Complex Numbers, Objectives, Complex Numbers, Study of Number Systems

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  • We started our study of number systems with the set of natural numbers, then the number zero was included to form the system of whole numbers; negative of natural numbers were defined. Thus, we extended our number system to whole numbers and integers.

  • To solve the problems of the type we included rational numbers in the system of integers. The system of rational numbers has been extended further to irrational numbers as all lengths cannot be measured in terms of lengths expressed in rational numbers.

  • Rational and irrational numbers taken together are termed as real numbers. But the system of real numbers is not sufficient to solve all algebraic equations. There are no real numbers which satisfy the equation .

  • In order to solve such equations, i.e., to find square roots of negative numbers, we extend the system of real numbers to a new system of numbers known as complex numbers.

  • In this lesson the learner will be acquainted with complex numbers, its representation and algebraic operations on complex numbers.

Objectives

  • After studying this lesson, you will be able to:

  • Describe the need for extending the set of real numbers to the set of complex numbers;

  • Define a complex number and cite examples;

  • Identify the real and imaginary parts of a complex number;

  • State the condition for equality of two complex numbers;

  • Recognise that there is a unique complex number x + iy associated with the point P(x, y) in the Argand Plane and vice-versa;

  • Define and find the conjugate of a complex number;

  • Define and find the modulus and argument of a complex number;

  • Represent a complex number in the polar form;

  • Perform algebraic operations (addition, subtraction, multiplication and division) on complex numbers;

  • State and use the properties of algebraic operations (closure, commutativity, associativity, identity, inverse and distributivity) of complex numbers; and

  • State and use the following properties of complex numbers in solving problems:

(i) and

(ii)

(iii)

(iv)

(v)

To find the square root of a complex number.

Complex Numbers

Consider the equation

This can be written as or

But there is no real number which satisfy .In other words, we can say that there is no real number whose square is . In order to solve such equations, let us imagine that there exist a number which equal to .

In 1748, a great mathematician, L. Euler named a number as Iota whose square is . This Iota or is defined as imaginary unit. With the introduction of the new symbol , we can interpret the square root of a negative number as a product of a real number with .

Therefore, we can denote the solution of (A) as

Thus,

Conventionally written as .

So, we have

  • are all examples of complex numbers.

Example:

Consider another quadratic equation:

Solution:

or

Or or

Any number which can be expressed in the form where are real numbers and , is called a complex number.

A complex number is, generally, denoted by the letter.

i.e. is called the real part of and is written as and is called the imaginary part of and is written as .

If and , then the complex number becomes which is a purely imaginary complex number.

and are all examples of purely imaginary numbers.

If and then the complex number becomes which is a real number.

and are all examples of real numbers.

If and , then the complex number becomes (zero). Hence the real numbers are particular cases of complex numbers.

Example:

Simplify each of the following using .

(i) (ii)

Solution:

(i)

(ii)

Developed by: