Math 2.0

definition

Modulus of a complex number

Definition

Let zz be a complex number with real part a=Re⁡za = \operatorname{Re} z and imaginary part b=Im⁡zb = \operatorname{Im} z. The modulus, or absolute value, of zz is the nonnegative real number ∣z∣=a2+b2.|z| = \sqrt{a^2 + b^2}.

Remarks

For a real number z=az = a, the modulus is a2=∣a∣\sqrt{a^2} = |a|, the absolute value of aa. In the complex plane, ∣z∣|z| is the distance from zz to 00, by the Pythagorean theorem. The distance between zz and ww is ∣z−w∣|z - w|. For example, ∣3+4i∣=5|3 + 4 i| = 5.

Explanations

L1 human

The modulus of a complex number is its length, that is, its distance from 00 in the plane. The point 3+4i3 + 4 i is the corner of a right triangle with sides 33 and 44, so its distance from 00 is 55.

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Complex numbers, object, no evidence levelComplex numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definition
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  1. Complex numbers object
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  9. Modulus of a complex number definition

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Sources

  1. Web https://en.wikipedia.org/wiki/Absolute_value, Complex numbers
  2. Web https://en.wikipedia.org/wiki/Complex_number, Complex conjugate, absolute value, argument and division

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