complexNumbers, functions, factoring

# complexNumbers, functions, factoring - Complex numbers the...

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Unformatted text preview: Complex numbers, the exponential function, and factorization over C 1 Complex Numbers Recall that for every non-zero real number x , its square x 2 = x · x is always positive. Consequently, R does not contain the square roots of any negative number. This is a serious problem which rears its head all over the place. It is a non-trivial fact, however, that any positive number has two square roots in R , one positive and the other negative; the positive one is denoted √ x . One can show that for any x in R , | x | = √ x · x. So if we can somehow have at hand a square root of − 1, we can find square roots of any real number. This motivates us to declare a new entity, denoted i , to satisfy i 2 = − 1 . One defines the set of complex numbers to be C = { x + iy | x, y ∈ R } and defines the basic arithmetical operations in C as follows: ( x + iy ) ± ( x ′ + iy ′ ) = ( x ± x ′ ) + i ( y ± y ′ ) , and ( x + iy )( x ′ + iy ′ ) = ( xx ′ − yy ′ ) + i ( xy ′ + x ′ y ) . 1 There is a natural one-to-one function R → C , x → x + i. , compatible with the arithmetical operations on both sides. It is an easy exercise to check all the field axioms, except perhaps for the existence of multiplicative inverses for non-zero complex numbers. To this end one defines the complex conjugate of any z = x + iy in C to be z = x − iy. Clearly, R = { z ∈ C | z = z } . If z = x + iy , we have by definition, z z = x 2 + y 2 . In particular, z z is either 0 or a positive real number. Hence we can find a non-negative square root of z z in R . Define the absolute value , sometimes called modulus or norm , by | z | = √ z z = √ x 2 + y 2 . If z = x + iy is not 0, we will put z − 1 = z z z = x x 2 + y 2 − i y x 2 + y 2 . It is a complex number satisfying z ( z − 1 ) = z z z z = 1 . Done. It is natural to think of complex numbers z = x + iy as being ordered pairs ( x, y ) of real numbers. So one can try to visualize C as a plane with two perpendicular coordinate directions, namely giving the x and y parts. Note in particular that 0 corresponds to the origin O = (0 , 0), 1 to (1 , 0) and i with (0 , 1). Geometrically, one can think of getting from − 1 to 1 (and back) by rotation about an angle π , and similarly, one gets from i to its square − 1 by rotating by half that angle, namely π/ 2, in the counterclockwise direction. 2 To get from the other square root of − 1, namely − i , one rotates by π/ 2 in the clockwise direction. (Going counterclockwise is considered to be in the positive direction in Math.) Addition of complex numbers has then a simple geometric interpretation: If z = x + iy , z ′ = x ′ + iy ′ are two complex numbers, represented by the points P = ( x, y ) and Q = ( x ′ , y ′ ) on the plane, then one can join the origin O to P and Q , and then draw a parallelogram with the line segments OP and OQ as a pair of adjacent sides. If R is the fourth vertex of this parallelogram, it corresponds to z + z ′ . This is called the....
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complexNumbers, functions, factoring - Complex numbers the...

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