Complex Numbers

Complex numbers extend the concept of the one-dimensional number line to a two-dimensional complex plane. They are essential in engineering for representing alternating currents, oscillations, and roots of polynomials that do not intersect the x-axis.

The Imaginary Unit

The foundation of complex numbers is the imaginary unit, denoted as ii (or jj in electrical engineering), defined as the square root of negative one.

Note

i=−1textandi2=−1\begin{aligned} i = \sqrt{-1} \quad \\text{and} \quad i^2 = -1 \end{aligned}

Powers of i

  • i1=ii^1 = i
  • i2=−1i^2 = -1
  • i3=i2⋅i=−ii^3 = i^2 \cdot i = -i
  • i4=(i2)2=(−1)2=1i^4 = (i^2)^2 = (-1)^2 = 1

Note

The powers of ii cycle every four powers: i,−1,−i,1i, -1, -i, 1. To find ini^n, divide nn by 44 and look at the remainder.

Standard Form

A complex number zz is written in standard rectangular form as z=a+biz = a + bi.

Components

  • Real Part: textRe(z)=a\\text{Re}(z) = a
  • Imaginary Part: textIm(z)=b\\text{Im}(z) = b
  • Complex Conjugate: zˉ=a−bi\bar{z} = a - bi. Multiplying a complex number by its conjugate always yields a real number: (a+bi)(a−bi)=a2+b2(a+bi)(a-bi) = a^2 + b^2.

Interactive Complex Plane (Argand Diagram)

Complex numbers can be plotted on a 2D plane where the x-axis is real and the y-axis is imaginary. Explore the polar representation (magnitude and angle) of a complex number.

Complex Plane (Argand Diagram)

r = 5.00|θ = 53.1°

Rectangular Form

z=3+4iz = 3 + 4i
Real Part (a)3
Imaginary Part (b)4

Polar Form

z=5.00(cos⁡53.1∘+isin⁡53.1∘)z = 5.00(\cos 53.1^\circ + i\sin 53.1^\circ)
Re(z)Im(z)

Polar Form and Euler's Formula

Complex numbers can also be expressed using a magnitude (rr) and an angle (θ\theta), which is extremely useful for multiplication and division.

Polar and Exponential Forms

z=r(cos⁡θ+isin⁡θ)=reiθ z = r(\cos\theta + i\sin\theta) = re^{i\theta}
Where r=a2+b2r = \sqrt{a^2 + b^2} and θ=tan⁡−1(ba)\theta = \tan^{-1}\left(\frac{b}{a}\right).

De Moivre's Theorem

This theorem makes it easy to compute powers and roots of complex numbers.

De Moivre's Theorem

A formula useful for finding powers and roots of complex numbers.

[r(cos⁡θ+isin⁡θ)]n=rn(cos⁡(nθ)+isin⁡(nθ))[r(\cos\theta + i\sin\theta)]^n = r^n(\cos(n\theta) + i\sin(n\theta))

Variables

SymbolDescriptionUnit
rrThe modulus (magnitude) of the complex number-
θ\thetaThe argument (angle) of the complex number in radians-
nnThe integer power to which the complex number is raised-
iiThe imaginary unit, satisfying i² = -1-

Finding n-th Roots

Extending De Moivre's Theorem to find all n distinct roots of a complex number.

zk=r1/n[cos⁡(θ+360circkn)+isin⁡(θ+360circkn)]\begin{aligned} z_k = r^{1/n} \left[ \cos\left(\frac{\theta + 360^{\\circ} k}{n}\right) + i \sin\left(\frac{\theta + 360^{\\circ} k}{n}\right) \right] \end{aligned}

Variables

SymbolDescriptionUnit
zkz_kThe k-th root of the complex number-
nnThe degree of the root (e.g., n=3 for cube roots)-
kkAn integer counter from 0 to n-1-
360∘ or 2π360^{\circ} \text{ or } 2\piA full rotation ensuring all roots are found within one cycle-
Key Takeaways
  • The i Cycle: Powers of ii repeat every four cycles (i,−1,−i,1i, -1, -i, 1).
  • Division: To divide complex numbers, multiply the top and bottom by the complex conjugate of the denominator.
  • Geometry: Multiplying by ii geometrically represents a 90^\\circ counterclockwise rotation in the complex plane.
  • Polar Power: Polar form (reiθre^{i\theta}) makes multiplying and finding powers of complex numbers significantly easier via De Moivre's Theorem.
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