I've been teaching my 10yo son some (for me, anyway) pretty advanced mathematics recently and he stumped me with a question. The background is this.

In the domain of natural numbers, addition and multiplication always generate natural numbers, staying in the same domain.

However subtraction of a large number from a smaller one needs to "escape" into the domain of integers, and division may result in escape to the real domain (like `3 / 5 -> 0.6`

).

It was a simple step from there to taking the square root of a negative number, hence requiring the escape into the complex domain, such as `4+7i`

.

He quite easily picked up that each of these domains was a superset of another, `natural -> integer -> real -> complex`

.

However, he then asked if an operation on a complex number would require yet another escape, a question I had to investigate. Now, it turns out that the square root of a complex number is simply another complex number along the lines of mathematical distribution: `(a+bi)`

, from memory.^{2} -> a^{2} + 2abi - b^{2}

But I'm wondering if there are *other* mathematical operations performed on complex numbers (or any of its subset domains) that can't be represented within the complex domain.

Apologies if I've used the wrong terms, it's been about 30 years since I did University level math.