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From the point of view of an axiomatic foundation of complex numbers, it is desirable to treat a complex number as an ordered pair a; b of real numbers a and b subject to certain operational rules which turn out to be equivalent to those above. For example, we de ne a; b c; d a c; b d , a; b c; d ac bd; ad bc , m a; b ma; mb , and so on. We then nd that a; b a 1; 0 b 0; 1 and we associate this with a bi, where i is the symbol for 0; 1 .
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POLAR FORM OF COMPLEX NUMBERS If real scales are chosen on two mutually perpendicular axes X 0 OX and Y 0 OY (the x and y axes) as in Fig. 1-2 below, we can locate any point in the plane determined by these lines by the ordered pair of numbers x; y called rectangular coordinates of the point. Examples of the location of such points are indicated by P, Q, R, S, and T in Fig. 1-2.
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Fig. 1-2
Fig. 1-3
Since a complex number x iy can be considered as an ordered pair x; y , we can represent such numbers by points in an xy plane called the complexp Argand diagram. Referring to Fig. 1-3 plane or above we see that x  cos , y  sin  where  x2 y2 jx iyj and , called the amplitude or argument, is the angle which line OP makes with the positive x axis OX. It follows that z x iy  cos  i sin  2
called the polar form of the complex number, where  and  are called polar coordintes. It is sometimes convenient to write cis  instead of cos  i sin . If z1 x1 iyi 1 cos 1 i sin 1 and z2 x2 iy2 2 cos 2 i sin 2 and by using the addition formulas for sine and cosine, we can show that z1 z2 1 2 fcos 1 2 i sin 1 2 g z1 1 fcos 1 2 i sin 1 2 g z2 2 zn f cos  i sin  gn n cos n i sin n 3 4 5
where n is any real number. Equation (5) is sometimes called De Moivre s theorem. We can use this to determine roots of complex numbers. For example, if n is a positive integer, z1=n f cos  i sin  g1=n &    '  2k  2k 1=n  cos i sin n n 6 k 0; 1; 2; 3; . . . ; n 1
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[CHAP. 1
from which it follows that there are in general n di erent values of z1=n . Later (Chap. 11) we will show that ei cos  i sin  where e 2:71828 . . . . This is called Euler s formula.
MATHEMATICAL INDUCTION The principle of mathematical induction is an important property of the positive integers. It is especially useful in proving statements involving all positive integers when it is known for example that the statements are valid for n 1; 2; 3 but it is suspected or conjectured that they hold for all positive integers. The method of proof consists of the following steps: 1. 2. 3. 4. Prove the statement for n 1 (or some other positive integer). Assume the statement true for n k; where k is any positive integer. From the assumption in 2 prove that the statement must be true for n k 1. This is part of the proof establishing the induction and may be di cult or impossible. Since the statement is true for n 1 [from step 1] it must [from step 3] be true for n 1 1 2 and from this for n 2 1 3, and so on, and so must be true for all positive integers. (This assumption, which provides the link for the truth of a statement for a nite number of cases to the truth of that statement for the in nite set, is called The Axiom of Mathematical Induction. )
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