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Math 4B Lecture 9 April 29, 2013 Doug Moore

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Announcements. 1. Midterm this Wednesday. You should know linear DE’s (dif- ferential equations), exact differentials, Cauchy-Euler polygon and homogeneous linear second order DE’s with constant coef- ficients. No calculators, computers, notes or books allowed. 2. No blue books needed. Exams will be on paper provided. 3. Bring identification. You will need to enter your TA’s name on your exam. 4. Homework 5 is short and will be due on Wednesday, May 1 at 6pm (the day of the midterm).
In mathematics, it is important to be specific about the type of numbers being used in a given problem. In most of calculus, one deals with real numbers, representable by arbitrary decimal expansions. But in some problems the “field” R of real numbers is not large enough to encompass all solutions. This sometimes occurs, for example, when solving quadratic equations. Recall that the solutions to 2 + + c = 0 are λ = - b ± q b 2 - 4 ac 2 a , and the expression within the square root may be negative. Al- though the square root of a negative real number does not exist as a real number, we can take square roots of negative numbers within the larger field C of complex numbers.

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The idea behind complex numbers is to introduce an “imagi- nary number” i to represent the square root of - 1. A complex number is a number of the form x + iy , where x and y are real numbers. Any nonzero complex number has two square roots. Complex numbers are often represented by points in the ( x, y )- plane. We call x the real part and y the imaginary part of the complex number x + iy . Addition and multiplication of complex numbers are defined by the formulae ( x 1 + iy 1 ) + ( x 2 + iy 2 ) = ( x 1 + x 2 ) + i ( y 1 + y 2 ) , ( x 1 + iy 1 )( x 2 + iy 2 ) = ( x 1 x 2 - y 1 y 2 ) + i ( x 1 y 2 + x 2 y 1 ) .
For example, (3+4 i )+(2+5 i ) = 5+9 i, (3+4 i )(2 - i ) = = 10+5 i. Addition is just another version of the usual “vector addition” used in navigation, i.e.

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