BASIC PROPERTIES OF COMPLEX NUMBERS

WHAT ARE COMPLEX NUMBERS:

The necessity of complex numbers comes with the introduction of Imaginary numbers since there are imaginary numbers along with the real numbers, these numbers must also be represented in some form there comes the necessity of the complex numbers. For the question What are complex numbers? The solution is the combination of both real numbers as well as imaginary numbers.

Complex numbers are usually represented as follows:

z=a+ib 

The complex numbers are introduced to solve the equations which have no proper roots:

For example:x2+5 = 0

On solving the above equation

We get x=√-5, the solution is an imaginary root, with the same root answer 2,23606798 i.

Operations on Complex numbers:

Addition 

Subtraction

Multiplication

Division

Addition of two complex numbers:

Addition can be easily done by adding the separate parts of the complex number, like adding the real part with the real part of another complex number and adding the imaginary part of a complex number with the imaginary part of another complex number

Example :

Complex number-1  1+i9  

Complex number-2  2+i3

Result : 3+12i

Subtraction of two complex numbers:

Subtraction can be done in the normal way as like as an addition

Example : 

Complex number 1 – 3+2i

Complex number 2 – 1 + i

Result : 2 + i

Multiplication of two complex numbers:

Two complex numbers are multiplied in the following manner,

Consider 

Z1 = (a+ib) and Z2 = (j+ik)

Follow these steps:

Step 1: Multiply two real  numbers  – a x j

Step 2: Multiply the real of the first complex number with the imaginary part of the second number : (a х k )i

Step 3: Multiply the imaginary part of the first complex number with the second real number (bx j)i

Step 4: Multiply two imaginary parts of the complex numbers

(b x k) (-1)

Totally,(a x j)+[(a x k)+(b x j)]i+ -1(b x k)

Division of two complex numbers:

The division mainly needs rationalizing, the rationalizing factor is the opposite symbol of the imaginary part.

Consider the two complex numbers

(a+ib),(j+ik)

(a+ib)/(j+ik)

(a+ib)/(j+ik) = (a+ib)/(j+ik) x (j – ik)/(j-ik)

(2+i3)/(1+i2)=(2+i3)/(1+i2) x (1-i2)/(1-i2)

        =(8-i)/5

PROPERTIES OF THE COMPLEX NUMBERS:

ADDITIVE IDENTITY OF COMPLEX NUMBERS:

The additive identity state that for complex numbers if the real part and the imaginary part  coefficient have both zeros in them then the result of that complex number is 0

Example

Here a=0,b=0 ; a+ib = 0

0+0.i = 0

COMMUTATIVE LAW OF COMPLEX NUMBERS:

This law states that if a,b,c,d are the read numbers in the form of complex number

 (a+ib),(c+id)

Here a + ib = c + id then a = c and b = d.

This law is similar to the commutative law of the normal integers with the addition

For example, consider two complex numbers (2+5i) and (1+2i)

Add the two numbers we get (3+7i)

So, (2+1)=3

(5+2)i = 7i

Hence satisfies the Commutative law.

ADDITIVE IDENTITY:

Additive identity property is satisfied by the complex numbers,a + ij +0 =a +ij

For example ; z+0 = z

5+6i + 0 =5+6i

CONJUGATE PROPERTY:

The conjugate property of a complex number is nothing but if the conjugate is added to the same number then the result should be a real  number

This complex conjugate is only done for the imaginary part, for example, the conjugate of the 9i is -9i

Example: Find the complex conjugate of 1+9i

Ans:1-9i

So if we add 1+9i + (1-9i)

The result is 2 hence the conjugate results in the real number which satisfies the conjugate property

ADDITIVE INVERSE:

The complex numbers satisfy the additive inverse too. Additive inverse law states that if we add the opposite sign for the same number then the result must be zero

The additive inverse of the complex number (2+ij) is (-2-ij)

Hence adding the above two results in zero, which is the additive inverse

z=a+bi, then −z=−a−bi.

ASSOCIATIVE LAW OF ADDITION:

The complex numbers satisfy the associative law of addition too that means

  a+ (b+c) = (a+b) + c  

COMPLEX NUMBERS SATISFY THE MULTIPLICATIVE PROPERTIES:

COMMUTATIVE LAW OF MULTIPLICATION:

Consider two complex numbers z,w the commutative law of multiplication is :

zw=wz

Where z and w are two complex numbers

ASSOCIATIVE LAW OF MULTIPLICATION 

Consider three complex numbers, z,w,v.According to the Associative law of multiplication

z(wv)=(zw)v

Where z,w,v are the three complex numbers.

MULTIPLICATION IDENTITY:

Multiplication identity is nothing but multiplying with 1 results in the same number

(a+ib).1 = (a+ib)

Complex numbers satisfy the multiplicative identity.

There is one more property of the complex numbers:

Consider two complex numbers z1 and z2, for the given two numbers 

|z1+z2|≤ |z1|+|z2|

Conclusion:

Complex numbers satisfy almost all the properties of real numbers. The major difference between real numbers and complex numbers is the existence of imaginary parts.

The quadratic equations with have no real and distinct roots result in the formation of complex numbers. The distinct roots are nothing but the real numbers, imaginary roots are obtained for the equations with no real roots. These are imaginary but still, consist of an imaginary plane and a real plane to define the real numbers. We use the b2-4ac concept to find the quality of roots in quadratic equations. These are often represented in a graph by considering the imaginary part on Y-axis and the real part on X-axis.