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Complex Numbers - The Complex Number System The need for...

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The Complex Number System The need for extending the real number system is evident when considering solutions of simple equations. For example, the equation x 2 + 1 = 0 has no real number solutions, for if x is any real number, then x 2 0 and so x 2 + 1 1. We extend the real number system to the system of complex numbers in such a way that the arithmetic of the real numbers is preserved. A complex number is an ordered pair of real numbers ( a, b ). The set of complex numbers is denoted by C . Just as the real numbers can be viewed as the points on the number line (real line), the complex numbers can be viewed as the points of the coordinate plane. Arithmetic of the Complex Numbers Equality: If z 1 = ( a 1 , b 1 ) and z 2 = ( a 2 , b 2 ) are complex numbers, then z 1 = z 2 if and only if a 1 = a 2 and b 1 = b 2 . Addition: If z 1 = ( a 1 , b 1 ) and z 2 = ( a 2 , b 2 ) are complex numbers, then z 1 + z 2 = ( a 1 , b 1 ) + ( a 2 , b 2 ) = ( a 1 + a 2 , b 1 + b 2 ) Properties of Addition: 1. z 1 + z 2 = z 2 + z 1 (Commutative) 2. ( z 1 + z 2 ) + z 3 = z 1 + ( z 2 + z 3 ) (Associative) Let z = ( a, b ). 3. z + (0 , 0) = (0 , 0) + z = z (0 , 0) is the Additive Identity. (Note: It is conven- tional to use 0 to represent the additive identity. The context will make it clear whether we mean the real number 0 or the complex number (0 , 0).) 4. z + ( - a, - b ) = ( - a, - b ) + z = 0 ( - a, - b ) = - z is the Additive Inverse Subtraction: To subtract z 2 from z 1 , we add the additive inverse of z 2 to z 1 . That is, z 1 - z 2 = z 1 + ( - z 2 ) = ( a 1 , b 1 ) + ( - a 2 , - b 2 ) = ( a 1 - a 2 , b 1 - b 2 ) . Multiplication: If z 1 = ( a 1 , b 1 ) and z 2 = ( a 2 , b 2 ) are complex numbers, then z 1 · z 2 = ( a 1 , b 1 ) · ( a 2 , b 2 ) = ( a 1 a 2 - b 1 b 2 , a 1 b 2 + a 2 b 1 ) Properties of Multiplication: 283
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1. z 1 · z 2 = z 2 · z
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