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Complex Numbers 2 The Argand Diagram
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Representing Complex Numbers
Real numbers are usually represented as positions on a horizontal number line. -3 -2 -1 1 2 3 4 5 Real Addition, subtraction, multiplication and division with real numbers takes place on this number line.
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The Argand Diagram Complex numbers also have an imaginary part so another dimension needs to be added to the number line 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7 -8 Real Imaginary -1 Complex numbers can be represented on the Argand diagram by straight lines. Putting complex numbers on an Argand diagram often helps give a feel for a problem.
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Some examples Imaginary w u Real z v 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7
-8 Real Imaginary -1 u v w z
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Complex numbers and their conjugates
1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7 -8 Real Imaginary -1 w z w* z*
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Addition 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7 -8 Real Imaginary -1 w z
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Subtraction 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7 -8 Real Imaginary -1
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The modulus of a complex number
Real O Imaginary y x x + yj
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The argument of a complex number
1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7 -8 Real Imaginary -1 θ z=2 + 3j w=-3 - 5j α between -180o and 180o
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Radians
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Loci using complex numbers
1 2 3 4 5 6 7 8
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The distance to a point Imaginary Real 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6
-7 -8 Real Imaginary -1
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Loci using arguments Re Im Re Im Re Im
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