How are complex numbers used in real life?

The complex number is derived because of the roots of any nth equation. If the equation has real roots it is fine else equation may have roots other than real means the roots which are not real in this confusion of roots that are not real complex numbers are taken birth. If the nth power equation has real roots and also a combination of real and imaginary roots. Those real and imaginary roots combination we called complex numbers.

Complex numbers

Complex numbers are in the form of a+ib which is also called a general form of complex numbers. In a+ib form a is a real part of a complex number, b is the imaginary part of a complex number and i is defined as √(-1). There are many forms of complex numbers. They are,

1. General form (Z = a + ib).
2. Polar form (Z = r(cosθ + isinθ)).
3. Exponential form (Z = re^iθ).

General form of a complex number

A complex number has a general form that looks like Z=a+ib where a is the real part and b is the imaginary part and i is called iota which is √(-1). 

Complex Plane Representation

The polar form of a complex number

A complex number has a polar form that looks like Z = r(cosθ + isinθ) where rcosθ is the real part and rsinθ is the imaginary part and i is called iota which is √(-1). 

The Exponential form of a complex number

A complex number has an exponential form that looks like Z=reiθ where eiθ can also be written as cosθ+isinθ which looks like a polar form at the end.

Note: We can also be able to convert general form to polar form, exponential vice versa, also able to convert polar form to general form, exponential form vice versa, and also able to convert exponential to general, polar form vice versa. Now we have to discuss where these complex numbers are useful in real life.

Complex numbers in Real life

Complex numbers are used in many varieties of fields in real life. Below given are the most important uses of complex numbers, and their proper explanation is also explained.

1. Complex number is used in Electronics.
2. Complex number is used in Electromagnetism.
3. Complex number is used to simplify the unknown roots if roots are not real for quadratic equations.
4. Complex numbers are used in computer science engineering.
5. Complex number is used in mechanical and civil engineering.
6. Complex numbers are used in control systems.

• Complex numbers in Electronics

In electronics, we are used to representing the general form of a complex number in the circuit having voltage and current. In Electronics circuit is mainly based on current and voltage. Those two elements are put together as single complex numbers. Z = V+ iI is the complex representation of a circuit having both current and voltage where V is the real axis part and I is the imaginary axis part so that we can able to see the comparison of both V and I by representing as a complex number in electronics. Sometimes in RC circuits or RLC circuits, if we want to combine two elements, say for example resistor and inductor we can write it as R + jX_L as well as if in case resistor and capacitor complex number representation would be R + j X_c where X_L = jwL and X_c = 1/jwc.

• Complex numbers in Electromagnetism

In electromagnetism, the main elements are the electric field and the magnetic field. Those elements are represented in the form of complex numbers where the real number or real axis is represented by an electric field, the magnetic field is represented as an imaginary part of an imaginary axis.

• Complex numbers to find roots in Quadratic equation

A quadratic equation is used to find the roots of the equation. If the quadratic equation has real roots it is fine. If the quadratic equation cannot able find the real roots then roots are said to be not real roots named as imaginary roots. Imaginary roots can be find by the formula (-b + √(b² – 4ac))/2a, (-b – √(b² – 4ac))/2a are two complex imaginary roots if quadratic equation cannot able to find real roots. In this application, we can use complex numbers.

• Complex numbers used in Computer science engineering

In Computer science, data place a major role. The data cannot be seen visually because it is in the form of CSV files. Those CSV file data can be seen by using visual representation methods in computer science. That Visual representation is in the real axis and imaginary axis only so complex numbers are used to represent the data in a visual format for computer science technology. In 2D images also we can use complex numbers. Rotation of a point having real part and imaginary part and translation of a point in a 2D image represents the complex number.

• Complex numbers in Mechanical and Civil Engineering

In mechanical and civil engineering designing is the main for cars and buildings. To do such a thing we have to use 2D designing concepts that mainly depend on complex numbers only. Rotations are also used in drawing for a point is represented by a complex number only.

• Complex numbers in control systems

Control systems conversion of the time-domain system to frequency domain is required that is to be done using Laplace transform. In that poles and zeros of the system are addressed using a complex plane having a real and imaginary axis. This is the reason and uses for the complex number used in the control system.

