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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.

User Guide

Welcome to nubtrek.

The content is presented in small-focused learning units to enable you to
think,
figure-out, &
learn.

Just keep tapping (or clicking) on the content to continue in the trail and learn.

User Guide

To make best use of nubtrek, understand what is available.

nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

nub,

trek,

jogger,

exercise.

User Guide

nub is the simple explanation of the concept.

This captures the small-core of concept in simple-plain English. The objective is to make the learner to think about.

User Guide

trek is the step by step exploration of the concept.

Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step.

User Guide

jogger provides the complete mathematical definition of the concepts.

This captures the essence of learning and helps one to review at a later point. The reference is available in pdf document too. This is designed to be viewed in a smart-phone screen.

User Guide

exercise provides practice problems to become fluent in the concepts.

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge.

summary of this topic

### Algebra of Complex Numbers

Voice

Voice

Home

Basis of Complex Number Arithmetic

»  Modeling sine waves
→  same frequency omega

→  different amplitude r

→  different phase phi

»  amplitude and phase

→  sin1.1x is shown in blue

→  2.5sin(1.1x+0.7) is shown in orange
2.5 amplitude is the size change.
0.7 phase is the horizontal shift.

The arithmetic is explained in next page.

### Complex Number Arithmetic

plain and simple summary

nub

plain and simple summary

nub

dummy

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.

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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

Keep tapping on the content to continue learning.
Starting on learning "Complex Number Arithmetic". ;; This page brings up the question, what are the applications of complex numbers and outlines those application scenarios.

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
• 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.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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.

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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

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

The answer is 'Amplitude and phase of sine waves'

Progress

Progress

In case of real numbers, Arithmetic operations makes sense as quantities are measured in real numbers. ;; For example ;; add and subtract : 3 point 3 meter per second plus 2 point 1 meter per second speed ; ; multiply and divide : 4 point 2 meter multiplied by 1 point 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 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;phase
Amplitude and phase
nothing;common
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 (that has amplitude and phase);; passive elements (that modifies 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 sine 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+i b 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.
What is an example of complex number serving as mathematical model?
amplitude;phase;sine;waves
Amplitude and phase of sine waves
just;abstraction
It is just abstraction
The answer is 'Amplitude and phase of sine waves'

we are not perfect yet...