Why complex numbers are represents as a+ib and can't be as (a,b)?

Question

I am confused as why do we need to represent the complex numbers with the imaginary y-axis if we can simply represent them as (x,y) ?

I've read that Multiplication by i is an anti-clockwise rotation of a quarter-circle over y-axis.

Multiplying 1 by i gives i. Multiplying i, by i once more, does another quarter-circle and gives -1. So, multiplying by -1 means a rotation of a half-circle. That is the meaning of i*i=-1.

So what does that supposed to mean ?

Suppose, I am solving an equation and I ended up with an answer like 3i does that mean i've moved from x-axis to y-axis by half-circle counter clock wise ? I couldn't be able to visualize this properly

Answer 1

Yes, in signal processing, complex numbers are usually visualized on the complex plane, as you have said.

The reason is that if you put them on a plane, then you are able to measure two important quantities:

1) Magnitude, which is $\sqrt{x^2 + y^2}$

2) Phase angle between your point and the origin, given by $\tan^{-1} \frac{y}{x}$.

If you simply left them as a point, ($x$,$y$), you would not be able to concretize and have a frame work for those quantities.

You may ask, why are those quantities, in turn, important? In signal processing, we are of course dealing with signals, and physically, we are dealing with 'real' signals. However, though a nice trick, an constant oscillation of a quantity in 'real' life, (like a cosine wave), is equivalent to two phasors, rotating around in opposite directions on the complex plane, and adding up together. With this framework, we can see that the phase angles 'cancel' each other out, and that the magnitudes of their resultant give us the magnitude of our 'real' signal.

In fact this is what one of euler's formulas captures. That is:

$$ \cos(2\pi ft) = \frac{e^{j2\pi ft} + e^{-j2\pi ft}}{2} $$

You can see here how we can easily relate a 'real' world concept, like an oscillating cosine wave, with the 'complex' world of phasors, as they exist and rotate around in the complex plane.

This is one of the corner stones of DSP.

Answer 2

For one definition of complex numbers, the symbology "a + ib" and "(a,b)" are equivalent representations as long as the operations on those symbols completely follow the set of rules for complex arithmetic (including multiplication implying a rotation).

The meaning is that complex arithmetic using such arithmetic rules actually simplifies a whole bunch of theorems and computations (including solutions of polynomial roots, infinite series convergence, etc.). The behavior of pairs of real quantities in the real world can sometimes be closely approximated by models using arithmetic under such rules, and then by calling one of the quantities "imaginary" to match the computational symbology used in the model.

Consider it a mathematical "trick" that is too helpful to not use. e.g. Cardano and other Renaissance-era Italian mathematicians attempted to solve cubic equations without the use of complex or imaginary numbers, and their solutions were tons more long-winded because of that.

Answer 3

Complex numbers are different from 2D vectors when it comes to differentiation. The derivative of a function of a complex variable has a stronger continuity requirement than 2D of real numbers. Thus, much stronger statements can be made in complex analysis than real analysis. In many simpler scenarios these sorts of properties don't come into play and they're functionally equivalent.

Answer 4

One way to think about complex numbers is to view $i$ as a "unit vector" in the direction of the imaginary axis.

In fact, the use of complex numbers as unit vectors later became the basis for quanternions, which were used to represent vector quantities before the development of modern vector analysis by Gibbs/Heaviside.

Answer 5

TL;DR: some notations refers to mathematical structures. If $(a,b)$ denotes vectors, then with a vector space one does not fully take into account the structure of complex numbers. However, the commutative filed of complex values $\mathbb{C}$ is way more interesting.

"3i" means that:

• with $i$, you move 90° degree counterclockwise from some position
• the $3$ factor, you take 2 more steps (the step size is the distance between the origin and the original position) into that direction.

You may have mixed $3i$ with $i^3=-i$, the later being the true "do the rotation thrice".

It combines a rotation composed with an homothetia (or dilation). And they commute: you can dilate first then rotate, or the other way, getting the same result. If you consider points in the plane given by $(x,y)$, one model to both rotate and dilate is to write them as vectors $[x,y]$, and combine them with action matrices of the parametric form

$$ \pmatrix{a& b \\-b &a}$$

And those matrices are close under addition and multiplication. The field of complex numbers $a+ib$, with this weird $i^2=1$, was discovered to a natural analogue of the above matrices. Known as Argand diagrams, this interpretation of complex numbers is due to Wessel and Argand. While initially, complex numbers were investigated to solve real polynomial equations that lacked "real solutions". Apparently, nothing to do with rotations and shears. But this are linked in a way that makes "complex numbers" so natural, despite iIt took a long time in the history of science to embrace them.

Similarly, real linear and time-invariant systems "really" need complex numbers to be dealt with. There is a lot in common between convolution and polynomial products.

• Multiplication of Complex Numbers is a Rotation Composed with a Dilation