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Absolute value

The absolute value of the complex number z = x + i y = r · e^iφ with r = 1 is

|e^iφ| = 1

 

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Explanation

You can draw a unit circle in the complex plane. All points on this circle conform to Euler's formula

e^iφ = cos φ + i sin φ

and therefor they give the points shown in the Argand diagram

Substitution of φ in the formula gives the values

	→
	→
	→
	→

As the only real values are 1 and –1, the absolute value is 1.

 

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