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Thought-Process to Discover Knowledge

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

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User Guide

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summary of this topic

### Introduction to Complex Numbers

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Home

Generic Form of Complex Numbers

»  All complex numbers are in the form a+ib

»  Example: Solutions of x^3=-1

→  (root(3)(-1))_(1st) =e^((i pi)/3) = 1/2 + i sqrt(3)/2

→  (root(3)(-1))_(2nd) =e^(i pi) = -1

→  (root(3)(-1))_(3rd) =e^((i 5pi)/3) = 1/2 - i sqrt(3)/2

### Generic Form of Complex Numbers

plain and simple summary

nub

plain and simple summary

nub

dummy

Complex numbers are in the form a+bi.

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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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.

Keep tapping on the content to continue learning.
Starting on learning "Generic form of complex numbers". ;; Irrational numbers are specified with the operations such as plus, square root, 5th root et cetera. ;; Unlike irrational numbers, complex numbers have an uniform representation. This page explains this in detail.

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

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
• solution does not exist

The answer is 'third root of -1'.

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

• 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
• 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
• 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
• 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

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?

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

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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

• 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.

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

• 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.

Progress

Progress

A quadratic equation of the form p x squared + q x + r= 0 can be re-arranged to x+ q by 2p squared, equals, minus r by p plus q by 2 p squared . What is the form in which the roots are?
a
a
i
i
b
i b
plus
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 cube equals minus 1?
third;root;minus
third root of minus 1
solution;does;not;exist
solution does not exist
The answer is third root of minus 1.
How many solutions are there for x cube equals minus 1?
3
three
1
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 ; square root of 2, third root of 5, pi, 3 minus 2 times seventh root of 5, et cetera. ;; In case of complex numbers defined for quadratic equations, a new number was defined i and the numbers are presented in a numerical expression form a+i b. ;; Do we have to define a number of new elements (like i= square root of minus 1) for each of the following? three solutions of x cube equals minus 1 ;; four solutions of x power 4 equals minus 1 ;; n solutions of x power n equals minus 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 cube = minus 1. How to find the three roots of this equation of degree 3?
use;r;are;or;theta;i;form
Use r e power i theta form for the real number
only;one;solution
Only one solution is possible
The answer is "Use r e power i theta form for the real number"
The equation x cube = minus 1 is given as, x cube = 1 e power i 2n +1 pi, where n=0,1,2,3, et cetera. ;; Solving that we get the three solutions. ;; e power i pi by 3 ;; e power i pi ;; e power i 5 pi by 3.
Generalizing what we have learned : For any algebraic expression resulting in a complex number, the solution can be equivalently given in the form a+b i where i equals square root of minus 1. This representation is named as 'complex number'.
What does the word 'complex' mean?
consisting;many;different;parts
consisting of many different parts
unified;one;item
unified one item
The answer is 'consisting of many different parts'.
A complex number a+b i 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?
reality;existing;thing
actually existing as a thing in reality
imagined;not;exist
that is imagined and does not exist
The answer is 'actually existing as a thing in reality'.
What does the word 'imaginary' mean?
reality;existing;thing
actually existing as a thing in reality
imagined;not;exist
that is imagined and does not exist
The answer is 'that is imagined and does not exist'.
For a complex number a+b i ;; 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 ?
real
In the complex number 3+8.1i , what is the part 8.1 ?
imaginary
Complex numbers are in the form a+b i.
Generic Form of Complex Numbers are in the form a+b i, where a, b are real numbers, and i equals square root of minus 1.
is square root 2 plus i fifth root 7 a complex number?
yes;s;real
Yes. The real and imaginary parts can be real numbers.
no;irrational;not
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.
is square root minus 2 plus i 5 a complex number?
yes;s;can;represented;form
Yes. It can be represented in a+ib form.
no;square;root;minus;2
No, square root minus 2 is not a real number
The answer is 'Yes. It can be represented in a+i b form.' Simplify the same as square root of 2 i+i 5 which is 0+i 5+ square root 2. The result is in a+i b form.

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