Question

Prove that the following number is irrational.

`4 - sqrt(3)`

Answer 1

We need to prove that `4 - sqrt(3)` is irrational.

Step 1: Recall the definition

A number is rational if it can be written as `p/q`, where p, q ∈ ℤ, q ≠ 0.

A number is irrational if it cannot be expressed in this form.

Step 2: Assume the contrary

Suppose `4 - sqrt(3)` is rational.

That means we can write `4 - sqrt(3) = p/q`, where p, q ∈ ℤ, q ≠ 0

Step 3: Rearrange

`sqrt(3) = 4 - p/q`

Since 4 is rational and `p/q` is rational, their difference must also be rational.

Thus, `sqrt(3)` would be rational.

Step 4: Contradiction

But it is a well-established fact that `sqrt(3)` is irrational.

This contradiction shows that our assumption was false.

Step 5: Conclusion

Therefore, the number `4 - sqrt(3)` cannot be rational.

`4 - sqrt(3)` is irrational.