Showing complex numbers as a point on an Argand diagram allows further properties to be found.
The modulus of a complex number is the length of the line which joins it to the origin (0, 0). From the diagram below, using Pythagoras' Theorem, it can be seen that:
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The modulus of the number a + bi r = √(a2 + b2) |
The modulus of a complex number z is written IzI
The argument (θ) of a complex number is the angle formed at the origin between the positive x-axis and the line which indicates the modulus. This angle is usually measured in radians, and is given for values between -π < θ ≤ π.The argument of a complex number z is written arg(z)
From the diagram it can be seen that:
a = r.cos θ and b = r.sin θ
Thus the complex number a + bi can be written:
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cis is an abbreviation for cos θ + i sin θ.
Summary
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a + bi is called the RECTANGULAR or REAL-IMAGINARY form r cisθ is called the POLAR or MODULUS-ARGUMENT form. |
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e.g. For the complex number z = 1 + i Modulus IzI = √ (12 + 12) = √2 Arg(z) = cos-1 (1/√2) = Therefore 1 + i = √2 cis |
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