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## Job Shankar # Study in Detail of Properties of Complex Number

Complex numbers are a mathematical concept that helps in finding the square root of a negative number. The concepts and properties of complex number were first introduced by a Greek mathematician, Hero of Alexendria. He was the first person to find out the square root of a negative number. Furthermore, in the 16th century Italian mathematician Gerolamo Cardano came up with the real identity of a complex number. He was actually finding the negative roots of cubic and quadratic polynomial expressions.

In today’s scientific era, complex numbers have several applications in signal processing, finding simple interest , electromagnetism, vibration analysis, and quantum mechanics. In this guide let us understand more properties and information about complex numbers. So, without further delay let us roll the intro.

## Graphing Complex Number

Complex numbers comprise a real and an imaginary part and are ordered as (Re(z),Im(z)). Those numbers are represented as coordinate points in the Euclidean plane. It is a plane with several complex numbers as a complex plane or an Argand Plane. This plane is named after Jean Robert Argand. If we talk about its representation, the graphical representation will be z = a + ib. In this representation, a is the real part with reference to the x-axis, and the ib which is an imaginary part with the reference to the y axis. This is graphical interpretation or graphing of complex numbers.

## What Are The Properties Of Complex Numbers?

There are certain properties of complex numbers that are useful for students to know how arithmetic operations happen through complex numbers. Keep reading below to know what properties they have.

1. Equality of complex numbers

When two complex numbers z1=a1+ib1 and z2=a2+ib2 are said to be equal if a1 = a2 (which are complex numbers) and b1 = b2 that are imaginary numbers are equal. And this property is called the equality of complex numbers. This equality of complex numbers’ property is somewhat similar to the equality of real numbers only. And if both of the complex numbers have the same magnitude and their argument (in simpler words angle) differs by integral multiple, they are equal in the polar form.

1. Addition of Complex Numbers

If you want to know how to add complex numbers, well it is something similar to adding natural numbers. To add complex numbers, the real part of it is added to its real part and the imaginary part will be added with the imaginary part. For instance, if there are two complex numbers z1=a1+ib1 and z2=a2+ib2 the over result will be z1+z2= (a1+a2) +i(b1+b2). This is how the addition of complex numbers works. Certain laws like closure law, commutative law, additive identity, additive inverse, and associative law are included in this addition of complex numbers.

1. Subtraction of Complex Numbers

Like the addition of complex numbers, the same process is followed for subtracting natural numbers. It means to subtract them, this operation is performed on the real part separately and the imaginary part separately. z1=a+ib and z2=c+id, the final subtraction operation will be z1-z2 = (a – c) + i(b – d)

1. Multiplying Complex Numbers

Multiplying complex numbers is comparatively different from other natural numbers’ multiplication. To make the multiplication process easier you need the formula of i2=−. For multiplying two complex numbers z1=a+ib and z2=c+id the final solved multiplied outcome will be z1xz2 = (ca – bd) + i(ad + bc)

1. Division of Complex Numbers

If you have read something about the reciprocal of a complex number formula, you can divide complex numbers easily. z1=a+ib and z2=c+id are two complex numbers and the final division will be z1/z2= (a+ib)×1/(c+id)=(a+ib)×(c−id)/(c2+d2).

To make these operations work efficiently you need to have the basics clear.