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**Unformatted text preview: **Professor P. Bishop MAC1105 (MDN) 1.3 Complex Numbers; Quadratic Equations with a Negative Discriminant The complex number system enables us to take even roots of negative numbers by means of the imaginary unit i , which is equal to the square root of 1; that is i 2 = -1 and i = 1- . By factoring 1 out of a negative expression, it becomes positive and an even root can be taken: -b = i b . Standard form for complex expression is a + bi , where a is the real part and bi is the imaginary part. All properties of exponents hold when the base is i , thus i 1 = i, i 2 = -1, i 3 = i 2 (i) = -1i = -i, i 4 = i 2 (i 2 ) = -1(-1) = 1. In general, for i n , divide n by 4: if the remainder is 0, i n = 1 ; if the remainder is 1, i n = i , if the remainder is 2, i n = -1 ; if the remainder is 3, i n = -i. The product of a complex number (a + bi) and its conjugate (a bi) is a nonnegative real number (a 2 + b 2 )....

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