So what are complex numbers? I thought numbers were already complex enough so what are the parameters that define a complex number?
A complex number is a number that holds both real and imaginary components in the form of a + bi, where a is a real number(ie a number that lies on the number line) and bi is an imaginary number.
So perhaps, we should start this by defining what an imaginary number is. An imaginary number, or sometimes referred to as ‘i’, is the √-1 . That’s the short answer but why do we need a separate number to express √-1 ? That’s because every number, regardless of whether it’s negative of positive – when squared, produces a positive number. Thus we can deduce that the square root of a positive number and been expressed as positive and negative, but that leaves the question of what happens when we square root a negative number? This question has baffled mathematicians for generations until we accepted the inclusion of a new constant; the imaginary number.
In fact, mathematicians have created an entire plane to represent the breadth of all the different types of number and thus the complex number was discovered!
Another way of visualising this:
COMPLEX NUMBER ARITHMETIC:
After clearing the basics of complex numbers, I would like to introduce complex number arithmetics via a few sample problems. Most of them are relatively similar to real number arithmetic except for a few exceptions.
Question 1:
What is (1+i) +(2+3i)?:
Answer is to add both the real parts and the imaginary parts of the complex number separately. So, the real components added would be be (2+1) which is 3 and the imaginary components added would equate to (3i+i) which is 4i. Thus, the resulting complex number is 3+4i!
We also apply the same principle with subtracting too!
Question 2:
What is (i+1)(3i+2)?:
The answer is to use the distributive property(more colloquially known as FOIL) to multiply out the brackets but beware of i squared which is -1.
So using FOIL, we obtain the polynomial 3i²+2i+3i+2. Since, we have defined i² = -1 ∴ 3i² = 3(-1) = -3.
∴we obtain the equation; 5i+2 -3 which is equal to 5i-1!
The same applies to division and exponents where ai = -a⁄i
Imaginary powers:
knowing i²=-1, so therefore what is i³?:
If we can decompose i³ into simpler terms, it would be i x i x i which i² x i which is -1 x i which is -i.
i⁴ would subsequently be i² x i² which is (-1)² which is 1. Can you see a pattern? i⁵ would be bring us i back again and this cycle would repeat indefinitely in steps of 4!
IMPORTANT NOTATION:
𝕽(x) denotes the real part of a complex number, x
𝕵(x) denotes an imaginary part of a complex number, x
|x| denotes the magnitude of the complex number, or in other words, the distance from the origin to the complex number. So, for example, the magnitude of any point in the graph shown above is 1.
x̅ denotes the same complex number but with -𝕵(x). So, suppose if x = a+bi so therefore x̅ = a -bi.
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