cannot be expressed as a ratio of two integers.

The number `sqrt(20)` lies strictly between 4 and 5 and is irrational, making it a suitable example of an irrational number between 4 and 5.

Therefore, `4 < sqrt(20) < 5` and `sqrt(20)` is irrational.