Representation of Complex Numbers

Introduction:

When we multiply real and imaginary numbers, their product is complex numbers. The symbol of the complex number is ‘z,’ and its formula is a + ib. This formula uses the variables a and b; we consider them real numbers and the variable ‘i’ as an imaginary number. Re(z) indicates the real component’ a’ and Im(z) with variable ‘b’ is the imaginary part. The imaginary number is ib.

There are many ways to represent a complex number in a complex plane; one way is a polar representation of complex numbers.

Ways of Complex Number Representation 

Complex numbers can calculate the negative number’s square root. Many scientific fields use these numbers, including signal processing, electromagnetism, fluid dynamics, quantum physics, and vibration analysis.

Complex number representation is on a rectangular coordinate plane, also known as a complex plane (denoted by C) with points (a,b). Point A is on a rectangular horizontal coordinate, and b is vertical. 

There are many ways of complex number representation on a complex plane:

A – Geometric Representation of Complex Numbers

We put a complex number on a complex plane in the geometric representation of complex numbers. The X-axis, also called the real axis, represents the complex number’s real side. We label it as ‘R’ or ‘Re.’ On the other hand, the Y-axis represents the imaginary part. It is also known as the imaginary axis. We label it as ‘iR’ or ‘Im’. The point from which coordinates origin is called ‘zero-point.’

The number 1 on a complex plane is one unit to the right side of the zero point on the real axis, X, and ‘i’ lies on one unit above the zero point on the imaginary axis Y. 

B- Matrix Representation of Complex Numbers

To understand the matrix representation of complex numbers, we should know that Matrix multiplication is noncommutative, which means that, in general, two matrix PQ ≠ QP. But there is a direct correspondence of all matrixes with the real and imaginary numbers. However, there is a difference between matrix identity and multiple identities of real numbers.

Further, a complex matrix, also known as a complex Vector, is a matrix of complex numbers. The addition of vector and matrix is the elementary addition, but the dot or the inner product requires a slight modification. We should define the length of a complex vector and measure the length from a point to its origin. Keeping the zero vector for zero is essential, and the fix is the sum of squares of the magnitude.

C-Polar Representation of Complex Numbers

The polar representation of complex numbers is another way to represent complex numbers on a complex plane. We know that the horizontal axis is the real axis, and the vertical axis is the imaginary axis. We define real and complex numbers in terms of r and θ, where r is the length of the vector and θ is the angle formed between the two axes. 

In polar representation of complex numbers we represent the complex number z=a+ib as z=r (Cosθ + iSinθ). In this polar representation of complex numbers, r is the modulus, and θ is the argument of complex numbers. 

When we divide or multiply complex numbers, the polar representation of complex numbers is advantageous.

Properties of Complex Numbers

To understand complex numbers in a better way and to perform mathematical operations using complex numbers, it is essential to understand their properties.

A- Complex Number Conjugate

The complex number’s conjugate is generated by taking the identical actual component of the complex number and altering the imaginary part to its additive inverse. When the sum and product of two complex numbers are actual, they are complex conjugate numbers.

B- The Reciprocal of a Complex Number

The reciprocal of a complex number helps divide one complex number by another complex number. The division of complex numbers is equal to the product of one complex number and the reciprocal of another complex number.

C- Complex Number Equality

Complex number equality is analogous to absolute number equality. The two complex numbers are equal if their real and imaginary part is equal. The magnitude of two complex numbers in the polar form is identical if their argument (angle) varies by an integral multiple of 2.

D- Complex Number Ordering

We cannot order complex numbers, though we can order real numbers and other related numbers. There is no ordering of complex numbers compatible with addition and multiplication.

Complex Number’s Operations

A- Addition

Complex number addition is analogous to natural number addition. In complex numbers, the real component is multiplied by the real part, while we multiply the imaginary number by the imaginary part.

B- Subtraction

The procedure of subtracting complex numbers is identical to subtracting natural numbers. In this case, for any two complex numbers, the subtraction is done individually over the real component and then across the imaginary part.

C- Multiplication

Complex numbers multiplication differs somewhat from the multiplication of natural numbers. Complex number multiplication in polar form varies slightly from the method mentioned above.

D- Division

We use the formula for the reciprocal of a complex number in complex number division.

Conclusion 

We can represent complex numbers on two distinct axes, horizontal and vertical. The horizontal axis is the real axis, while the vertical axis is the imaginary axis. Re(z) represents the real axis, while Im(z) represents the imaginary axis. We add, subtract, divide, and multiply the complex number’s natural and imaginary components in the same way as we do these operations in elementary algebra.