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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 NumbersIntroduction to Complex Numbers

Representation of Complex Numbers (incomplete)

In this page, as initial understanding of complex numbers, quadratic equations are examined. The solutions to quadratic equations is in the form `a+ib`, where `i=sqrt(-1)`.

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When using the real numbers, we come across problems that are mathematically modeled as `x^2=-1`. There is no solution to this in real number system. So, what is done?

  • the equation has no solution
  • the equation has no solution
  • number system is extended beyond real numbers

The answer is 'number system is extended beyond real numbers'. This number system that is over-and-above the real number system and is named as 'complex numbers'.

To include solutions to polynomials which number system is introduced?

  • Pronunciation : Say the answer once
    Spelling: Write the answer once

The answer is 'Complex Numbers'

Irrational numbers are represented with numerical expressions or symbols.
How a complex number is represented? In this topic an incomplete explanation to the form of representing complex number is discussed.

In the topic 'Generic form of Complex Numbers', the information discussed in here is further developed to complete the explanation.

In irrational number system, the solution to `x^2=2` is given as `+-sqrt(2)`.
Learning from that, what could be the solution to `x^2=-1`?

  • `+-sqrt(-1)`
  • `+-sqrt(-1)`
  • `1`
  • `-1`

The answer is '`+-sqrt(-1)`'. The solution is represented as a numerical expression.

Note to students: In the attempt to develop knowledge in stages, the problems given in here are specific to quadratic equations. A more generic discussion will follow once first level of knowledge is acquired.

What is the solution to `x^2 = -4`?

  • `2`
  • `-2`
  • `+-2sqrt(-1)`
  • `+-2sqrt(-1)`
  • `+-2`

The answer is '`+-2sqrt(-1)`'

What is the solution to `(x+3)^2 = -4`?

  • `-1`
  • `5`
  • `-3+-2sqrt(-1)`
  • `-3+-2sqrt(-1)`
  • `+-sqrt(-1)`

The answer is '`-3+-2sqrt(-1)`'

It is noted that a quadratic equation of the form `px^2+qx+r= 0` can be re-arranged to `(x+q/(2p))^2 = -r/p + (q/(2p))^2`.
For any equation in this form, we can arrive at a solution in the form `a+b sqrt(-1)`.

`sqrt(-1)` is represented with a letter `i`
The solution to a quadratic equation is in the form `a+bi`.

Note : The said explanation covers only solutions to quadratic equations. Let us examine this representation in detail and then later generalize this for complex numbers.

[Initial understanding] Solution to quadratic equation is in the form `a+bi`, where `i=sqrt(-1)`.

Form of complex number : Solution to quadratic equation is `a+bi` where `a, b in RR` and `i=sqrt(-1)`

Solved Exercise Problem:

What are the solutions to the equation `(x-1)^2 = -16`?

  • `+-4`
  • `+-4i`
  • `1+-4i`
  • `1+-4i`
  • `+-1+-4i`

The answer is '`1+-4i`'.

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