Why does square rooting a negative number never give us an answer?

Question

I used to do this on my calculators and it never worked! I think it's because you can't multiply any number by itself to get a negative number. Is that even true? I think it is! I've tried it out and it never worked! Look here:$$0.5\cdot0.5=0.25$$$$0\cdot0=0$$$$-1\cdot-1=1$$$$2.1\cdot2.1=4.41$$$$-7.5\cdot-7.5=56.25$$I never get a negative for multiplying negative numbers because a negative times a negative is a positive and no matter what I do, I always get a nonnegative number! Tell me what you think! Is this why?

Answer 1

For every real $x$ such that $x\gt 0$, we have $$ \sqrt{-x}=\sqrt{(-1)x}=\sqrt{-1}\sqrt{x} =i\sqrt{x} $$

Answer 2

You are (almost) correct!

It is a property of the real numbers that any number squared is positive, i.e. $\forall x \in \mathbb{R}, x^2 \ge 0$. So all positive real numbers have two square roots - a positive one and a negative one (although the square root function is defined by convention to be the positive value) and negative numbers have none.

But.

A large amount of mathematics is about "what if". In this case, "what if we pretended that $\sqrt{-1}$ was a number and tried to treat it like one?" So we define $i$ to be one of the solutions to $i^2=-1$, and see where it takes us. As it turns out, it leads us to the Complex numbers, a concept that opens up whole new possibilities in a number of fields, including ones as far reaching as number theory and electrical engineering.

I won't go into details here, but in moving from the real numbers to the complex numbers you gain some things - for example, every polynomial over the real numbers has a solution in the complex numbers - and you lose others, like the good old ordering that lets you say that one number is bigger than another.

Then there are other extensions of the real numbers, like the quaternions, which get even weirder and more interesting because you exchange what seem to be even more fundamental properties (like the idea that it doesn't matter what order you multiply numbers) in exchange for different ones.