Complex Numbers - Introduction and Examples (with Solutions), Quantitative Aptitude - UPSC PDF Download

Introduction

Complex numbers are the numbers which are expressed in the form of a+ib where i is an imaginary number called iota and has the value of (√-1). 

Example: 3 + 5i is a complex number, where 3 is a real number and 5i is an imaginary number. Therefore, the combination of both numbers is a complex one.

See the table below to differentiate between a real number and an imaginary number:

• The main application of these numbers is to represent periodic motions such as water waves, alternating current, light waves, etc., which relies on sine or cosine waves etc. 
• There are certain formulas which are used to solve the problems based on complex numbers. 
• Also, the mathematical operations such as addition, subtraction and multiplication are performed on these numbers.

The key concepts are highlighted in this lesson will include the following:

• Introduction
• Algebraic Operation on Complex Numbers
• Formulas
• Power of iota (i)
• Identities
• Modulus and Conjugate of a Complex Number
• Examples
• Argand Plane & Polar Representation of Complex Number

Complex Numbers Definition

The complex number is basically the combination of a real number and an imaginary number. 

The real numbers are the numbers which we usually work on to do the mathematical calculations. But the imaginary numbers are not generally used for calculations but only in the case of imaginary numbers. 

Let us check the definitions for both the numbers:
1. What are Real Numbers?

• Any number which is present in a number system such as positive, negative, zero, integer, rational, irrational, fractions, etc. are real numbers. It is represented as Re().
  Example: 12, -45, 0, 1/7, 2.8, √5 are all real numbers.

2. What are Imaginary Numbers?

• The numbers which are not real are imaginary numbers. When we square an imaginary number, it gives a negative result. It is represented as Im().
  Example: √-2, √-7, √-11 are all imaginary numbers.
• In the 16th century, the complex numbers were introduced, which made it possible to solve the equation x²  +1 = 0. The roots of the equation are of form x = ±√-1 and no real roots exist. Thus, with the introduction of complex numbers, we have Imaginary roots.
• We denote √-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.

Algebraic Operation on Complex numbers

There can be four types of algebraic operation on complex numbers.
The four operations on the complex numbers include:

1. Addition
2. Subtraction
3. Multiplication
4. Division

Quadratic Equations: Complex Numbers

When we solve a quadratic equation of the form ax²+bx+c = 0, the roots of the equations can be determined in three forms:

1. Two Distinct Real Roots
2. Similar Root
3. No Real roots (Complex Roots)

Complex Number Formulas

1. Addition
   (a + ib) + (c + id) = (a + c) + i(b + d)
2. Subtraction
   (a + ib) – (c + id) = (a – c) + i(b – d)
3. Multiplication
   When two complex numbers are multiplied by each other, the multiplication process should be similar to the multiplication of two binomials. It means that FOIL method (Distributive multiplication process) is used.
   (a + ib) (c + id) = (ac – bd) + i(ad + bc)
4. Division
   (a + ib) / (c + id) = (ac+bd)/ (c² + d²) + i(bc – ad) / (c² + d²)
   Question for Complex Numbers - Introduction and Examples (with Solutions), Quantitative Aptitude

   Try yourself:What type of number is  −17i?

   • A.

     Real
   • B.

     Imaginary
   • C.

     Irrational Number
   • D.

     Rational

   Explanation

   First, let's write the number  −17i  in the form of  a+ib
   = 0+(−17)i
   ∵  −17i can be written in the form of  a+bi  where a and b are real numbers, it is a complex number. 
   Also, since a = 0, it is also an imaginary number. 
   −17i is complex and imaginary

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   Question for Complex Numbers - Introduction and Examples (with Solutions), Quantitative Aptitude

   Try yourself:What is (4 + 3i) + (2 + 2i)?

   • A.

     5 + 6i
   • B.

     ​8 + 6i
   • C.

     6 +​ 5i
   • D.

     4 + 5i

   Explanation

   = 4 + 3i + 2 + 2i
   = (4 + 2) + (3 + 2)i
   = 6 + 5i

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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 an integral value (positive or negative).Similarly, we can find for the negative power of i, which are as follows:

• i^-1 = 1 / i
  Multiplying and dividing the above term with i, we have:
  i^-1 = i / i  ×  i/i  × i^-1  = i / i² = i / -1 = -i / 1 = -i

Note: √-1 × √-1 = √(-1  × -1) = √1 = 1 contradicts to the fact that i² = -1.

Therefore, for an imaginary number, √a × √b is not equal to √ab.

Identities

Let us see some of the identities:

• (z₁ + z₂)² = (z₁)² + (z₂)² + 2 z₁ × z₂
• (z₁ – z₂)² = (z₁)² + (z₂)² – 2 z₁ × z₂
• (z₁)² – (z₂)² =  (z₁ + z₂)(z₁ – z₂)
• (z₁ + z₂)³ = (z₁)³ + 3(z₁)² z₂  +3(z₂)² z₁ + (z₂ )³
• (z₁ – z₂)³ = (z₁)³ – 3(z₁)² z₂  +3(z₂)² z₁ – (z₂ )³

Modulus and Conjugate

Let z=a+ib be a complex number.

The Modulus of z is represented by |z|:
 
The conjugate of “z” is denoted by:
Mathematically,

Argand Plane and Polar Representation

• 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.
  

Problems and Solutions

Example 1: Simplify 
a) 16i + 10i(3-i)
b) 11i + 13i – 2i
Solution: 

a) 16i + 10i(3-i)

= 16i + 10i(3) + 10i (-i)
= 16i +30i – 10i²
= 46 i – 10 (-1)
= 46i + 10

b) 11i + 13i – 2i = 22i

Example 2: Express the following in a+ib form.

And then find the Modulus and Conjugate of the complex number.
Solution:
Given


Modulus:

Conjugate: