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Complex Numbers (Euler’s Formula and Phasors)

Learn complex numbers, Euler’s formula, polar form, and how phasors simplify sinusoidal physics (AC circuits and oscillations).

  • University Physics Year 1
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Complex numbers are not “extra math for no reason” — they are a compact way to represent amplitude and phase, which shows up everywhere in physics (oscillations, waves, AC circuits, and quantum mechanics).

Why this matters + quick links

You meet complex numbers whenever physics has a “rotation in time/phase” idea:

1) Basics

A complex number is

z = a + ib,

where a and b are real and i² = -1.

  • Real part: Re(z) = a
  • Imaginary part: Im(z) = b
  • Complex conjugate: z* = a-ib
  • Modulus: |z| = square root of (a² + b²)

Geometric picture: plot z as a point in the complex plane (horizontal axis = real part, vertical axis = imaginary part). Then:

  • |z| is the distance from the origin.
  • arg(z) is the angle from the positive real axis.

Quick example (conjugate + modulus)

Let z = 3-4i.

  • z* = 3 + 4i
  • |z| = square root of (3² + (-4)²) = 5
  • Useful identity: zz* = |z|² (here: (3-4i)(3 + 4i) = 25)
Key benefit (why conjugates show up in physics)

Conjugates let you isolate magnitudes cleanly. For example, phasor power formulas often contain V tilde I tilde * so that the average power comes from a real part.

2) Polar / exponential form

Write z using a magnitude r = |z| and an angle θ = arg(z):

z = r(cos θ + i sin θ).

Euler’s formula says:

e^iθ = cos θ + i sin θ,

so the same number can be written:

z = re^iθ.

Why this is useful:

  • Multiplication becomes easy:
    (r₁e^iθ₁)(r₂e^iθ₂) = (r₁r₂)e^(i(θ₁ + θ₂)).
  • Division becomes easy:
    r₁e^iθ₁/r₂e^iθ₂ = (r₁/r₂)e^(i(θ₁-θ₂)).
Pitfall: angles and quadrants

When you convert z = a + ib to (r,θ), don’t use θ = tan⁻¹ (b/a) blindly: it can put you in the wrong quadrant. Use the correct quadrant (think “atan2”) and keep track of whether your calculator is in degrees or radians.

3) Powers and roots (De Moivre’s idea)

Using z = re^iθ, powers are straightforward:

zⁿ = rⁿ e^inθ.

Roots come from the fact that angles differing by 2π give the same complex number:

z = re^iθ = re^(i(θ + 2π k)).

So the nth roots are:

z^(1/n) = r^(1/n)e^(i(θ + 2π k)/n), k = 0,1,…,n-1.

Worked example A (power using polar form)

Compute (1 + i)⁵.

First write 1 + i in polar form:

1 + i = square root of 2 e^(iπ/4).

Then:

(1 + i)⁵ = (square root of 2)⁵ e^(i5π/4) = 4 square root of 2 (cos 5π/4 + i sin 5π/4).

Since cos(5π/4) = sin(5π/4) = -(square root of 2)/2:

(1 + i)⁵ = 4 square root of 2 (-(square root of 2)/2-i(square root of 2)/2) = -4-4i.

Worked example B (cube roots)

Solve z³ = 8.

Write 8 = 8eⁱ⁰. Then:

z = 8^(1/3)e^(i(0 + 2π k)/3) = 2e^(i2π k/3), k = 0,1,2.

So the three roots are:

2, 2e^(i2π/3), 2e^(i4π/3).

4) Why physics loves this (sinusoids and phasors)

You can represent a cosine as the real part of a complex exponential:

cos(ω t + φ) = Re(e^(i(ω t + φ))).

Two useful identities (derived from Euler’s formula):

cos θ = (e^iθ + e^(-iθ))/2, sin θ = (e^iθ-e^(-iθ))/2i.

So a general sinusoid can be written as:

A cos(ω t + φ) = Re(Ae^iφₚₕₐₛₒᵣe^(iω t)).

The phasor Ae^iφ stores “how big” and “how phase-shifted” the signal is.

Key benefit

You do algebra with phasors (complex numbers), then take the real part at the end.
This turns trig-heavy problems into much cleaner calculations.

Derivatives/integrals become algebra

If you choose the convention Re(X tilde e^(iω t)) for a sinusoid, then:

(d/dt)(X tilde e^(iω t)) = iωX tilde e^(iω t), ∫ X tilde e^(iω t) dt = ((X tilde)/iω)e^(iω t) + C.

This is why phasors simplify linear ODEs in circuits and oscillations.

Pitfall: you must choose a convention and stick to it

Some texts use e^(iω t), others use e^(-iω t). Both work, but it changes whether “differentiate” corresponds to + iω or -iω, and it changes the sign conventions for impedances and phase angles.

5) Worked example (adding sinusoids with the same frequency)

Write 3 cos(ω t)-4 sin(ω t) as a single cosine.

Use complex exponentials:

3 cos(ω t)-4 sin(ω t) = Re((3 + 4i)e^(iω t)).

Write the complex number in polar form:

3 + 4i = 5e^iα, α = tan⁻¹ (4/3).

So:

Re((3 + 4i)e^(iω t)) = Re(5e^(i(ω t + α))) = 5 cos(ω t + α).

Therefore:

3 cos(ω t)-4 sin(ω t) = 5 cos(ω t + α), α = tan⁻¹ (4/3).

6) Worked example (phasors in a circuit: amplitude + phase)

Consider a series R–L circuit driven by:

v(t) = V₀ cos(ω t).

Use phasors with the Re(V tilde e^(iω t)) convention:

V tilde = V₀, Z = R + iω L.

Ohm’s law in phasor form is V tilde = ZI tilde, so:

I tilde = (V tilde)/Z = V₀/(R + iω L).

Magnitude (current amplitude):

|I tilde | = V₀/(square root of (R² + (ω L)²)).

Phase (how much current lags voltage):

∠I tilde = - tan⁻¹ ((ω L)/R).
Quick checks (limits)
  • Low frequency ω → 0: Z → R, so |I tilde | → V₀/R and phase → 0 (purely resistive).
  • High frequency ω → ∞: |I tilde | → 0 and phase → -90° (inductor dominates).

  • UY1: Phasors & Alternating Currents

    Applies complex numbers to AC circuits and phase relationships.

    Open lesson

  • UY1: L-R-C Series Circuit

    Where complex impedance turns a differential equation into algebra.

    Open lesson

  • Second-order differential equations

    Complex roots lead to sinusoidal solutions.

    Open lesson

  • Matrices and linear algebra

    Eigenvalues and oscillations connect directly to complex numbers.

    Open lesson

Back to Mathematics for Undergraduate Physics