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MAT1341-L01-ComplexNumbers

# MAT1341-L01-ComplexNumbers - COMPLEX NUMBERS JOSE...

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COMPLEX NUMBERS JOS ´ E MALAG ´ ON-L ´ OPEZ What is a Number? Numbers are “symbols” used for measuring and counting purposes. Examples of numbers are: Natural Numbers: N = { 0 , 1 , 2 , 3 , . . . } ; Integers: Z = { 0 , ± 1 , ± 2 , ± 3 , . . . } ; Rationals: Q = { a b | a and b are integers } . All these numbers are part of a larger collection of such symbols, the Real Numbers , denoted by R . Are There More Numbers? From Calculus we know that there is another collection of numbers, the irrationals. We can say, in short, that the irrationals are the real numbers that are not rationals. For a more precise definition we need to consider limits, which we will avoid since is not part of this course. Examples of irrational numbers are: 2, π , e , etc. In general, numbers have been “obtained” when we want to consider the solution set of a given type of equations. For example, the integers are obtained from equations of the form x + a = 0 , a N , or the rationals, obtained from equations of the form ax = b, a, b Z . In this course we will consider two extensions of the real numbers, the complex numbers and matrices. Complex Numbers The complex numbers are the solutions of equations of the form x 2 = a, a R . 1

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Fortunately, we only need to be concerned with the solution of x 2 = 1. Definition. The symbol ı = 1 will be called the imaginary unit . Notice that ı 2 = 1. Definition. The complex numbers , denoted by C , are defined as the collection of ALL the expressions of the form a + b ı, a, b R .
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MAT1341-L01-ComplexNumbers - COMPLEX NUMBERS JOSE...

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