Question

Is the square root of 5 plus the square root 5 equal to the square root of 10?

Answer 1

No:

`sqrt(5)+sqrt(5) = 2sqrt(5) = sqrt(4)sqrt(5) = sqrt(20) ~~ 4.472135955`

`sqrt(10) = sqrt(2*5) = sqrt(2)sqrt(5) ~~ 3.16227766`

Explanation:

In the above answer, I use `sqrt(ab) = sqrt(a)sqrt(b)` - which is true for any `a, b >= 0`.

Generalisation

Suppose `f` is a function that takes any Real number and gives us another Real number.

Under what circumstances would we expect the following:

`f(a+b) = f(a) + f(b)` for all `a` and `b`?

Actually the only kind of functions that behave like this are all linear functions which take the form:

`f(x) = mx + c`, where `m` and `c` are constants.

In addition `f(0) = 0`, so `c = 0` and `f(x)` must take the form:

`f(x) = mx` for some constant `m`.

The function `f(x) = sqrt(x)` is not linear, so does not behave in this way.