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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

Generic Form of Complex Numbers

Irrational numbers are specified with the operations such as `+`, `sqrt( )`, `root(5)( )` etc. Unlike irrational numbers, complex numbers have an uniform representation. This page explains this in detail.



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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`. What is the form in which the roots are?

  • `a`
  • `i`
  • `i b`
  • `a+i b`
  • `a+i b`

The answer is '`a+i b`'

We had derived the complex number notation `a+i b` specifically for quadratic equations. Let us examine other cases where complex numbers are defined.

What is the solution of `x^3 = -1`?

  • third root of `-1`
  • third root of `-1`
  • solution does not exist

The answer is 'third root of `-1`'.

How many solutions are there for `x^3 = -1`?

  • three
  • three
  • one

The answer is 'three'. There are `3` possible solutions to the equation of degree `3`.

The irrational numbers are defined as numerical expressions of various kinds
 •  `sqrt(2)`
 •  `root(3)(5)`
 •  `pi`
 •  `3-2root(7)(5)`



In case of complex numbers for defined quadratic equations, a new number was defined `i` and the numbers are presented in a numerical expression form `a+ib`.

Do we have to define a number of new elements (like `i=sqrt(-1)`) for each of the following?
 •  three solutions of `x^3=-1`
 •  four solutions of `x^4=-1`
 •  n solutions of `x^n=-1`

That is not necessary as the Euler's representation solves this problem and provides a generic form for all complex numbers.

Consider the example `x^3=-1`. How to find the three roots of this equation of degree 3?

  • Use `r e^(i theta)` form for the real number
  • Use `r e^(i theta)` form for the real number
  • Only one solution is possible

The answer is 'Use `r e^(i theta)` form for the real number'. `

The equation `x^3=-1` is given as `x^3 = 1 e^(i (2n+1)pi)` where `n=0,1,2,3,4...`

The solution is found by taking third root of right-hand-side and substituting `n=0,1,2`.

Solving that we get the three solutions
 •  `e^((i pi)/3)`
 •  `e^(i pi)`
 •  `e^((i 5pi)/3)`

For values `n>=3`, the solution is same as that of `n=0,1,2`. For example, `n=3`, we get `e^((i 7pi)/3)` `= e^((i pi)/3)` -- which is same as the solution for `n=0`.

Generalizing what we have learned : For any algebraic expression resulting in a complex number, the solution can be equivalently given in the form `a+bi` where `i=sqrt(-1)`. This representation is named as 'complex number'.

What does the word 'complex' mean?

  • consisting of many different parts
  • consisting of many different parts
  • unified one item

The answer is 'consisting of many different parts'.

A complex number `a+bi` is called complex as it has two parts to it
 •  `a` - called the real part
 •  `b` - called the imaginary part.

What does the word 'real' mean?

  • actually existing as a thing in reality
  • actually existing as a thing in reality
  • that is imagined and does not exist

The answer is 'actually existing as a thing in reality'.

What does the word 'imaginary' mean?

  • actually existing as a thing in reality
  • that is imagined and does not exist
  • that is imagined and does not exist

The answer is 'that is imagined and does not exist'.

For a complex number `a+bi`
 •  `a` is the real part
 •  `b` is the imaginary part.

Historically, the real number solutions for polynomials were easily found, and the rest of the solutions were named with meaning "imaginary". This name stuck.

Later in this course, in the page "Modeling Sine Waves using Complex Numbers" the significance of real and imaginary parts are explained.

In the complex number `3+8.1i`, what is the part `3`?

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

The answer is 'Real Part'

In the complex number `3+8.1i`, what is the part `8.1`?

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

The answer is 'Imaginary Part'

Complex numbers are in the form `a+bi`.

Generic Form of Complex Numbers : are in the form `a+bi` where `a, b in RR` and `i=sqrt(-1)`.

Solved Exercise Problem:

Is `sqrt(2)+i root(5)(7)` a complex number?

  • Yes. The real and imaginary parts can be real numbers.
  • Yes. The real and imaginary parts can be real numbers.
  • No. The real and imaginary parts can not be irrational numbers.

The answer is 'Yes. The real and imaginary parts can be real numbers.'. Irrational numbers are real numbers.

Solved Exercise Problem:

Is `sqrt(-2)+i5` a complex number?

  • Yes. It can be represented in `a+ib` form.
  • Yes. It can be represented in `a+ib` form.
  • No. `sqrt(-2)` is not a real number

The answer is 'Yes. It can be represented in `a+ib` form.' Simplify the same as `sqrt(2)i+i5` which is `0+i(5+sqrt(2))`. The result is in `a+ib` form.

                            
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