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MIT 18 03 - Complex Numbers

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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 = 0, or x 2 + 2x + 5 = 0, had no solutions. The problem was with certain cubic equations, for example 3 x − 6x + 2 = 0. 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 don’t 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 = 0 or b = 0, they are omitted (unless both are 0); thus we write a + i0 = a, 0 + ib = ib, 0 + i0 = 0 . 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 i2 = −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 c2 + d2 c2 + d2 This division procedure made use of complex conjugation: if z = a + ib, we define the complex conjugate of z to be the complex number (6) z¯ = a − ib (note that zz¯ = a 2 + b2 ). The size of a complex number is measured by its absolute value, or modulus, defined by (7) |z| = a + ib = a2 + b2; (thus : zz¯ = z2 ).| | | |Remarks. One can legitimately object to defining complex numbers simply as formal expressions a + ib, on the grounds that “formal expression” is too vague a concept: even if people can handle it, computers cannot. For the latter’s sake, we therefore define a complex number to be simply an ordered pair (a, b) of real numbers. With this definition, the arithmetic laws are then defined in terms of ordered pairs; in particular, multiplication is defined by (a, b)(c, d) = (ac − bd, bc + ad) . The disadvantage of this approach is that this definition of multiplication seems to make little sense. This doesn’t bother computers, who do what they are told, but people do better at multiplication by being told to calculate as usual, but to use the relation i2 = −1 to get rid of all the higher powers of i whenever they occur. Of course, even if you start with the definition using ordered pairs, you can still introduce the special symbol i to represent the ordered pair (0, 1), agree to the abbreviation (a, 0) = a, and thus write (a, b) = (a, 0) + (0, 1)(b, 0) = a + ib . 2. Polar representation. Complex numbers are represented geometrically by points in the plane: the number a+ib is represented by the point (a, b) in Cartesian coordinates. When the points of the plane are thought of as representing complex numbers in this way, the plane is called the complex plane. By switching to polar coordinates, we can write any non-zero complex number in an alternative form. Letting as usual x = r cos α, y = r sin α, we get the polar form for a non-zero complex number: assuming x + iy = 0, ≤(8) x + iy = r(cos α + i sin α) . �  � 3 C. COMPLEX NUMBERS When the complex number is written in polar form, we see from (7) that r = x + iy , (absolute value, modulus) | |We call α the polar angle or the argument of x + iy. In symbols, one sometimes sees α = arg (x + iy) (polar angle, argument) . The absolute value is uniquely determined by x + iy, but the polar angle is not, since it can be increased by any integer multiple of 2λ. (The complex number 0 has no polar angle.) To make α unique, one can specify 0 � α < 2λ principal value of the polar angle. This so-called principal value of the angle is sometimes indicated by writing Arg (x + iy). For example, Arg (−1) = λ, arg (−1) = ±λ, ±3λ, ±5λ, . . . . Changing between Cartesian and polar representation of a complex number is the same as changing between Cartesian and polar coordinates. Example 1. Give the polar form for: −i, 1 + i, 1 − i, −1 + i�3 . Solution. −i = i cos 32 � 1 + i = �2 (cos � + i sin � 4 )4 −1 + i�3 = 2 (cos 2� + i sin 23 � ) 1 − i = �2 (cos −4 � + i sin −� )3 4 The abbreviation cis α is sometimes used for cos …


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MIT 18 03 - Complex Numbers

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