MATH1110 Lecture Notes - Lecture 4: Gerolamo Cardano, An Imaginary Tale, Cartesian Coordinate System
18 Jun 2018
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MATH1110: Mathematics 1
Complex Numbers(I): Introduction
1 Introduction to Complex Numbers
Consider the equation
z2−1 = 0
To find the solution we rearrange and take square roots:
z2= 1
=⇒z=±1
What about
z2+ 1 = 0?
If we try to do the same thing we are asked to solve
z2=−1
This has no real solution!
Indeed, ever since Descartes’ introduction of the Cartesian plane, we could
see the absence of real solutions by plotting f(z) = z2+ 1:
The function doesn’t cross the horizontal axis - it never equals zero for any
real number input.
Here our story would end, except for a tangled tale of secrecy in the 1500’s
concerning cubic equations. (See Paul J Nahin’s book An Imaginary Tale:
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Document Summary
To nd the solution we rearrange and take square roots: z2 1 = 0. If we try to do the same thing we are asked to solve z2 = 1. Indeed, ever since descartes" introduction of the cartesian plane, we could see the absence of real solutions by plotting f (z) = z2 + 1: The function doesn"t cross the horizontal axis - it never equals zero for any real number input. Here our story would end, except for a tangled tale of secrecy in the 1500"s concerning cubic equations. (see paul j nahin"s book an imaginary tale: The long and the short of it is: girolamo cardano and. Rafael bombelli made a bold move, and said, lets treat as if it was a number, just other surd like 2, and see what happens. z = 1. 500 years later, electrical engineers are grateful. (exercise: nd out why. ) In the meantime, we give a formal de nition:
