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

Sep 9, 2015

No:

$\sqrt{5} + \sqrt{5} = 2 \sqrt{5} = \sqrt{4} \sqrt{5} = \sqrt{20} \approx 4.472135955$

$\sqrt{10} = \sqrt{2 \cdot 5} = \sqrt{2} \sqrt{5} \approx 3.16227766$

#### Explanation:

In the above answer, I use $\sqrt{a b} = \sqrt{a} \sqrt{b}$ - which is true for any $a , b \ge 0$.

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:

$f \left(a + b\right) = f \left(a\right) + f \left(b\right)$ for all $a$ and $b$?

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

$f \left(x\right) = m x + c$, where $m$ and $c$ are constants.

In addition $f \left(0\right) = 0$, so $c = 0$ and $f \left(x\right)$ must take the form:

$f \left(x\right) = m x$ for some constant $m$.

The function $f \left(x\right) = \sqrt{x}$ is not linear, so does not behave in this way.