College Algebra Textbook Supplement -Acosta & Karwowski - Fall 2013

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College Algebra with Current Interesting Applications and Factsby Acosta & Karwowski, Kendall Hunt, 2012Textbook SupplementContentsA.Complex Numbers2B.Distance Formula, Midpoint Formula, and Circles9C.Synthetic Division23Topic Expansion from Content Already in TextbookD.More on Rational Functions33E.Non-Linear Systems of Equations47F.Other Types of Equations61G.Solving Polynomial and Rational Inequalities71
2Section A:Complex NumbersSuppose we wish to solve the equationx2=25. Since we know that the square of a real numberis either zero or positive, there is no real number that would satisfy this equation. To solve thisproblem, mathematicians created a number system that is based upon a new number: theimaginary unit,commonly referred to as "i."Little Facts: It is said that the name "imaginary number" was originally coined by René Descartes in theseventeenth century as a derogatory term, because at that time such numbers were regarded by some as fictitious oruseless. Swiss mathematician Leonhard Euler introduced the letter "i" to represent the square roots of negatives in1777. In modern times these numbers have essential, concrete applications in math, physics, electrical engineering,and many other scientific and related areas. Sources: The Imaginary UnitTheimaginary unit,i, is defined with the following propertiesi=√−1andi2=√−1√−1=1We can use the imaginary unit to rewrite the square root of a negative number.Rewriting the Expression√−Ifa> 0,then√−=i√Let us rewrite√−25as a product of a real number andi:√−25=(−1)(25)=i√25= 5iWe can check the answer by squaring 5i:(5i)2= 52i2= (25)(1)=25Example 1Rewrite√−28in terms ofi, and simplify if possible.Solution√−28=i√28=i√47= 2i√7Note:It is also acceptable to write an expression like 2i√7as 2√7i,but we must be sure to write the "i"outside the radical symbol. To avoid being read as being under the radical, we generally write the answerwith "i" in front of the radical.When mathematicians added a real number to multiples of imaginary units, the set ofcomplexnumberswas formed.Complex NumberA complex number is one of the forma+bi, whereaandbare real numbers.In a complex number, we callathe real part andbisthe imaginary part.Two complex numbersa+biandc+diare equal if and only ifa=candb=d.Any real number,a, can be written as a complex number asa+ 0i. In this case,b= 0. If acomplex number hasb0, then we calla+bi, animaginary number (nonreal complex
3number). On the other hand, ifb0 buta= 0, thena+bi= 0 +bi=biand we call this apureimaginary number. Some examples of complex numbers are:2 + 5iimaginary number,a0b07ipureimaginary number,a= 012real number,b= 0A complex number written in the forma+biis said to be instandard form.

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Term
Fall
Professor
RUSSEL

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