Complex modulus

The complex modulus of a number is its nonnegative distance from zero in the complex plane. For z = a + bi, it is |z| = √(a² + b²).

Complex plane: A plane in which complex numbers are represented by points with real and imaginary coordinates. The modulus measures a complex number’s distance from the plane’s origin.
Complex number: A number expressible as a + bi, where a and b are real and i² = −1. The modulus assigns a nonnegative magnitude to every complex number.
Modulus of a complex number: The nonnegative magnitude assigned to a complex number, equal to its distance from zero. This alternate name appears in mathematical writing for the same quantity.
Argument of a complex number: The angle a nonzero complex number makes with the positive real axis, defined modulo 2π. Argument describes direction, while modulus describes distance from the origin.
Pythagorean theorem: A theorem stating that the square of a right triangle’s hypotenuse equals the sum of the squares of its legs. Applied to the real and imaginary coordinates, it gives the modulus formula.
Real part: The real part of a + bi is the real number a. It supplies one coordinate in the right triangle defining the modulus.
Complex logarithm: A multivalued complex function whose values have real part equal to the logarithm of the input modulus. The modulus determines the real component of each logarithm value.
Complex distance: The distance between complex numbers z and w, defined as the modulus of z − w. It generalizes modulus from distance to zero to distance between any two points.
Complex conjugate: The complex conjugate of a + bi is a − bi, obtained by reversing the sign of the imaginary part. Multiplying a number by its conjugate gives the square of its modulus.
Imaginary part: The imaginary part of a + bi is the real coefficient b of i. It supplies the other coordinate in the right triangle defining the modulus.

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