Fundamentals of Complex Numbers: Operations and Applications

This document provides a comprehensive guide to complex number operations. It begins by defining complex numbers and their real and imaginary parts. The document then explains how to add and subtract complex numbers by separating the real and imaginary components. It proceeds to detail the multiplication of complex numbers, demonstrating the distribution of terms and simplification using the property of i^2. The final section focuses on the division of complex numbers, emphasizing the use of complex conjugates to rationalize the denominator and obtain a real number, enabling easy division. General formulas are also provided for each operation, making it a complete guide for understanding complex number arithmetic.

Complex numbers: real and imaginary parts
Every complex number has the following shape: a+ib.
The first part of that complex number is real: the real constant a. The second part is
imaginary: the real constant b multiplied by i.
The constant a is referred to as "the real part", not too controversially, and the constant b
is referred to as "the imaginary part".
NOTICE: the imaginary part is just the constant b, not ib.
Adding and subtracting complex numbers
Add
If you know how to add and subtract real numbers then you can add and subtract complex
numbers. All you have to remember is to keep the real parts and the imaginary parts
separate. Here's what I mean.
Suppose we want to add the two complex numbers 2+3i and 4-i. What do you think the
answer will be? Adding 2+3i and 4-i
First look at the real parts of each of the complex numbers.
The first complex number, 2+3i, has real part 2.
The second complex number, 4-i, has real part 4.
Add the real parts together, to get the real part of the result.
So the result has real part 6.
Next, look at the imaginary parts of each complex number.
The first complex number, 2+3i, has imaginary part 3. (NOT 3i)
The second complex number, 4-i, has imaginary part -1. (NOT -i)
Add the imaginary parts together, to get the imaginary part of the result.
So the result has imaginary part 2.
Thus our result is: (2+3i)+(4-i)=6+2i.

Subtraction
Similarly, we can subtract two complex numbers by dealing with their real parts and
imaginary parts separately. For example, what's 2+3i minus 4-i? Write down what you
think it should be
2+3i minus 4-i
By subtracting the real part of the second complex number from the real part of the first,
we get the real part of the result to be 2-4=-2.
By subtracting the imaginary part of the second complex number from the imaginary part
of the first, we get the imaginary part of the result to be 3-(-1)=4.
So the full result is: (2+3i)-(4-i)=-2+4i.
Multiplying complex numbers
Suppose we have the two complex numbers: 2+3i and 1+4i and we want to multiply them
together. If I write them like this: (2+3i)(1+4i)
(2+3i)(1+4i)
We multiply them by multiplying out the brackets as follows.
For each of the parts in the first brackets, we multiply by each of the parts in the second
brackets. This gives: (2+3i)(1+4i)=2x1+2x4i+3ix1+3ix4i.
Now we can simplify that: (2+3i)(1+4i)=2+8i+3i-12. (NOTE: the "-12" at the end is
negative because we have i2, which is -1.)
Simplifying still more, by gathering together all the real parts and all the imaginary parts,
we obtain the final result: (2+3i)(1+4i)=-10+11i.
Division of complex numbers
We have looked at adding, subtracting and multiplying complex numbers, so naturally we
now turn to dividing complex numbers. This is slightly less straightforward than the other
operations.

Suppose we have two complex numbers: 2+6i and 4+i, and we want to know their ratio:
(2+6i)/(4+i). It is not obvious how we can divide by the complex number 4+i.
It would be easy to do the division if the number on the bottom were real, for instance
(2+6i)/2 is simply 1+3i: we divide the real part by 2 and we divide the imaginary part by
2. But what if the number on the bottom is not real?
The solution to this question lies with complex conjugates. Recall that when you multiply
a complex number by its complex conjugate the result is REAL. So ....... if we multiply
the bottom number of our division by its complex conjugate, we'll end up with something
real on the bottom, and then we can do the division.
Of course, if we multiply the bottom of the ratio by anything then we have to multiply the
top by the same thing, otherwise we're changing the value of the whole expression.
So let's try that plan with our example: (2+6i)/(4+i).
First, write down the complex conjugate of 4+i. That's 4-i.
Second multiply both the top and bottom by that number.
The bottom: (4+i)(4-i)=16-4i+4i+1 = 17. (real as expected).
The top: (2+6i)(4-i)=8-2i+24i+6 = 14+22i.
Third, carry out the division, now that the bottom is real.
The ratio is now: (14+22i)/17, so that's just 14/17 + 22i/17.
So that's the answer.
If we make our numbers more general, we can write down a general formula
(a+ib)/(c+id)=(ac+bd)+i(bc-ad))/(c2+d2)