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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue
mathsComplex NumbersAlgebra of Complex Numbers

Complex Number Arithmetic

This page brings up the question, what are the applications of complex numbers and outlines those application scenarios.

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In case of real numbers, Arithmetic operations makes sense as quantities are measured in real numbers.
 •  add and subtract : `3.3` m/sec + `2.1` m/sec speed
 •  multiply and divide : `4.2` meter `xx 1.1` coins per meter

It is so far explained that complex numbers are solutions to polynomials. In that context, what does it mean to
 •  add and subtract complex numbers
 •  multiply and divide complex numbers

This topic explains that from application perspective.

Most authors directly discuss the abstraction without any need to understand an application perspective. And, when one comes across the application where the concepts are relevant, at that time he/she has to put in extra effort to understand the abstraction in the application context.

As part of this course one application context is explained.

Complex numbers serve as mathematical model for
 •  Alternating Current(AC) sine waves with amplitude and phase, and the passive elements in the AC circuits that modify amplitude and phase of the input.
 •  Analysis of Systems that modify amplitude and phase of input.
 •  Analysis of Signals that can be expressed as sum of sine waves of different frequencies, amplitude and phase.
 •  Abstracted 2D plane : called complex plane, and abstract problems in that.

Note the key points in the following :
 •  AC with amplitude and phase
 •  Systems that change amplitude and phase of input.

What is common in the above applications?

  • Amplitude and phase
  • Amplitude and phase
  • nothing common in them

The answer is 'Amplitude and phase'. The frequency of sine waves is a constant in these.

Complex numbers model amplitude and phase for
 •  sine waves ( amplitude and phase)
 •  passive elements (modification of amplitude and phase)

A sine wave of frequency `f` with amplitude `r` and phase `theta` is equivalently a complex number `r cos theta + i r sin theta`.

An element acts on an incoming sine wave and affects the amplitude and phase. The factor by which it modifies `r` and the amount by which it shifts the phase `theta` is equivalently a complex number `r cos theta + i sin theta`.

Note 1: These statements are explained in topic 'Physics : Alternating Current'

Note 2: For the super-scientific readers -- The passive elements does rate of change and accumulation of input sine wave. In the case of sine waves, those are modeled as differentiation and integration, and effectively result in a phase shift.

The upcoming page "Complex Number : Modeling sine waves" provides proofs and detailed explanation on how complex number is used in sine wave with amplitude and phase.

When the frequency of sine wave is constant :
 •  Addition of two sine waves with different amplitude and phase results in one sine wave of result-amplitude and result-phase. This is modeled by complex number addition / subtraction.
 •  Interaction of one sine wave input by a passive component that changes amplitude and phase, results in one sine wave of result-amplitude and result-phase. This is modeled by complex number multiplication and division.

Complex numbers provide the mathematical model for amplitude and phase of sine waves and how they are modified.

Complex numbers are a tool to approach problems in 2D complex plane.

Complex Arithmetic is understood from these applications perspective.

Application Scenario: Complex number `a+ib` model
 •  a sine wave of amplitude and phase
 •  an element that modifies amplitude and phase of sine waves
Mathematical operations (like addition, subtraction, multiplication, division) between complex numbers are defined for that model.

Solved Exercise Problem:

What is an example of complex number serving as mathematical model?

  • Amplitude and phase of sine waves
  • Amplitude and phase of sine waves
  • It is just abstraction

The answer is 'Amplitude and phase of sine waves'

slide-show version coming soon