Complex Number Formula

A combination of a real number and an imaginary number forms a complex number. The concept of the two-dimensional complex plane is explained in detail with the help of a complex number by using the horizontal axis for the real part and the vertical axis for the imaginary part. Let us learn complex number formula here.

What is a Complex Number?

A complex number is a number having both real and imaginary parts that can be expressed in the form of a + bi, where a and b are real numbers and i is the imaginary part, which should satisfy the equation i² = −1. In complex number, a is the real part and b is the imaginary part of the complex number.

Equality of Complex Number Formula

Take this equation into consideration. a+bi=c+di. Here, real part is equal with each other and imaginary parts are equal i.e. a=c and b=d

• Addition of Complex Numbers: (a+bi)+(c+di) = (a+c) + (b+d)i
• Subtraction of Complex Numbers: (a+bi)−(c+di) = (a−c) + (b−d)i
• Multiplication of Complex Numbers: (a+bi)×(c+di) = (ac−bd) + (ad+bc)i
• Multiplication Conjugates: (a+bi) × (a+bi) = a²+b²
• Division of Complex Numbers: \(\frac{(a+bi)}{(c+di)} = \frac{a+bi × c−di}{c+di × c−di}\)

Source:en.wikipedia.org

Terms used in Complex Numbers:

•  Argument – Argument is the angle we create by the positive real axis and the segment connecting the origin to the plot of a complex number in the complex plane.
• Complex Conjugate –  For a given complex number a + bi, a complex conjugate is a – bi.
• Complex Plane – It is a plane which has two perpendicular axis, on which a complex number a + bi is plotted having the coordinate as (a, b).
• Imaginary Axis  –  The axis in the complex plane that in usual practice coincides with the y-axis of the rectangular coordinate system, and on which the imaginary part bi of the complex number a + bi is plotted.
• Modulus – In the complex plane, the modulus is the distance between the plot of a complex number and the origin.
• Polar Form Of A Complex Number – Let,  z be the complex number  a + bi,
• Polar form of z = r\((\cos(\Theta) + i\sin(\Theta))\), where r = | z| and \(\Theta\) is the argument of z.
• Real Axis  –  The real axis in the complex plane is that which coincides with the x-axis of the rectangular coordinate system. On the real axis, the real part of a complex number a + bi is plotted.

De Moivre’s Theorem

De Moivre’s theorem generalizes the relation to show that to raise a complex number to the n^th power, the absolute value is raised to the n^th power and the argument is multiplied by n.

Let \(z = r(\cos(\Theta) + i\sin(\Theta)\)

Then \(z^n = [r(\cos(\Theta) + i\sin(\Theta)]^n\)

= \(r^n(\cos(n\Theta) + i\sin(n\Theta)\)

Where, n is any positive integer

Proof of De Moivre’s Theorem:

De Moivre’s theorem states that \((\cos\Theta + i\sin\Theta)^n = \cos(n\Theta) + i\sin(n\Theta)\)

Assuming n = 1

\((\cos\Theta + i\sin\Theta)^1 = \cos(n\Theta) + i\sin(1\Theta)\)

Assume n = k is true

So, \((\cos\Theta + i\sin\Theta)^k = \cos(n\Theta) + i\sin(k\Theta)\)

Letting n = k + 1

\((\cos\Theta + i\sin\Theta)^(k+1) = \cos(n\Theta) + i\sin((k+1)\Theta)\)

Assuming n = k, we get

= \((\cos(k\Theta) + i\sin(k\Theta)) x (\cos\Theta + i\sin\Theta)\)

= \(\cos(k\Theta)\cos(\Theta) + i\cos(k\Theta)\sin(\Theta) + i\sin(k\Theta)\cos(\Theta) – sin(k\Theta)\sin(\Theta)\)

Now we know that,

\(\sin(a + b) = \sin(a)\cos(b) + \sin(b)\cos(a)and\)

\(\cos(a + b) = \cos(a)\cos(b) – \sin(a)\sin(b)\)

\(= \cos(k\Theta + \Theta) + i\sin(k\Theta + \Theta)\)

\(=\cos((k + 1)\Theta) + i\sin((k + 1)\Theta) \)

Solved Examples for Complex Number Formula

Q.1: Simplify 6i + 10i(2-i)

Solution: 6i + 10i(2-i)

= 6i + 10i(2) + 10i (-i)

= 6i +20i – 10 i²

= 26 i – 10 (-1)

= 26i + 10