Algebraic Operations on Complex Numbers

Algebraic Of Complex Numbers

Have you ever heard of complex numbers? Do you know what an iota is? What kind of number is \[\sqrt{-2}\]?  Does the number even exist? To get all your answers, let’s first understand the entire number system.

Number System

The number system is broadly divided into two parts: Real Numbers and Complex Numbers. 

  1. Real numbers 

Real numbers are those which can be shown on a number line. On the other hand, complex numbers are those which can not be expressed on a number line or be experienced in real life. Real Numbers are further divided into two categories called rational and irrational numbers. 

Rational numbers are numbers which can be expressed as fractions and their denominators are not equal to 0. All the real numbers which are not rational are Irrational numbers. Rational numbers are made by dividing two integers. Integers include all negative and positive natural numbers along with zero. Integers are further divided into two sub-categories: whole numbers and natural numbers. Whole numbers are positive counting numbers along with 0. When you remove 0 from whole numbers, we obtain positive counting numbers which are known as natural numbers.

  1. Complex Numbers

Complex numbers are also known as Imaginary numbers. Now that we know the definition of complex numbers and that complex numbers are the part of the Number System, let’s see some examples. 

All the negative numbers under root are imaginary numbers. 

\[\sqrt{-2}\] and \[\sqrt{-2}\] are two very different things. The first one is a real number. Since it’s a negative number under root the second one is a complex number. A complex number is represented in the following way: a+bi, where a is the real part and b is the imaginary part.

You can write the complex number \[\sqrt{-2}\]  in  a+bi form. 0+2i is equal to \[\sqrt{-2}\]. You must be wondering why are we using the symbol ’i’? What does it mean? Well, it is iota. Have you ever heard of iota? If not, then is all that you need to know about iota.


IOTA

Iota is a greek letter which is used to represent the imaginary part of a complex number. Iota(i) is considered to be the square root of -1. It may also be defined as a number whose square is -1.

i=\[\sqrt{-1}\]

i²=-1

i³=i

i⁴=1


Algebraic Operations On Complex Numbers:

Four types of algebraic operations can be done on complex numbers. These four algebra of complex numbers are:

  • Addition

  • Subtraction

  • Multiplication

  • Division

There are several properties that algebra on imaginary numbers follow:


Closure law

The sum or product of two imaginary numbers will always get you an imaginary number.


Commutative Law

If you change the order of imaginary number while adding or multiplying the result will not change that is the answer you get will always be the same. 


Associative Law

If you add or multiply any three complex numbers in any order the result will always remain the same. 


Existence of Additive Identity

This property tells us that if we add zero to any complex we will get the same complex number. This shows that there’s a number that can be added to get the same number back. It is also known as zero complex number and is denoted as 0 (or 0 + i0).


Existence of Additive Inverse

A complex number has the opposite sign for its both real and imaginary parts. This is known as the Existence of  Additive inverse.


Multiplicative Identity

Multiplicative Identity is a property which talks about the existence of a complex number that when multiplied to another will get the same result. it is denoted as 1 (or 1 + i0)


Multiplicative Inverse

It is a property of any non- zero complex number to have a reciprocal. This is known as the multiplicative inverse.


Distributive Property

When you split the multiplication of a complex number by another term this property is known as the distributive property.

Note: Subtraction follows all the properties followed by addition.


Fun Facts:

  1. Both real and imaginary parts are present in the square root of i.

  2. The N-th root can have N number of unique solutions and any root of i has multiple unique solutions

  3. The result may vary depending on whether i is present in the numerator or denominator in an imaginary fraction. 

  4. When you raise i to the i power, the number you get is a real number.

  5.  Numbers like \[\pi \], i and e are all related to one another.

FAQ (Frequently Asked Questions)

1. What Is A Complex Number? Discuss About The Powers Of IOTA.

Complex numbers are those which can not be expressed on a number line or be experienced in real life. Complex numbers are also known as Imaginary numbers. All the negative numbers under root are imaginary numbers. A complex number is represented in the following way: a+bi, where a is the real part and b is the imaginary part. 

For Example : 

1. 4+5i 

2. 10-9i

3. √-2 

Iota(i) is considered to be the square root of -1. It may also be defined as a number whose square is -1.

i=√-1

i²=-1

i³=i

i⁴=1

2. What All Algebraic Operations Can Be Done On Complex Numbers? What Is Argand Plane?

Four types of algebraic operations can be done on complex numbers. These four operations on the complex numbers are:

  • Addition

  • Subtraction

  • Multiplication

  • Division

Addition

(w + ix) + (y + iz) = (w + y) + i(x + z)

Subtraction

(w + ix) – (y + iz) = (w – y) + i(x – z)

Multiplication

(w + ix). (y + iz) = (wy – xz) + i(wz + xy)

Division

(w + ix) / (y + iz) = (wy+xz)/ (y2 + z2) + i(xy – wz) / (y2 + z2)

The way we have the cartesian plane and the XY plane in it to represent real numbers. Similarly, we have the Argand plane which has a system of rectangular coordinates to represent the complex numbers which are written in the form  a+ib. Here a and b are the coordinates.