2 Modulus, complex conjugates, and exponential form.pdf -...

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Math 446: Lecture 2 (Complex Numbers) Wednesday, August 26, 2020 Topics: Moduli Complex conjugates Exponential form Math 446: Lecture 2 (Complex Numbers)
Moduli Definition The modulus (absolute value) of z = x + iy C is defined by | z | := p x 2 + y 2 . Properties. (1) Re z 6 | Re z | 6 | z | and Im z 6 | Im z | 6 | z | . (2) |- z | = | z | . (3) | z | > 0 and | z | = 0 ⇐⇒ z = 0. In particular | z 1 - z 2 | = 0 ⇐⇒ z 1 = z 2 . (4) | z 1 - z 2 | = distance between z 1 and z 2 = p ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 . (5) | z 1 + z 2 | 6 | z 1 | + | z 2 | (triangle inequality) (5 0 ) | z 1 | - | z 2 | 6 | z 1 - z 2 | . (6) | z 1 z 2 | = | z 1 || z 2 | . (7) 1 z = 1 | z | if z , 0 and z 1 z 2 = | z 1 | | z 2 | if z 2 , 0. Math 446: Lecture 2 (Complex Numbers) .
Moduli Proof. (1) follows from x 6 | x | 6 p x 2 + y 2 and y 6 | y | 6 p x 2 + y 2 . (2) is clear: p ( - x ) 2 + ( - y ) 2 = p x 2 + y 2 . (3) is clear: p x 2 + y 2 = 0 ⇐⇒ x 2 + y 2 = 0 ⇐⇒ x = y = 0 ⇐⇒ z = 0. (5) follows from | z 1 + z 2 | = | OB | 6 | OA | + | AB | = | z 1 | + | z 2 | . (5 0 ) By (5) | z 1 - z 2 | + | z 2 | > | ( z 1 - z 2 ) + z 2 | = | z 1 | , so | z 1 | - | z 2 | 6 | z 1 - z 2 | , and respectively | z 1 - z 2 | + | z 1 | = | z 2 - z 1 | + | z 1 | > | ( z 2 - z 1 ) + z 1 | = | z 2 | , so | z 2 | - | z 1 | 6 | z 1 - z 2 | . Math 446: Lecture 2 (Complex Numbers)
Moduli (6) z 1 = x 1 + iy 1 , z 2 = x 2 + iy 2 = z 1 z 2 = ( x 1 x 2 - y 1 y 2 ) + i ( x 1 y 2 + x 2 y 1 ) = ⇒ | z 1 z 2 | = p ( x 1 x 2 - y 1 y 2 ) 2 + ( x 1 y 2 + x 2 y

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