Question on surds rule Through the textbook, I've been taught the rule $\frac{\sqrt a}{\sqrt b} = \sqrt\frac{a}{b}$, however I realized that if all numbers are assumed to be real, and $a<0 ,b<0$, then the rule is not true as $\frac{\sqrt{-a}}{\sqrt{-b}} = \sqrt\frac{a}{b}$, whereas in $\frac{\sqrt{-a}}{\sqrt{-b}}$, the process makes it invalid as square rooting negative numbers is impossible. So does this mean that the surds rule (multiplication of surds included) is invalid for negative numbers? Or does it mean complex numbers have to be introduced in all cases?

The main problem is $\sqrt{\cdot}$ isn't defined for negative values as you've pointed out. While you can find numbers that square to negative numbers using the complex numbers, the problem is that there's no natural way to choose them. That is to say, $i$ and $-i$ are essentially indistinguishable (I mean mathematically. Our notation allows us to tell them apart, but we essentially just picked one of the two at random and called that one $i$ and the other one $-i$. It didn't matter which one we named which.) Thus while you can define a continuous function, $s: \Bbb{R}\to\Bbb{C}$, such that $s(x)^2 = x$ for all $x\in \Bbb{R}$, say by 
$$s(x) := \begin{cases}\sqrt{x} & x\ge 0 \\ i\sqrt{|x|} & x < 0, \end{cases}$$ 
if you check to see whether this satisfies $s(xy)=s(x)s(y)$, or $s(x/y)=s(x)/s(y)$, you'll see that it fails to satisfy either except in special cases.
