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Richa Nandra, Lovely Professional University Unit 1: Complex Numbers

Unit 1: Complex Numbers Notes

CONTENTS
Objectives

Introduction
1.1 Geometry
1.2 Polar coordinates
1.3 Summary
1.4 Keywords

1.5 Self Assessment
1.6 Review Questions
1.7 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss the meaning of geometry

Explain the polar coordinates

Introduction

Let us hark back to the first grade when the only numbers you knew were the ordinary everyday
integers. You had no trouble solving problems in which you were, for instance, asked to find a
number x such that 3x = 6. You were quick to answer “2”. Then, in the second grade, Miss Holt
asked you to find a number x such that 3x = 8. You were stumped—there was no such ”number”!
You perhaps explained to Miss Holt that 3 (2) = 6 and 3 (3) = 9, and since 8 is between 6 and 9, you
would somehow need a number between 2 and 3, but there isn’t any such number. Thus, you
were introduced to “fractions.”
These fractions, or rational numbers, were defined by Miss Holt to be ordered pairs of integers—
thus, for instance, (8, 3) is a rational number. Two rational numbers (n, m) and (p, q) were defined
to be equal whenever nq = pm. (More precisely, in other words, a rational number is an equivalence
class of ordered pairs, etc.) Recall that the arithmetic of these pairs was then introduced: the sum
of (n, m) and (p, q) was defined by
(n, m) + (p, q) = (nq + pm, mq),
and the product by
(n, m) (p, q) = (np, mq).

Subtraction and division were defined, as usual, simply as the inverses of the two operations.
In the second grade, you probably felt at first like you had thrown away the familiar integers and
were starting over. But no. You noticed that (n, 1) + (p, 1) = (n + p, 1) and also (n, 1)(p, 1) = (np, 1).
Thus, the set of all rational numbers whose second coordinate is one behave just like the integers.

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