What is the complex modulus?

Complex Analysis

The complex modulus (also called the complex norm or complex absolute value) is the length (i.e., the absolute value) of a complex number in the complex plane. It is usually denoted |z|, but you might also see the notation mod z.

How to find the Complex Modulus

Complex modulus is analogous to the absolute value for real numbers, but as complex numbers aren’t ordered, we can’t use the method we would normally use for real numbers. However, as the absolute value of a complex number is simply the (Euclidean) distance from the origin to the number’s position in the complex plane, we can use the Pythagorean theorem to calculate it. As this is a “distance,” the complex modulus is always real and non-negative [1].

To find the modulus of a complex number z = a + ib, solve √(x2 + y2), where x and y are real numbers.

Example question: What is the Complex Modulus of z = 1 + 2i?
Solution: |z| = √(12 + 22) = √(5).

Properties of Complex Modulus

1. z equals 0 only if |z| = 0.
2. In terms of complex conjugation, the modulus can be written as |z| = √(z · z). Geometrically, z is obtained by reflecting z over the real axis. Therefore, the modulus of a complex conjugate z is the same as that of the complex number z.
3. If z = a + 0i is real, then |z| = |a| (a complex number z is real if and only if z = z [2]).
4. The distance between any two numbers z and w can be calculated by |z – w|.

Disambiguation: Note that the term “complex modulus” has another (unrelated) definition: the ratio of stress to strain under vibratory conditions in materials engineering.

References

[1] Chapter 5: Complex Numbers. Retrieved November 9, 2021 from: http://www2.hawaii.edu/~robertop/Courses/TMP/5_Complex_Numbers.pdf
[2] Sivakumar, N. Chapter 1: Preliminaries.

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Definition of Modulus of a Complex Number:

Let z = x + iy where x and y are real and i = √-1. Then the non negative square root of (x\(^{2}\)+ y \(^{2}\)) is called the modulus or absolute value of z (or x + iy).

Modulus of a complex number z = x + iy, denoted by mod(z) or |z| or |x + iy|, is defined as |z|[or mod z or |x + iy|] = + \(\sqrt{x^{2} + y^{2}}\) ,where a = Re(z), b = Im(z)

i.e., + \(\sqrt{{Re(z)}^{2} + {Im(z)}^{2}}\)

Sometimes, |z| is called absolute value of z. Clearly, |z| ≥ 0 for all zϵ C.

For example:

(i) If z = 6 + 8i then |z| = \(\sqrt{6^{2} + 8^{2}}\) = √100 = 10.

(ii) If z = -6 + 8i then |z| = \(\sqrt{(-6)^{2} + 8^{2}}\) = √100 = 10.

(iii) If z = 6 - 8i then |z| = \(\sqrt{6^{2} + (-8)^{2}}\) = √100 = 10.

(iv) If z = √2 - 3i then |z| = \(\sqrt{(√2)^{2} + (-3)^{2}}\) = √11.

(v) If z = -√2 - 3i then |z| = \(\sqrt{(-√2)^{2} + (-3)^{2}}\) = √11.

(vi) If z = -5 + 4i then |z| = \(\sqrt{(-5)^{2} + 4^{2}}\) = √41

(vii) If z = 3 - √7i then |z| = \(\sqrt{3^{2} + (-√7)^{2}}\) =\(\sqrt{9 + 7}\)  =  √16 = 4.

Note: (i) If z = x + iy and x = y = 0 then |z| = 0.

(ii) For any complex number z we have, |z| = |\(\bar{z}\)| = |-z|.

Properties of modulus of a complex number:

If z, z\(_{1}\) and z\(_{2}\) are complex numbers, then

(i) |-z| = |z|

Proof:

Let z = x + iy, then –z = -x – iy.

Therefore, |-z| = \(\sqrt{(-x)^{2} +(- y)^{2}}\) = \(\sqrt{x^{2} + y^{2}}\) = |z|

(ii) |z| = 0 if and only if z = 0

Proof:

Let z = x + iy, then |z| = \(\sqrt{x^{2} + y^{2}}\).

Now |z| = 0 if and only if \(\sqrt{x^{2} + y^{2}}\) = 0

⇒ if only if x\(^{2}\) + y\(^{2}\) = 0 i.e., a\(^{2}\) = 0and b\(^{2}\) = 0

⇒ if only if x = 0 and y = 0 i.e., z = 0 + i0

⇒ if only if z = 0.

(iii) |z\(_{1}\)z\(_{2}\)| = |z\(_{1}\)||z\(_{2}\)|

Proof:

Let z\(_{1}\) = j + ik and z\(_{2}\) = l + im, then

z\(_{1}\)z\(_{2}\) =(jl - km) + i(jm + kl)

Therefore, |z\(_{1}\)z\(_{2}\)| = \(\sqrt{( jl - km)^{2} + (jm + kl)^{2}}\)

= \(\sqrt{j^{2}l^{2} + k^{2}m^{2} – 2jklm  + j^{2}m^{2} + k^{2}l^{2} + 2 jklm}\)

= \(\sqrt{(j^{2} + k^{2})(l^{2} + m^{2}}\)

= \(\sqrt{j^{2} + k^{2}}\) \(\sqrt{l^{2} + m^{2}}\), [Since, j\(^{2}\) + k\(^{2}\) ≥0, l\(^{2}\) + m\(^{2}\) ≥0]

= |z\(_{1}\)||z\(_{2}\)|.

(iv) |\(\frac{z_{1}}{z_{2}}\)| = \(\frac{|z_{1}|}{|z_{2}|}\), provided z\(_{2}\) ≠ 0.

Proof:

According to the problem, z\(_{2}\) ≠ 0 ⇒ |z\(_{2}\)| ≠ 0

Let \(\frac{z_{1}}{z_{2}}\) = z\(_{3}\)

⇒ z\(_{1}\) = z\(_{2}\)z\(_{3}\)

⇒ |z\(_{1}\)| = |z\(_{2}\)z\(_{3}\)|

⇒|z\(_{1}\)| = |z\(_{2}\)||z\(_{3}\)|, [Since we know that |z\(_{1}\)z\(_{2}\)| = |z\(_{1}\)||z\(_{2}\)|]

⇒ \(\frac{|z_{1}}{z_{2}}\) = |z\(_{3}\)|

⇒ \(\frac{|z_{1}|}{|z_{2}|}\) = |\(\frac{z_{1}}{z_{2}}\)|, [Since, z\(_{3}\) = \(\frac{z_{1}}{z_{2}}\)]

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The modulus of a complex number

, also called the complex norm, is denoted

and defined by

If

is expressed as a complex exponential (i.e., a phasor), then

The complex modulus is implemented in the Wolfram Language as Abs[z], or as Norm[z].

The square

of

is sometimes called the absolute square.

Let

and

be two complex numbers. Then

so

Also,

so

and, by extension,

The only functions satisfying identities of the form

are

,

, and

(Robinson 1957).

Absolute Square, Absolute Value, Complex Argument, Complex Number, Imaginary Part, Maximum Modulus Principle, Minimum Modulus Principle, Real Part http://functions.wolfram.com/ComplexComponents/Abs/ Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972.Krantz, S. G. "Modulus of a Complex Number." §1.1.4 n Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 2-3, 1999.Robinson, R. M. "A Curious Mathematical Identity." Amer. Math. Monthly 64, 83-85, 1957.Complex Modulus

Weisstein, Eric W. "Complex Modulus." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ComplexModulus.html

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