M1410204 - 2.35 SECTION 2.4 COMPLEX NUMBERS Let a b c and d...

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2.35 SECTION 2.4: COMPLEX NUMBERS Let a , b , c , and d represent real numbers. PART A: COMPLEX NUMBERS i , the Imaginary Unit We define: i = 1 . i 2 = 1 If c is a positive real number c R + ( ) , then c = i c . Note: We often prefer writing i c , as opposed to c i , because we don’t want to be confused about what is included in the radicand. Examples 15 = i 15 16 = i 16 = 4 i 18 = i 18 = i 9 2 = 3 i 2 Here, we used the fact that 9 is the largest perfect square that divides 18 evenly. The 9 “comes out” of the square root radical as 9 , or 3. Standard Form for a Complex Number a + bi , where a and b are real numbers a , b R ( ) . a is the real part; b or bi is the imaginary part. C is the set of all complex numbers, which includes all real numbers. In other words, R C .
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2.36 Examples 2, 3 i , and 2 + 3 i are all complex numbers. 2 is also a real number. 3 i is called a pure imaginary number, because a = 0 and b 0 here. 2 + 3 i is called an imaginary number, because it is a nonreal complex number. PART B: THE COMPLEX PLANE The real number line (below) exhibits a linear ordering of the real numbers. In other words, if c and d are real numbers, then exactly one of the following must be true: c < d , c > d , or c = d .
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