Complex Numbers

Complex numbers are numbers which can be represented in the form of a+ib where i is an imaginary number known as iota with the value of (√-1). The key concepts highlighted in this lesson will include the following:

  • Complex Numbers Introduction
  • Algebraic Operation on Complex numbers
  • Power of Iota (i)
  • Complex Number Identities
  • Modulus and Conjugate of a Complex number
  • Argand Plane & Polar Representation of Complex Number
  • Quadratic Equation and Complex Numbers
Complex Number

Complex Number

While solving the equation of the form \(ax^{2}+bx+c = 0\), the roots of the equations can take three forms which are as follows:

      • Two Distinct Real Roots
      • Similar Root
      • No Real roots (Complex Roots)

Complex Numbers Introduction

The introduction of complex numbers in the 16th century made it possible to solve the equation \(x^{2}+1 = 0\). The roots of the equation are of the form \(x = \pm \sqrt{-1}\) and no real roots exist. Thus, with the introduction of complex numbers, we have Imaginary roots.

We denote \(\sqrt{-1}\) with the symbol i, where i denotes Iota (Imaginary number).

An equation of the form z= a+ib, where a and b are real numbers, is defined to be a complex number. The real part is denoted by Re z = a and the imaginary part is denoted by Im z = b.

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Algebraic Operation on Complex numbers:

There can be four types of algebraic operation on complex numbers which are mentioned below. Visit the linked article to know more about these algebraic operations along with solved examples. The four operations on the complex numbers include:

  • Addition of two complex numbers
  • Subtraction of two complex number
  • Multiplication of two complex number
  • Division of two complex number

Power of Iota (i)

Depending upon the power of “i”, it can take the following values

\(i^{4k+1} = i\) \(i^{4k+2} = -1\) \(i^{4k+3} = -i\) \(i^{4k} = 1\)

where k can have any integral value (positive or negative).

Similarly, we can find for the negative power of i, which are as follows

\(i^{-1}=\frac{1}{i}\)

Multiplying and dividing the above term with i, we have

\(i^{-1}=\frac{1}{i} \times \frac{i}{i}\) \(i^{-1}=\frac{i}{i^{2}} = \frac{i}{-1} = -i\)

Note: \(\sqrt{-1}\times \sqrt{-1} = \sqrt{-1 \times -1} = \sqrt{1} = 1 \) contradicts to the fact that \(i^{2}= -1\).

Therefore for an imaginary number, \(\sqrt{a}\times \sqrt{b} \neq \sqrt{ab}\).

Complex Number Identities:

(i) \(\left ( z_{1}+z_{2} \right )^{2} =z_{1}^{2}+z_{2}^{2}+2z_{1}z_{2}\)

(ii) \(\left ( z_{1}-z_{2} \right )^{2} =z_{1}^{2}+z_{2}^{2}-2z_{1}z_{2}\)

(iii) \(z_{1}^{2}-z_{2}^{2} = \left ( z_{1} + z_{2} \right )\left ( z_{1} – z_{2} \right )\)

(iv) \(\left ( z_{1}+z_{2} \right )^{3} = z_{1}^{3}+3z_{1}^{2}z_{2}+3z_{1}z_{2}^{2}+z_{2}^{3}\)

(v) \(\left ( z_{1}-z_{2} \right )^{3} = z_{1}^{3}-3z_{1}^{2}z_{2}+3z_{1}z_{2}^{2}-z_{2}^{3}\)

Modulus and Conjugate of a Complex number:

Let z=a+ib be a complex number.

The Modulus of z is represented by \(\left | z \right |\).

Mathematically, \(\left | z \right |= \sqrt{a^{2}+b^{2}}\)

The conjugate of “z” is denoted by \(\bar{z}\).

Mathematically, \(\bar{z}= a – ib\)

      

Example: Express the following in a+ib form.

\(\frac{5+ \sqrt{3}i}{1-\sqrt{3}i}\).

And then find the Modulus and Conjugate of the complex number.

Solution:

Given \(\frac{5+ \sqrt{3}i}{1-\sqrt{3}i}\)

z= \(\frac{5+ \sqrt{3}i}{1-\sqrt{3}i} \times \frac{1+\sqrt{3}i}{1+\sqrt{3}i} = \frac{5-3+6\sqrt{3}i}{1+3}=\frac{1}{2} + \frac{3}{2}i\)

Modulus, \(\bar{z} = \sqrt{\left ( \frac{1}{2}\right )^{2} + \left (\frac{3}{2}\right )^{2}}\) \(\bar{z} = \sqrt{\left ( \frac{10}{4}\right )}\) \(\bar{z} = \frac{\sqrt{10}}{2}\)

Conjugate, \(\bar{z} = \frac{1}{2} – \frac{3}{2}i\)

Argand Plane and Polar Representation of Complex Number:

Similar to the XY plane, the Argand(or complex) plane is a system of rectangular coordinates in which the complex number a+ib is represented by the point whose coordinates are a and b.

We find the real and complex components in terms of r and θ, where r is the length of the vector and θ is the angle made with the real axis. Check out the detailed argand plane and polar representation of complex numbers in this article and understand this concept in a detailed way along with solved examples.

Additional Articles on Complex Numbers:

Quadratic Equation:

The Quadratic equation is of the form \(ax^{2}+bx+c=0\)

where the roots of the equation are given as

\(\large x= \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\)

or, \(\large x= \frac{-b \pm \sqrt{D}}{2a}\)

where Discriminant (D)\( = b^{2}-4ac\)

For an equation having complex roots, D<0

      

Example:

Find the roots of the given quadratic equation \(27x^{2}-10x+1=0\)

Solution:

Given \(27x^{2}-10x+1=0\)

The roots of the equation are given as-

\(\large x= \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\) \(\large x= \frac{-(-10)\pm \sqrt{(-10)^{2}-4\times 27 \times 1}}{2 \times 27}\) \(\large x= \frac{10\pm \sqrt{-8}}{54}\) \(\large x= \frac{5}{27} \pm \frac{\sqrt{2}i}{27}\)

Learn more about the Identities, Conjugate of complex number, and other complex numbers related concepts at BYJU’S. Also, get additional study materials for various maths topics along with practice questions, examples, and tips to be able to learn maths more effectively.


Practise This Question

If  A, B, C are represented by 3 + 4i, 5 - 2i , -1 + 16i, then A, B, C are
[RPET 1986]