Sample Problems

Problem1: Solve the quadratic equation X² + 5X + 3 = 0 and verify whether the roots are imaginary or not?

Solution:  

3X² + 5X + 3 = 0 looks like ax² + bx + c = 0

a = 3, b = 5, c = 3

Formula for roots of quadratic equation = (-b + √(b² – 4ac))/2a, (-b – √(b² – 4ac))/2a

= (-5 + √25 – 4 × 3 × 3)/2 × 3, (-5 – √25 – 4 × 3 × 3)/2 × 3

= (-5 + √25 – 36)/6, (-5 – √25 – 36)/6

= (-5 + √-11)/6, (-5 – √-11)/6 

= (-5+i 3.3)/6, (-5- i 3.3)/6

= -0.83 + i 0.55, -0.83 – i 0.55

The roots -0.83 + i 0.55, -0.83 – i 0.55 are in the form of a+ib and a-ib so they are complex numbers. The roots of the quadratic equations may be complex numbers. In this application of finding roots of quadratic equations complex numbers are used.

Problem 2: Calculate the impedance of an RC circuit if the resistor has a resistance of 2 ohms and the capacitor has a capacitance of 4 farads in a circuit which are in series and frequency f = 1kHz?

Solution:

Given that R = 2 ohm

C = 4 Farads

Impedance Z = R + X_c where X_c = 1/jwc

Z = R + (1/j(2 × π × f × C)

Substitute R, f, C in the above equation we get the impedance of the RC circuit.

Z = 2-j(1/ 2 × 3.14 × 1 × 4)

Z = 2 – j(1/6.28 × 4)

Z = 2 – j(0.0398) 

Impedance Z = 2 – j(0.0398) which is also in the form of complex numbers only. So that is why complex numbers are used in calculating the impedance of circuit.

Problem 3: If z₁ = 2 + 5i and z₂ = 7 + 9i. Calculate z₁ × z₂?

Solution:

Given that z₁ = 2 + 5i and z₂ = 7 + 9i 

z₁ × z₂ = (2 + 5i)(7 + 9i)

= 14 + 18i + 35i + (i² ) × 45

= 14 + 53i – 145

= -31 + 53i.

z₁ × z₂ which is in the form of complex number (a + ib) = -31 + 53i.

Problem 4: If If z₁ = 5.5 + 5i and z₂ = 7.7 + 7i. Calculate z₁ + z₂?

Solution:

Given that z₁ = 5 + 5i and z₂ = 7 + 7i 

z₁ + z₂ = 5.5 + 5i + 7.7 + 7i 

= 13.2 + 12i

z₁ + z₂ which is in the form complex number (a + ib) = 12 + 12i.

Problem 5: If If z₁ = 6 + 5i and z₂ = 6 + 7i. Calculate z₁ – z₂?

Solution:

Given that z₁ = 6 + 5i and z₂ = 6 + 7i

z₁ – z₂ = 6 + 5i – 6 + 7i

= 0 + 12i

z₁ – z₂ which is in the form complex number (a + ib) = 0 + 12i.

Problem 6: If If z₁ = 1 + 5i and z₂ = 2 + 5i. Calculate z₁/z₂?

Solution:

Given that z₁ = 1 + 5i and z₂ = 2 + 5i

z₁/z₂ = 1 + 5i/2 + 5i

Multiply and divide with 2 – 5i

z₁/z₂ = 1 + 5i × (2 – 5i)/(2 – 5i)(2 + 5i)

z₁/z₂ = (2 – 5i + 10i + 25)/(4 + 25)

z₁/z₂ = (27 + 5i)/29

z₁/z₂ = 0.93 + i 0.17

z₁/z₂ which is in the form of complex number (a + ib) = 0.93 + i 0.17.

Problem 7: How to represent complex number 9+10 i in the 2D complex plane?

Solution: 

Given that complex number a + ib = 9 + 10i.

Where a = 9 and b = 10 which means the real part is 9 and the imaginary axis part is 10