Complex Numbers

$i = - 1$

Imagine $i$ is a variable
- For instance, $3 \times i = 3 i = 9 \times - 1 = - 9$
Now allow addition and subtraction of imaginary numbers $(a + bi) + (c + d i) = (a + c) + (b + d) i$
Now allow multiplication of imaginary numbers

$(a + bi) (c + d i) = a c + b c i + b c i + b d i^{2} = a c + 2 b c i - b d$

The conjugate of an imaginary number $z$ is $\overline{z}$
- the conjugate of $a + bi$ is $a - bi$
To divide, multiply the numerator and denominator by the conjugate of the denominator, and simplify from there

Imaginary numbers can be represented in a plane
Here, the real component is represented as an x-coordinate
The imaginary component is represented as a y-coordinate
Multiplying the imaginary number by $i$ rotates it 90˚ anti-clockwise about the origin

Going Rogue (Not in Course Content)

In the 1970s, a small group of mathematicians made a discovery in the field of complex numbers, which must rank as one of the most startling, mysterious, and awe-inspiring in the history of mathematics
Remember sequences, e.g. recursive formula for an arithmetic sequence, $T_{1} = 2, T_{n + 1} = T_{n} + 3$
Think of a number, $n$
- Square the number, and add the original number $n$
- Square the result, and add the original number $n$
- Square the result, and add the original number $n$
- Repeat another 5 times
  - For $- 2 \leq 0 \leq 0.25$ , the number appears to approach a finite value
  - For $n = - 1$ , the result oscillates between 0 and $n$
  - For $1 < n$ , $n$ approaches infinity
- So, for some values of $n$ , e.g. 2, sequences of numbers which approach infinity are generated
- For others, e.g. -1 or 0.2, sequences with finite values are generated
The process we just used can be defined as $T_{1}, T_{n + 1} = T_{n}^{2} + T_{1}$
Exactly which values of $T_{1}$ give a sequence whose $T_{n}$ values remain small forever?
- For $- 2 \leq 0 \leq 0.25$ , the number appears to approach a finite value
You can do the same thing with complex numbers