Solving Simultaneous Equations with Complex Numbers

Question

I've just started studying Computer Science at university and have been thrown into the deep end with mathematics. I haven't done maths since 2013 so it is like learning it all over again.

We have began with complex numbers which I have never done before and is leaving me confused.

In one of the questions we were given, we are asked the following:

Solve for $a$ and $b$

$(a - 3bi) + (b - 2ai) = 4 + 6i$

($i^2=-1$)

Could someone give me a break down on how to solve this as I haven't been able to move past it.

Thank you in advance!

Answer 1

The most important concept for you to grasp is that a complex number carries two pieces of information: the real part and the imaginary part. A real number only carries one piece of information: its location on the number line.

So when you add two complex numbers, you need to keep track of the real parts and the imaginary parts separately. In your case, $ a + b = 4$ and $-2a -3b = -6$. So $a =6$ and $b=-2$.

In most cases, it's actually better to think of complex numbers as having magnitude and direction (measured as a counterclockwise angle), but that discussion is not for here.

Answer 2

$z=a-3bi$ , $d=b-2ai$ and $x=4+6i$

What $\text{Re}(z)$ represents is the reel part of $z$ and what $\text{Im}(z)$ represents is the imaginary part

$$\text{Re}(z)=a, \quad \text{Im}(z)=-3b$$ $$\text{Re}(d)=b, \quad \text{Im}(d)=-2a$$ $$\text{Re}(x)=4, \quad \text{Im}(x)=6$$ sonra we get $$ a+b=4$$ and $$-3b-2a=6$$ write $b$ in terms of $a$ and...