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Complex Numbers, Iota, Algebraic operations, Geometrical Representation, Definition, Practice Problems & FAQs

Complex Numbers, Iota, Algebraic operations, Geometrical Representation, Definition, Practice Problems & FAQs

Going back to history, we know that the discovery of positive and negative numbers along with and variables have simplified mathematics.These numbers are used to solve equations like , and . Further, rational numbers were needed to solve equations of the type .

Now, let us consider,

This equation cannot be solved using the existing rational numbers. Hence irrational numbers came into picture.The set of numbers were then extended to a larger set of numbers known as real numbers which consist of rationals and irrationals.

However, solving equations like was not possible because we have studied that the square of a real number is always greater than or equal to zero.

To solve such kind of equations great mathematician “Leonhard Euler” introduced imaginary number iota represented by , where whose square is which is a real number which is not positive.

So, can be solved as which can also be written as


Let's proceed further to understand more about complex numbers and some important concepts related to a complex number with the help of some examples in detail.

Table of contents

Definition of Complex Numbers

A number of the form is known as a complex number, where are real and A complex number is denoted by and written as .

Here, is known as the real part of and is denoted as , is known as the imaginary part of and is denoted by .

The set of complex numbers can be defined as . This set of complex numbers is the largest set of numbers.


Note : 

  • The set of real numbers is a proper subset of the set of complex numbers, because any real number can be written in the form of a complex number, like 2 can be written as

Here, can be referred to as a purely real number. Whereas, numbers like

are referred to as purely imaginary numbers.

Integral Powers of Iota

To summarize we can say that,

For , we get-

Note : Sum of any four consecutive powers of is zero, i.e.,

Algebraic Operations with Complex Numbers

Let us assume two complex numbers & such that

Addition Multiplication

 

Subtraction Division 

,

 

Multiply and divide by .

,

Properties of Algebraic Operations on Complex Numbers

Let be any three complex numbers 

  • Addition of complex numbers is commutative

i.e.,  

  • Addition of complex numbers is associative

 

  • Multiplication of complex numbers is commutative and associative

i.e., and

  • Multiplication of complex numbers is distributive over addition and subtraction

i.e., and

  • If then but for any complex numbers if , then not necessarily,

For example: and , then

  • If the product of two complex numbers is , then at least one will have to be .
  • Additiveinverseof a complex number is
  • Multiplicativeinverseof a complex number is

Note : 

As in reals, associative property and commutative properties do not hold true for subtraction and division of complex numbers.

Algebraic Identities:

The algebraic identities that we have learnt for real numbers hold good for complex numbers as well. Some of these are listed below:

Geometrical Representation of a Complex Number

To understand how we can represent a complex number geometrically, we need to first understand the argand plane.

Argand Plane

Similar to the plane, the Argand(or complex) plane is a system of rectangular coordinates in which the complex number is represented by the point whose coordinates are ( .


The horizontal axis is known as the ‘Real axis’ and the vertical axis is known as the ‘Imaginary axis’.

A complex number, , can be represented on the argand plane by a unique point .


Let , point is represented on an argand plane for different quadrants.

  • As the name decribes all the points on the real axis are purely real. A purely real number , i.e., , is represented by the point on the Real axis.
  • As the name decribes all the points on the Imaginary axis are purely imaginary. A purely imaginarynumber , i.e., , is represented by the point on the Imaginary axis.

Practice Problems

Example : Find the value of  
<math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>&#x2211;</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>=</mo><mn>13</mn></mrow></munderover><mfenced><mrow><msup><mi>i</mi><mi>n</mi></msup><mo>+</mo><msup><mi>i</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>,</mo><mi>w</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>e</mi><mo>&#xA0;</mo><mi>i</mi><mo>=</mo><msqrt><mo>-</mo><mn>1</mn></msqrt></math>

