COMPLEX_NUMBER

COMPLEX_NUMBER - COMPLEX NUMBER Real number Natural number...

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Unformatted text preview: COMPLEX NUMBER Real number : Natural number, zero, integer (positive; negative), rational number, irrational number, real number Imaginary number : i = 1 − Complex number : z = a + b i ; where a and b are real numbers Re(z) = a and Im(z) = b Two complex numbers z 1 = a 1 + b 1 i and z 2 = a 2 + b 2 i are equal if a 1 = a 2 and b 1 = b 2 Conjugate : = z a- b i Exercise: Show that Re(z) = ) z z ( + 2 1 and Im(z) = ) z z ( i − 2 1 Modulus z : z 2 = a 2 + b 2 Argument θ : θ = arg( z ) = arctan ( a b ) Principal Argument : Arg( z ) ε [- π , π ] Graphical Representation z-plane (Argand diagram) Polar Form : z = r ( cos θ + i sin θ ) where r = 2 2 b a + Exercise : Represent the following complex numbers in the Argand diagram : i ; i ; i ; i 14 7 2 2 5 12 4 3 + − − − − + Multiplication by a real number : i Ab Aa ) i b a ( A Az z 1 1 1 1 1 2 + = + = = Addition/substraction : z 1 + z 2 = ( a 1 + b 1 i ) ± ( a 2 + b 2 i ) = ( a 1 + a 2 ) ± ( b 1 + b 2 ) i Exercise : Given i z 3 2 1 + = and i z 5 4 2 + = . Compute 2 1 3 2 z z + . Represent the complex numbers in the Argand diagram. numbers in the Argand diagram....
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COMPLEX_NUMBER - COMPLEX NUMBER Real number Natural number...

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