Skip to content

Complex modulus

Measure the size of a complex number, connect its modulus to the conjugate identity, and interpret a synthetic Fourier coefficient without losing track of phase or scaling.

The complex modulus is the magnitude of a complex number: its distance from zero in the complex plane. For z=a+biz=a+bi, where a,ba,b are real and i2=−1i^2=-1,
∣z∣=a2+b2.\lvert z\rvert=\sqrt{a^2+b^2}.
The result is a nonnegative real number, with ∣z∣=0\lvert z\rvert=0 exactly when z=0z=0. For a real input it reduces to ordinary absolute value. In quantitative work, it lets us measure the size of a complex transform coefficient while keeping that question separate from its phase.

A length, not a sum of coordinates

Plot z=3+4iz=3+4i at the point (3,4)(3,4). The real and imaginary components form perpendicular sides of a right triangle, so

∣3+4i∣=32+42=5.\lvert3+4i\rvert=\sqrt{3^2+4^2}=5.

Adding the components gives 3+4=73+4=7, which does not measure the diagonal distance. The same geometry gives ∣z−w∣\lvert z-w\rvert as the distance between two complex numbers zz and ww.

Complex-plane diagram with real component 3 and imaginary component 4 meeting at a right angle. The diagonal from zero to 3 plus 4i has modulus 5, rather than coordinate sum 7.
Complex-plane diagram with real component 3 and imaginary component 4 meeting at a right angle. The diagonal from zero to 3 plus 4i has modulus 5, rather than coordinate sum 7.
Original dimensionless example. Modulus measures the diagonal length; the real and imaginary components remain distinct coordinates.

Modulus discards direction. The numbers 3+4i3+4i, 3−4i3-4i and −3−4i-3-4i all have modulus 5, despite occupying different points. Knowing magnitude alone cannot reconstruct a complex number.

Why the conjugate appears in the formula

The complex conjugate of z=a+biz=a+bi is z‾=a−bi\overline z=a-bi. Multiplying them cancels the imaginary terms:
zz‾=(a+bi)(a−bi)=a2+b2=∣z∣2.z\overline z=(a+bi)(a-bi)=a^2+b^2=\lvert z\rvert^2.
Thus ∣z∣=zz‾\lvert z\rvert=\sqrt{z\overline z}, taking the nonnegative real square root. For the example, (3+4i)(3−4i)=25(3+4i)(3-4i)=25, so the modulus is 5. The product is the squared modulus. Squaring the original number instead gives (3+4i)2=−7+24i(3+4i)^2=-7+24i, which is a different operation.

Two useful rules follow from this length interpretation and identity:

∣zw∣=∣z∣∣w∣,∣z+w∣≤∣z∣+∣w∣.\lvert zw\rvert=\lvert z\rvert\lvert w\rvert, \qquad \lvert z+w\rvert\leq\lvert z\rvert+\lvert w\rvert.

The second is the triangle inequality. Magnitudes multiply, but generally do not add: z+(−z)=0z+(-z)=0 even when each term has magnitude 5.

For a real angle θ\theta, eiθ=cos⁡θ+isin⁡θe^{i\theta}=\cos\theta+i\sin\theta has modulus 1 because cos⁡2θ+sin⁡2θ=1\cos^2\theta+\sin^2\theta=1. Multiplying by it rotates a complex number without changing its magnitude.

Reading an amplitude from a complex coefficient

Suppose an analyst constructs a fictional periodic component of P&L, purely to illustrate frequency analysis. Its complex coefficient is A=(3+4i) USDA=(3+4i)\,\text{USD}. Let tt be elapsed time in days and ω=2π\omega=2\pi radians per day, so the period is one day. Define the real signal by
p(t)=Re⁡ ⁣(Aeiωt)=(3cos⁡(ωt)−4sin⁡(ωt)) USD.\begin{aligned} p(t)&=\operatorname{Re}\!\left(Ae^{i\omega t}\right)\\ &=\left(3\cos(\omega t)-4\sin(\omega t)\right)\,\text{USD}. \end{aligned}

The two components combine into a single oscillation:

p(t)=5cos⁡(ωt+θ) USD,θ=atan2⁡(4,3)≈0.9273 radians.p(t)=5\cos(\omega t+\theta)\,\text{USD}, \qquad \theta=\operatorname{atan2}(4,3)\approx0.9273\text{ radians}.
Its amplitude is ∣A∣=5 USD\lvert A\rvert=5\,\text{USD}. The phase determines the timing of the peaks, while the modulus determines their distance from the zero baseline. At t=0t=0 the signal is USD 3, not USD 5; amplitude is a maximum excursion, not the current value.
Synthetic one-day periodic P&L component 3 cosine minus 4 sine, in USD. The signal starts at USD 3 and oscillates between a trough of negative USD 5 and a peak of USD 5; its complex coefficient has modulus USD 5.
Synthetic one-day periodic P&L component 3 cosine minus 4 sine, in USD. The signal starts at USD 3 and oscillates between a trough of negative USD 5 and a peak of USD 5; its complex coefficient has modulus USD 5.
Original synthetic signal, with angular frequency 2π2\pi radians per day. The modulus determines amplitude; this illustration contains no observed market data or estimated trading relationship.
Check the Fourier convention before interpreting a coefficient. Here we used p(t)=Re⁡(Aeiωt)p(t)=\operatorname{Re}(Ae^{i\omega t}). If the same signal is written as the two-sided Fourier pair
p(t)=c1eiωt+c−1e−iωt,c1=A2,c−1=A‾2,p(t)=c_1e^{i\omega t}+c_{-1}e^{-i\omega t}, \qquad c_1=\frac A2,\quad c_{-1}=\frac{\overline A}{2},

each coefficient has modulus USD 2.50; together they produce the USD 5 amplitude. A transform library's scaling also matters. A magnitude spectrum is therefore not automatically a list of peak amplitudes without knowing the representation.

This is one role of modulus in Fourier analysis. It does not turn a coefficient into a probability, option price or expected profit. A characteristic function is a different, dimensionless probability transform; the USD-valued coefficient above is not one. Replacing complex coefficients by their moduli also removes phase information and can change the reconstructed signal.

For the option-pricing application, Fourier Transform Pricing develops characteristic functions, inversion and explicit transform conventions. The public page is a preview; the complete lesson requires membership.

Sources

Last updated
  • complex numbers
  • modulus
  • conjugation
  • fourier analysis
  • amplitude
Keep connecting the ideas
← Back to the glossaryBack to top ↑