Solution : 
<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#xA0;</mo><munderover><mo>&#x2211;</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>=</mo><mn>13</mn></mrow></munderover><mfenced><mrow><msup><mi>i</mi><mi>n</mi></msup><mo>+</mo><msup><mi>i</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>=</mo><munderover><mo>&#x2211;</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>=</mo><mn>13</mn></mrow></munderover><mfenced><msup><mi>i</mi><mi>n</mi></msup></mfenced><mo>+</mo><munderover><mo>&#x2211;</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>=</mo><mn>13</mn></mrow></munderover><mfenced><msup><mi>i</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mfenced><mspace linebreak="newline"/><mo>=</mo><mfenced><mrow><mi>i</mi><mo>+</mo><msup><mi>i</mi><mn>2</mn></msup><mo>+</mo><msup><mi>i</mi><mn>3</mn></msup><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msup><mi>i</mi><mn>13</mn></msup></mrow></mfenced><mo>+</mo><mfenced><mrow><msup><mi>i</mi><mn>2</mn></msup><mo>+</mo><msup><mi>i</mi><mn>3</mn></msup><mo>+</mo><msup><mi>i</mi><mn>4</mn></msup><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msup><mi>i</mi><mn>14</mn></msup></mrow></mfenced><mspace linebreak="newline"/><mo>&#x2235;</mo><mi>s</mi><mi>u</mi><mi>m</mi><mo>&#xA0;</mo><mi>o</mi><mi>f</mi><mo>&#xA0;</mo><mi>n</mi><mo>&#xA0;</mo><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mi>s</mi><mo>&#xA0;</mo><mi>o</mi><mi>f</mi><mo>&#xA0;</mo><mi>a</mi><mo>&#xA0;</mo><mi>G</mi><mo>.</mo><mi>P</mi><mo>&#xA0;</mo><mi>w</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo>&#xA0;</mo><mi>f</mi><mi>i</mi><mi>r</mi><mi>s</mi><mi>t</mi><mo>&#xA0;</mo><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mo>&#xA0;</mo><mo>'</mo><mi>a</mi><mo>'</mo><mo>&#xA0;</mo><mi>a</mi><mi>n</mi><mi>d</mi><mo>&#xA0;</mo><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>o</mi><mi>n</mi><mo>&#xA0;</mo><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mo>&#xA0;</mo><mo>'</mo><mi>r</mi><mo>'</mo><mo>&#xA0;</mo><mi>i</mi><mi>s</mi><mo>&#xA0;</mo><mfrac><mrow><mi>a</mi><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>r</mi><mi>n</mi></msup></mrow></mfenced></mrow><mrow><mn>1</mn><mo>-</mo><mi>r</mi></mrow></mfrac><mspace linebreak="newline"/><mi>A</mi><mi>b</mi><mi>o</mi><mi>v</mi><mi>e</mi><mo>&#xA0;</mo><mi>s</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo>&#xA0;</mo><mi>i</mi><mi>s</mi><mo>&#xA0;</mo><mi>a</mi><mo>&#xA0;</mo><mi>G</mi><mo>.</mo><mi>P</mi><mo>&#xA0;</mo><mi>w</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo>&#xA0;</mo><mi>f</mi><mi>i</mi><mi>r</mi><mi>s</mi><mi>t</mi><mo>&#xA0;</mo><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mi>s</mi><mo>&#xA0;</mo><mo>'</mo><mi>i</mi><mo>'</mo><mo>&#xA0;</mo><mi>a</mi><mi>n</mi><mi>d</mi><mo>&#xA0;</mo><mo>'</mo><msup><mi>i</mi><mn>2</mn></msup><mo>'</mo><mo>&#xA0;</mo><mi>r</mi><mi>e</mi><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>l</mi><mi>y</mi><mo>&#xA0;</mo><mi>a</mi><mi>n</mi><mi>d</mi><mo>&#xA0;</mo><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>o</mi><mi>n</mi><mo>&#xA0;</mo><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>s</mi><mo>&#xA0;</mo><mo>'</mo><mi>i</mi><mo>'</mo><mspace linebreak="newline"/><mo>=</mo><mfrac><mrow><mi>i</mi><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>i</mi><mn>13</mn></msup></mrow></mfenced></mrow><mrow><mn>1</mn><mo>-</mo><mi>i</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mi>i</mi><mn>2</mn></msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>i</mi><mn>13</mn></msup></mrow></mfenced></mrow><mrow><mn>1</mn><mo>-</mo><mi>i</mi></mrow></mfrac><mo>&#xA0;</mo><mspace linebreak="newline"/><mo>=</mo><mfrac><mrow><mi>i</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mi>i</mi></mrow></mfenced></mrow><mrow><mn>1</mn><mo>-</mo><mi>i</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mi>i</mi></mrow></mfenced></mrow><mrow><mn>1</mn><mo>-</mo><mi>i</mi></mrow></mfrac><mspace linebreak="newline"/><mo>=</mo><mi>i</mi><mo>-</mo><mn>1</mn><mspace linebreak="newline"/></math>

As we know that sum of any four consecutive powers of iota is , then above series can be simplified to:

Hence value of given expression is .

