['Gautam Chekuri', 'gautam.chekuri@gmail.com', 'http://github.com/gautamc']
[Observe ∿ deconstruct ∿ Generalize ∿ Reflect ∿ Repeat]

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Understanding Complex Numbers

Initially drafted in 2013; updated in 2014 & March 2016

1. A complex number has the form a + bi where a and b are defined as two real numbers and i2=-1.

I’ve always wondered about the reasoning behind this definition of i - why i2=-1?

I have come to understand that the `+` between `a` and `bi` does not mean that we are somehow going to add `a` into `bi`.
The `+` is simply a convention – we must look at the complex number as a pair of two numbers `a` and `bi`.

The plot given below is the familiar complex plane. Each point on the plane is a complex number.

The following notes are my attempt to deconstruct and understand the complex number.

2. A complex number is a generalization derived from the concept of a number.

The concept of a number:

3. Given a set of things, we use numbers to count and order the things.
With the use of numbers, we could tell how many things our set contains, call a certain thing the first and some other the last.
4. Each thing in our set can be represented by a number.
Given a bag of apples, we can decide that Apple 1 is bigger than Apple 2.
Being in a bag – real or hypothetical – allows the apples to be counted and arranged relative to other apples in the bag.
It is not that the apple by itself was countable – the bag had something to do with enabling this ability in our mind.
5. The number is not the apple, we don’t eat the number; the number is an idea that allows us think about things in certain ways.

The generalization:

6. What if there was something, that could be represented (or thought about) better with a pair of numbers.
And what if, these pairs of numbers had their own rules for addition and multiplication.
- A pair of numbers with their own " mathematics", slightly differing from the mathematics of our everyday apple counting number.
If there are such things, we would assume them to be very different from the aforementioned apples.
7. English language definition: ComplexConsisting of many different and connected parts.

Two numbers, paired to act as one:

8. ‘Complex Number’ is the name given to a pair of numbers.
There are well defined methods for performing math operations on such pairs of numbers;
- afterall, what good are numbers – complex or otherwise – if there isn’t a straightforward way to add them.

But before everything else, one must ask the following questions:

What is it about these two numbers, that we want to pair them and treat the pair as a number?
Do these two numbers in the pair have special abilities?
Are they both alike – in terms of what each of them can do; or, are they very different from each other?

Both alike in abilities, but differing in one fundamental property – their multiplicative identity

9. One might ask what the phrase – “abilities of a number” means. Such a question is asking what it means to be a number.
Simply put, addition & multiplication are two abilities that a number has.
Multiplication is repeated addition.
Therefore, one can conclude that the ability to add is what makes the number – if I can add then I can also multiply.
From the notion of addition we can derive what it means to be greater and smaller when comparing two numbers.
All of these abilities, as noted previously, allow us to count and order things.
We could deconstruct and analyze what it means to be a number in more detail, but for now let us move forward without doing so.

The two numbers that are paired to create a complex number are named the Real part and the Imaginary part.
Real and Imaginary can each be used for counting and ordering the aforementioned apples
- everything that can be done with a Real number can also be done with an Imaginary number.
In this sense, they are alike.
10. While similar, there is one fundamental difference between a Real number and an Imaginary number.

To appreciate the nature of this difference we must first understand the ‘multiplicative identity’.
‘Multiplicative Identity’ is the name given to a number that when multiplied by does not result in any change.
Simply put, 1 is the Multiplicative Identity that we are familiar with:
5 × 1 = 5
76 × 1 = 76

The Multiplicative Identity for every Real number is 1. Hence, every Real number x, can be written as 1 x
For Imaginary numbers the Multiplicative Identity is defined as -1. Hence, every Imaginary number x, can be written as -1 x
11. The question that comes to mind at this point is: why -1?
It is helpful to think about this question by framing the following two questions:
Why is the Multiplicative Identity the square root of a negative number?
Why was -1 choosen as that negative number?

We all agree that a negative number cannot have a square root.
At this point, we should note that before mathematicians arrived at the concept of negative numbers, one might have found it impossible to imagine negative numbers.
Similarly, prior to the evolution of the concept of irrational numbers, it might not have been easy to think of numbers that cannot be represented as a fraction of integers.
The discovery of Zero itself, has been a relatively recent event in the history of mathematics.

In each case, the new discovery was able to represent something that was un-representable in the existing system.
For example, Rational numbers can express quantities that natural numbers alone cannot – ratios of various nature; ideas that evolved as people started to trade with each other.
The Greek mathematician Pythagoras initially thought that any number could be expressed as a fraction of two integers.
Later he learnt that this was not the case and discovered Irrational numbers.

The pattern of progress in (mathematical) thought: that which is undefinable will yield the answers we seek:

12. Therefore, if different kinds of numbers were arrived at previously by trying to define that which was undefined
then it would only be logical that someone would search for a similar pattern and identify the next undefined “thing” to build upon.
That undefined “thing” was the square root of a negative number.

The choice of -1 reminds us of 1; it is easy for us to accept 1 as the Multiplicative Identity.
How different would things be if the Multiplicative Identity of Imaginary numbers was defined as -2?
Would we consider -2 to be twice as undefined as -1? - I don’t think we can.
There can’t be degrees of undefinedness when comparing two or more undefined values;
- If such a case existed we could select one of those value as an ‘origin’ and define the rest relative to this origin – and they would not be undefined anymore.
Hence, I think that the choice of -1 is primarly motivated by the need to it make easier for us think of -1 as the Multiplicative Identity.

The symbol i, that is used to represent an imaginary number and which is defined as follows:
i 2 = -1
i = -1
is similar in spirit to the + and – symbols used to denote positive and negative integers.
It is also similar in spirit to the … symbol placed after an irrational number like 3.141592653589793…

In all these cases, the symbols allow us to imagine beyond the bounds that were previously established.
Therefore, the Imaginary number is not some kind of secret, difficult to fathom number.
13. Sita Aur Gita is a Indian lost & found movie about two twin sisters; separated at birth; both alike in appearance but exposed to different circumstances.
Eventually the two sisters meet and Gita introduces herself to Sita thus: “Do not be surprised sister Sita, I am not foreign to you – I am you in a different form”.
This is how we can understand the Real and Imaginary numbers – different forms of each other.
Each unimaginable from the other’s point of view, but both exist.

The philosopher mathematician Rene Descartes came up with the term “Imaginary number”.
It seems, his use of the word “Imaginary” was meant to be derogatory.
However, given all the useful applications it has found, while being part of the complex number, the Imaginary number has done quite well for itself.
14. When paired together as a Complex Number, Real and Imaginary numbers form a powerful mental tool.
This tool allows us work with things & concepts that are best represented & understood by two quantities.
Electrical impedance is an example of such a concept.
What resistance is in a DC circuit, impedance is in an AC circuit.
Impedance contains two numbers. Adding and multiplying impedance values is implemented as complex number addition and multiplication.
15. Treating a pair of numbers as a single number by defining addition/multiplication etc for these pairs is also done with vectors.
However vector arithmetic is defined differently from complex number arithmetic.
Some concepts are better understood with complex numbers, while others are more suitable for vectors.

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