Introduction to Complex Numbers

Complex Numbers Introduction

• Complex numbers were developed to solve equations lacking real solutions.
• Mahavira, an Indian mathematician, encountered such an equation (x² + 1 = 0).
• Mahavira recognized the limitations of real numbers in solving this equation.
• European mathematicians also studied this problem, realizing the need for a broader number system.
• Complex numbers encompass numbers beyond the real number system.
• Leonhard Euler formalized complex numbers in 1748.
• Euler introduced the imaginary unit i to represent √-1.

The Imaginary Unit i and its Properties

• The imaginary unit i is defined as the square root of -1 (√-1).
• i is not a real number, but a complex number.
• i² equals -1.
• The reciprocal of i is -i.
• i⁴ equals 1.
• Adding consecutive powers of i (from i¹ to i⁴) results in zero.
• Any four consecutive powers of i add up to zero.

Complex Numbers: Real and Imaginary Parts

• Complex numbers are denoted by the variable z.
• A complex number comprises a real part and an imaginary part.
• The standard form is z = a + bi, where a is the real part and b is the imaginary part.
• a and b are real numbers.
• The set of all complex numbers is represented by ℂ.
• Real numbers are a subset of complex numbers.

Complex Numbers: A Larger Number System

• All number systems preceding complex numbers (natural numbers, whole numbers, integers, rational numbers, irrational numbers) are subsets of complex numbers.
• Complex numbers encompass all other number systems.
• Complex numbers play a crucial role in mathematics.
• Applications span various fields like calculus, geometry, and engineering.
• Despite their imaginary nature, complex numbers have real-world applications.
• Euler demonstrated their utility in solving equations unsolvable using real numbers.
• Complex numbers are essential in electrical engineering for analyzing alternating current (AC) circuits.

Examples and Summary

• z = 5 + 3i exemplifies a complex number with a real part (5) and an imaginary part (3).
• The imaginary unit i (0 + i) and a real number 5 (5 + 0*i) can also be expressed as complex numbers.
• The set of complex numbers contains both real and imaginary numbers.
• Complex numbers are a fundamental mathematical concept.
• The inclusion of i significantly broadens the number system, enabling solutions to problems beyond the limitations of real numbers.