Example : Find the set of points on the complex plane such that is real and positive where .

Solution :


<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>a</mi><mi>s</mi><mi>e</mi><mo>&#xA0;</mo><mn>1</mn><mo>&#xA0;</mo><mo>:</mo><mo>&#xA0;</mo><mspace linebreak="newline"/><mi>P</mi><mi>u</mi><mi>t</mi><mi>t</mi><mi>i</mi><mi>n</mi><mi>g</mi><mo>&#xA0;</mo><mi>y</mi><mo>=</mo><mn>0</mn><mo>&#xA0;</mo><mi>i</mi><mi>n</mi><mo>&#xA0;</mo><mi>e</mi><mi>q</mi><mo>.</mo><mo>&#xA0;</mo><mfenced><mi>i</mi></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>&gt;</mo><mn>0</mn><mo>&#xA0;</mo><mspace linebreak="newline"/><mo>&#x21D2;</mo><mi>x</mi><mo>&#x2208;</mo><mi>R</mi><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mfenced><mrow><mo>&#x2235;</mo><mi>a</mi><mo>&gt;</mo><mn>0</mn><mo>,</mo><mo>&#xA0;</mo><mi>D</mi><mo>&lt;</mo><mn>0</mn></mrow></mfenced><mspace linebreak="newline"/><mo>&#x2234;</mo><mi>L</mi><mi>o</mi><mi>c</mi><mi>u</mi><mi>s</mi><mo>&#xA0;</mo><mi>i</mi><mi>s</mi><mo>&#xA0;</mo><mi>y</mi><mo>=</mo><mn>0</mn><mo>&#xA0;</mo><mo>&amp;</mo><mi>x</mi><mo>&#x2208;</mo><mi>R</mi><mspace linebreak="newline"/><mi>i</mi><mo>.</mo><mi>e</mi><mo>.</mo><mo>,</mo><mo>&#xA0;</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>i</mi><mi>n</mi><mi>i</mi><mi>t</mi><mi>e</mi><mo>&#xA0;</mo><mi>p</mi><mi>o</mi><mi>i</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo>&#xA0;</mo><mi>o</mi><mi>n</mi><mo>&#xA0;</mo><mi>x</mi><mo>-</mo><mi>a</mi><mi>x</mi><mi>i</mi><mi>s</mi><mspace linebreak="newline"/><mspace linebreak="newline"/></math>

i.e. all the points lying on the real axis.

1

Example : If is purely imaginary, then find the value of .

Solution: 

Given,

Multiplying and dividing by , we get,

 

To be purely imaginary,

We know that if ,  

Thus,

Example : Solve

Solution : 

