Introduction to Complex Numbers

Transcription

1 Introducton to Complex Numbers Let s revew the varous classfcaton of number we have encountered Number Systems Natural Numbers (Postve Integers) {1,, 3, 4, } Whole Numbers (Postve Integers plus zero) {0, 1,, 3, 4, } Integers (Whole numbers and ther oppostes) {-4, -3, -, -1, 0, 1, } Ratonal Numbers(Can be expressed as a rato of two numbers) 1 3,, 4,.1 Irratonal Numbers(cannot be expressed as a rato of two numbers) 7,, Real Numbers (Ratonal and Irratonal numbers) 1 3,, 4,.1, 7,, Up untl now, you've been told that you can't take the square root of a negatve number. That's because you had no numbers that when squared, produced a negatve number.. Every number was postve after you squared t. So you couldn't very well square-root a negatve and expect to come up wth anythng sensble. However what sort of mathematcs could you do f you pretended that there was such an answer for the square root of the smplest negatve number, -1. Ths new number s called "", standng for "magnary", because "everybody knew" that wasn't "real". (That's why you couldn't take the square root of a negatve number before: you only had "real" numbers; that s, numbers wthout the "" n them.) At ths tme, nobody beleved that any "real world" use would be found for ths new number, other than easng the computatons nvolved n solvng certan equatons, so the new number was vewed as beng a pretend number nvented for convenence sake.

2 The magnary Number The magnary number: Wth Important Note: Now, you may thnk you can do ths: But ths doesn't make any sense! You already have two numbers that square to 1; namely 1 and +1. And already squares to 1. So t's not reasonable that would also square to 1. Ths ponts out an mportant detal: When dealng wth magnares, you gan somethng (the ablty to deal wth negatves nsde square roots), but you also lose somethng (some of the flexblty and convenent rules you used to have when dealng wth square roots). In partcular, YOU MUST ALWAYS DO THE -PART FIRST! : Smplfy Smplfy Smplfy

3 Smplfy When we started defnng our number systems, we ended wth the Reals. We are now gong to expand to a larger system that ncludes both the Real Numbers and the Imagnary numbers. The Complex Numbers Complex Numbers (nclude the Reals plus the Imagnary Numbers) a b where a and b are real numbers and 1 We recall that 1 a b real part magnary part If a=0 then the complex numbers are known as pure magnary numbers Every complex number has a conjugate number. A conjugate complex number dffers from the complex number only n ts sgn. a b, a b We usually let a complex number be represented by z, z 4 3 Then the conjugate s denoted as z wth a bar over t, z 4 3 Gven a complex number wth a real part of 5 and an magnary part of 4, wrte as a complex number z z a b 5

4 Solve x x 1 0. x x 1 0 x x Graphng Complex Numbers Cartesan form When dealng wth complex numbers, the numbers along the x-axs are sad to be "real" numbers. The numbers on the y-axs are "magnary" numbers. The way ths s wrtten depends whether t s used wthn an engneerng envronment or a purely mathematcal envronment. Mathematcally plottng the value of 3+4 would represent the hypotenuse. Wthn the realms of engneerng, s used for current so the next avalable letter s used, j. Ths would be wrtten as 3 + j4. Ths format of expressng complex numbers s called "Rectangular form" or "Cartesan form"..

5 Argand dagram consder the followng dagram Here are plotted four Cartesan complex number values. The dagram for showng these numbers s called an Argand dagram. Named after, but not nvented by, Jean Robert Argand. It was, n fact, nvented some years earler by Casper Wessel.