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Unformatted text preview: C. Complex Numbers 1. Complex arithmetic. Most people think that complex numbers arose from attempts to solve quadratic equa- tions, but actually it was in connection with cubic equations they first appeared. Everyone knew that certain quadratic equations, like x 2 + 1 = , or x 2 + 2 x + 5 = , had no solutions. The problem was with certain cubic equations, for example 3 x 6 x + 2 = . This equation was known to have three real roots, given by simple combinations of the expressions (1) A = 3 1 + 7 , B = 3 1 7; one of the roots for instance is A + B : it may not look like a real number, but it turns out to be one. What was to be made of the expressions A and B ? They were viewed as some sort of imaginary numbers which had no meaning in themselves, but which were useful as intermediate steps in calculations that would ultimately lead to the real numbers you were looking for (such as A + B ). This point of view persisted for several hundred years. But as more and more applications for these imaginary numbers were found, they gradually began to be accepted as valid numbers in their own right, even though they did not measure the length of any line segment. Nowadays we are fairly generous in the use of the word number: numbers of one sort or another dont have to measure anything, but to merit the name they must belong to a system in which some type of addition, subtraction, multiplication, and division is possible, and where these operations obey those laws of arithmetic one learns in elementary school and has usually forgotten by high school the commutative, associative, and distributive laws. To describe the complex numbers, we use a formal symbol i representing 1; then a complex number is an expression of the form (2) a + ib, a, b real numbers . If a = or b = 0, they are omitted (unless both are 0); thus we write a + i = a, + ib = ib, + i = . The definition of equality between two complex numbers is (3) a + ib = c + id a = c, b = d . This shows that the numbers a and b are uniquely determined once the complex number a + ib is given; we call them respectively the real and imaginary parts of a + ib . (It would be more logical to call ib the imaginary part, but this would be less convenient.) In symbols, (4) a = Re ( a + ib ) , b = Im ( a + ib ) 1 2 18.03 NOTES Addition and multiplication of complex numbers are defined in the familiar way, making use of the fact that i 2 = 1 : (5a) Addition ( a + ib ) + ( c + id ) = ( a + c ) + i ( b + d ) (5b) Multiplication ( a + ib )( c + id ) = ( ac bd ) + i ( ad + bc ) Division is a little more complicated; what is important is not so much the final formula but rather the procedure which produces it; assuming c + id = 0, it is: a + ib ac + bd bc ad (5c) Division = a + ib c id = + i c + id c + id c id c 2 + d 2 c 2 + d 2 This division procedure made use of complex conjugation : if z = a + ib , we define the complex conjugate of...
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