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# Class Note for MATH 1310 at UH 3

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This 9 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15

1310 24 Complex numbers Solve x2 l440 Now solve x2 4 0 Big problems There is no solution in the Real numbers so we ll come up with a new number system Complex numbers The basic building block is l This new number is also called i it used to be considered imaginary but it s really not it s a basic fact of everyday life as in the equations that govern electric power generation 44 This new number 1 comes up as part of the solution to equation x2 4 0 It turns out that it s not a new number system it s an expansion of the real numbers into a system called the complex numbers A complex number is a number of the form a bi where the i is V l So that old familiar number 2 is 2 0i 3 is 3 0i and 05 is complex with zero times i too In fact every real number has b 0 in it s complex form 5 is01i There are two parts to a complex number the a called the real part and the b called the complex part Simplify 5 m W 43 You can add subtract multiply and divide complex numbers 7 the rules are just a little more well 7 forgive the pun complex When you add a truly complex number you add the two real parts and then you add the two complex parts 35i 1 2i With subtract don t forget to distribute 5 2i 3 6i Now to move on to multiplication we have to spend a bit of time exploring the properties of the number i Let s go on and do a cyclic chart for this starting with familiar numbers First back up and look at powers of 1 Now let s look at powers of l Now let s explore powers of i Note the 4 part cycle The number 4 gures in calculating very high powers of i using the Remainder Theorem 3911 1 i13 3920 1 3929 1 3957 1 3992 391002 5i3i2 9i5 4004 3999 1 Ok now let s discuss propemess It isn t proper to have an i in the denominator of a fraction Let s look at a situation and how to make the number a proper a bi number m 3i 2i Note that propemess impinges on diVision 7 it s not proper to divide by i Let s spend a little time multiplying 7 multiplying complex numbers uses distribution and FOIL but since there s an unfamiliar number involved you ll need to practice Put every answer into a bi standard form 3i5 3i 1 i5 2i 5 2i3 4i 2 3i2 3i 5 i5 i Now let s go back and look at some more improper fractions and each has a big hint preceding it 1 il i 2 i2 i 1i 2 i 5 i5 i 3 i3i i 3 i So you have to be clever with conjugates to make one of these fractions proper This IS division by complex numbers making it proper Let s solve some equations X240 X2160 x2 250 X2810 x21210 Use the quadratic formula to solve X2 iX20 Use CTS to solve X2 4X90 Use CTS to solve x2 8X 20

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