Given,


<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>(</mo><mn>1</mn><mo>+</mo><mi>i</mi><mo>)</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>(</mo><mn>2</mn><mo>-</mo><mi>i</mi><mo>)</mo><mi>x</mi><mo>-</mo><mn>5</mn><mo>-</mo><mn>3</mn><mi>i</mi><mo>=</mo><mn>0</mn><mspace linebreak="newline"/><mo>&#x21D2;</mo><mi>x</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mfenced><mrow><mn>2</mn><mo>-</mo><mi>i</mi></mrow></mfenced><mo>&#xB1;</mo><msqrt><mn>16</mn><msup><mfenced><mrow><mn>2</mn><mo>-</mo><mi>i</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>.</mo><mn>2</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow></mfenced><mfenced><mrow><mn>5</mn><mo>+</mo><mn>3</mn><mi>i</mi></mrow></mfenced></msqrt></mrow><mrow><mn>2</mn><mo>.</mo><mn>2</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow></mfenced></mrow></mfrac><mspace linebreak="newline"/><mo>&#x21D2;</mo><mi>x</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mfenced><mrow><mn>2</mn><mo>-</mo><mi>i</mi></mrow></mfenced><mo>&#xB1;</mo><msqrt><mn>16</mn><mfenced><mrow><mn>4</mn><mo>-</mo><mn>1</mn><mo>-</mo><mn>4</mn><mi>i</mi></mrow></mfenced><mo>+</mo><mn>8</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow></mfenced><mfenced><mrow><mn>5</mn><mo>+</mo><mn>3</mn><mi>i</mi></mrow></mfenced></msqrt></mrow><mrow><mn>4</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow></mfenced></mrow></mfrac><mspace linebreak="newline"/><mo>&#x21D2;</mo><mi>x</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mfenced><mrow><mn>2</mn><mo>-</mo><mi>i</mi></mrow></mfenced><mo>&#xB1;</mo><msqrt><mn>48</mn><mo>-</mo><mn>64</mn><mi>i</mi><mo>+</mo><mn>8</mn><mfenced><mrow><mn>5</mn><mo>+</mo><mn>8</mn><mi>i</mi><mo>-</mo><mn>3</mn></mrow></mfenced></msqrt></mrow><mrow><mn>4</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow></mfenced></mrow></mfrac><mspace linebreak="newline"/><mo>&#x21D2;</mo><mi>x</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mfenced><mrow><mn>2</mn><mo>-</mo><mi>i</mi></mrow></mfenced><mo>&#xB1;</mo><msqrt><mn>64</mn></msqrt></mrow><mrow><mn>4</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow></mfenced></mrow></mfrac><mspace linebreak="newline"/><mo>&#x21D2;</mo><mi>x</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mfenced><mrow><mn>2</mn><mo>-</mo><mi>i</mi></mrow></mfenced><mo>&#xB1;</mo><mn>8</mn></mrow><mrow><mn>4</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow></mfenced></mrow></mfrac><mspace linebreak="newline"/><mo>&#x21D2;</mo><mi>x</mi><mo>=</mo><mfrac><mrow><mfenced><mrow><mn>2</mn><mo>-</mo><mi>i</mi></mrow></mfenced><mo>&#xB1;</mo><mn>2</mn></mrow><mfenced><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow></mfenced></mfrac><mo>=</mo><mfrac><mrow><mn>4</mn><mo>-</mo><mi>i</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow></mfrac><mo>,</mo><mfrac><mrow><mo>-</mo><mi>i</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>i</mi><mo>&#xA0;</mo></mrow></mfrac><mspace linebreak="newline"/><mo>&#x21D2;</mo><mi>x</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mo>-</mo><mi>i</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow></mfrac><mo>&#xD7;</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi>i</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>i</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><mo>-</mo><mn>5</mn><mi>i</mi></mrow><mn>2</mn></mfrac><mspace linebreak="newline"/><mi>a</mi><mi>n</mi><mi>d</mi><mo>,</mo><mo>&#xA0;</mo><mi>x</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mi>i</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>i</mi><mo>&#xA0;</mo></mrow></mfrac><mo>&#xD7;</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi>i</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>i</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mfenced><mfrac><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow><mn>2</mn></mfrac></mfenced><mspace linebreak="newline"/><mo>&#x2234;</mo><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mo>-</mo><mn>5</mn><mi>i</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mo>&#xA0;</mo><mo>-</mo><mfenced><mfrac><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow><mn>2</mn></mfrac></mfenced><mspace linebreak="newline"/><mspace linebreak="newline"/><mspace linebreak="newline"/><mspace linebreak="newline"/></math>

FAQs

Q 1.Is zero a complex number?
Answer: Since can be written as , it is a purely real as well as purely imaginary number. Hence, we can say it is both real and complex.

Q 2.Is a real number a complex number too?
Answer: 
Yes, every real number is a complex number but every complex number may not be a real number.

Q 3.Does even and odd complex numbers exist?
Answer: 
A complex number can be an even number too if its real part is an even integer and its imaginary part is zero. A complex number can be an odd number too if its real part is an odd integer and its imaginary part is zero.

Q 4.What are the real life applications of complex numbers?
Answer:
Complex numbers are used in several areas such as signal processing, AC circuit analysis, Quantum mechanics etc.

Related Concept Links
Conjugate of a Complex Numbers Modulus of a Complex Number
nth roots of unity Argument of a Complex Number

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