Complex Numbers-Basic Properties

Complex numbers happen to be a delightful topic from the point of view that, on the one hand, they offer a great perspective to deal enrichment with pure mathematics, and on the other hand, they are a source of a great deal of physical understanding of reality. Today we are going to discuss complex numbers and their basic properties in detail.

This material talks about complex numbers. You will find brief information on the concept of complex numbers, a thorough explanation of properties of complex numbers, their functions and so on. So, let’s, start with the definition of complex numbers.

Definition of Complex Numbers

Complex numbers are defined to be the set of all symbols in the form x +iy, where x is a real number and y is an imaginary number. Here, i is the square root of minus -1 ( √-1). Remember X is a real number, where real means X square is non-negative.

C= { x +iy : x is real number and y is imaginary number}

A few examples of complex numbers are 1+2i, −4−5i, etc.

Basic Properties of Complex Number

In this section, you will find a thorough explanation of the basic properties of a complex number and its functions. So, let’s, learn these properties to expertise the knowledge of the complex numbers

1. Equality

The definition of equality for complex numbers is that the two complex numbers are equal if and only if their real parts are equal and their imaginary units are equal.

Consider this, x₁ + iy₁ = x₂ + iy₂

Where, x₁ =x₂ and y₁ = y₂

2. Conjugate

The conjugate of a complex number is keeping the real number of the equation as it is and changing the imaginary unit with its inverse. The conjugate is represented by  Z*  or ˉz sign mathematically.

For example,  if Z = x + iy, then its, conjugate would be

ˉz = x – iy

The addition and products of any two conjugates are real numbers then those would be called conjugates of complex numbers.

Z + ˉz =  ( x+ iy) + ( x-iy)

= 2x

And  Z * ˉz = ( x+iy) * (x -iy)

= x² + y²

3. Reciprocal

For the division of complex number with another complex number the reciprocal of a complex number is useful.

The reciprocal of Z = x + iy would be,

Z -1 = 1 / (x + iy)

= (x-iy )/ (x² + y² )

=  x / (x₂ + y₂ ) +  i (-y) / (x² + y² )

Hence, Z ≠ Z^-1

4. Ordering

The total ordering properties do not hold true in the case of complex numbers. The ordering property is satisfied with real numbers, but the counterpart imaginary unit does not satisfy the ordering properties.

Total ordering refers to  x ≠ y, it could be either X < Y or X > Y

Functions of Complex Numbers

As we know now, complex numbers are the set of all numbers in the form x+iy, where ‘x’ is a real number and ‘iy’ is the imaginary number. Will learn about how to perform the functions on complex numbers.

Addition of Complex Numbers

In addition, all we need to do is add similar units together, i.e., we need to group our real numbers together and our imaginary numbers together.

Let’s understand with simple scenario (a+bi) + (c +di). In this scenario, ‘a’ and ‘c’ are our real numbers groups & ‘bi’ and ‘di’ are our imaginary number groups.

This can be mathematically be expressed as (a + c) + (bi + di).

Example: (2 + 10i) + (2- 2i)

= (2+2) + (10i-2i)

= 4 + 8i

Subtraction of Complex Numbers

For subtraction will use a similar technique as we followed in addition of complex numbers, this time with a little deviation that, instead of addition, we will subtract the similar units.

Let’s have a look at case (a + bi) – (c + di), ). In this scenario, ‘a’ and ‘c’ shows real numbers & ‘bi’ and ‘di’ shows an imaginary number.

The solution can be can be (a – c) + (bi – di).

Example: ( 9 – 5i) – (11 + 2i)

= ( 9-11) + ( -5i – 2i)

= -2 + -7i

Multiplication of Complex Numbers

For multiplication of complex numbers, we need to multiply the ‘first term’ then the ‘outside term’ followed by multiplication of ‘inside term’ and “last term’ respectively.

Let’s see the scenario ( a + bi) (c + di),

Where a x c = ‘first term’, a x di = ‘outside term’, bi x c = ‘inside term’ and bi x di = ‘last term’

Mathematical expression = ac + adi + bci + bdi2, where,  i2 = ( √-1)2 = -1

Example: ( -4 + 2i) (5 + 7i)

=  (-4)*( 5 ) + (-4)*(7i) + (2i)*(5) + (2i)*(7i)

= -20 -28i + 10i + 14i2

=  -20 -28i + 10i -14 ( as we can refer above, i2 = ( √-1)2 = -1)

= -20-14 -28i +10i

= -34 -18i

Division of Complex Numbers

We can use complex conjugates for the division of complex numbers. The complex conjugate can be found by simply changing the sign of the complex number. We will focus on a conjugate of denominator only. This is because i2 = ( √-1)2  and we can’t have a square root in our denominator. A complex conjugate would allow removing a square root from the denominator. We will multiply and divide the conjugate of the denominator.

Consider this illustration,  (a+bi) / (c+di) *  (c-di)/ (c-di)

= (a+bi) (c-di) / (c+di) (c-di)

Example: -8+6i / 8-5i

First of all, we have to find the complex conjugate of the denominator 8-5i, which would be 8+5i to solve this will have to multiply and divide with this conjugate.

Therefore, new equation would be  (-8 + 6i) / (8-5i) * (8+5i) / (8+5i)

= (( -8 +6i) * (8+5i)) / ((8-5i) * (8+5i))

= [ (-8 * 8) + (-8 *5i) + (6i * 8) + ( 6i * 5i)] /[ (8-5i) * (8+5i)]

=  [-64 -40i + 48i -30 ]/ [(8)2 – (5i)2]

= [-94 +8i] /[ 64 +25]

= [-94 + 8i] /89

Conclusion

In this article describing complex numbers, we studied the concept of complex numbers and their basic properties in length. We covered several other topics such as the functions that can be performed on the complex numbers like addition, subtraction etc. and other related topics, along with the solved examples. We hope this study material helped you better understand complex numbers and their basic properties.