26 Jul 2025

Complex Numbers, Euler's Formula, De Moivre's Theorem, and Roots

Argand-plane geometry, polar form, complex multiplication, integer powers, and all roots of a complex number.

bsc semester-ii mathematical-physics complex-analysis complex-numbers

A complex number is

\[z=x+iy, \qquad i^2=-1,\]

and is represented by the point $(x,y)$ in the Argand plane. Its conjugate and modulus are

\[\bar z=x-iy, \qquad \lvert z\rvert=\sqrt{z\bar z}=\sqrt{x^2+y^2}.\]

For $z\ne0$, let $\theta$ be an argument of $z$. Then

\[x=r\cos\theta,\qquad y=r\sin\theta,\qquad r=\lvert z\rvert,\]

so

\[\boxed{z=r(\cos\theta+i\sin\theta)}.\]

Because $\theta$ and $\theta+2\pi k$ represent the same point, the argument is multivalued. A selected principal argument is commonly restricted to one interval of length $2\pi$.

Euler’s formula

Let

\[w(\theta)=\cos\theta+i\sin\theta.\]

Differentiation gives

\[\frac{dw}{d\theta} =-\sin\theta+i\cos\theta =i(\cos\theta+i\sin\theta) =iw,\]

with $w(0)=1$. The exponential $e^{i\theta}$ is the solution of the same equation $w^{\prime}=iw$ with the same initial value. Hence

\[\boxed{e^{i\theta}=\cos\theta+i\sin\theta}.\]

Therefore the polar form is

\[\boxed{z=re^{i\theta}}.\]

Multiplication now has a direct geometrical meaning:

\[z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}.\]

The moduli multiply and the arguments add.

For $z_2\ne0$, division follows by multiplying by $e^{-i\theta_2}/r_2$:

\[\boxed{\frac{z_1}{z_2} =\frac{r_1}{r_2}e^{i(\theta_1-\theta_2)}}.\]

Thus division divides moduli and subtracts arguments.

De Moivre’s theorem

For an integer $n$,

\[(\cos\theta+i\sin\theta)^n =(e^{i\theta})^n=e^{in\theta}.\]

Using Euler’s formula again,

\[\boxed{ (\cos\theta+i\sin\theta)^n =\cos(n\theta)+i\sin(n\theta)}.\]

For negative $n$, the result follows by taking reciprocals, since $\lvert e^{i\theta}\rvert=1$.

Roots of a complex number

Let

\[z^n=Re^{i\Theta}, \qquad R>0.\]

Write $z=re^{i\theta}$. Then

\[r^ne^{in\theta}=Re^{i(\Theta+2\pi k)}.\]

Equality of moduli and arguments gives

\[r=R^{1/n}, \qquad \theta=\frac{\Theta+2\pi k}{n}.\]

Only $k=0,1,\ldots,n-1$ give distinct roots, because increasing $k$ by $n$ adds $2\pi$ to $\theta$. Thus

\[\boxed{ z_k=R^{1/n} \exp\!\left[i\frac{\Theta+2\pi k}{n}\right], \quad k=0,\ldots,n-1}.\]

The roots \(z_k=R^{1/5}e^{i(\Theta+2\pi k)/5}\) lie at equal angular intervals \(2\pi/5\). Their polygon is equation-generated from the root formula.

For example, the cube roots of $8=8e^{i2\pi k}$ are

\[z_k=2e^{i2\pi k/3}, \qquad k=0,1,2,\]

or

\[\boxed{2,\quad -1+i\sqrt3,\quad -1-i\sqrt3}.\]

Cubing each value gives $8$, as verified in the Unit III Maxima worksheet.

Solved Problems

1. Write $z=-\sqrt3+i$ in polar form and evaluate $z^6$ using De Moivre’s theorem.

   Its modulus is

   \[r=\sqrt{(-\sqrt3)^2+1^2}=2.\]

   The point is in quadrant II, with reference angle $\pi/6$, so its principal argument is

   \[\Theta=\pi-\frac{\pi}{6}=\frac{5\pi}{6}.\]

   Hence

   \[z=2e^{i5\pi/6}.\]

   De Moivre’s theorem gives

   \[z^6=2^6e^{i5\pi} =64(\cos5\pi+i\sin5\pi) =\boxed{-64}.\]

   The modulus check is $\lvert z^6\rvert=2^6=64$, and the angle $5\pi$ is equivalent to $\pi$ modulo $2\pi$, so the result must lie on the negative real axis.

2. Find all solutions of $z^4=-16$ and verify their geometrical arrangement.

   Write

   \[-16=16e^{i(\pi+2\pi k)}.\]

   The fourth-root modulus is $16^{1/4}=2$, and the distinct arguments are

   \[\theta_k=\frac{\pi+2\pi k}{4} =\frac{\pi}{4}+\frac{k\pi}{2}, \qquad k=0,1,2,3.\]

   Therefore

   \[\boxed{ z=\sqrt2+i\sqrt2,\; -\sqrt2+i\sqrt2,\; -\sqrt2-i\sqrt2,\; \sqrt2-i\sqrt2}.\]

   Each root has modulus $2$, consecutive roots differ in argument by $\pi/2$, and raising any root to the fourth power gives modulus $16$ and argument $\pi$ modulo $2\pi$.

Descriptive Questions

1. Derive Euler’s formula from the differential equation satisfied by $\cos\theta+i\sin\theta$.
2. Explain geometrically why complex multiplication multiplies moduli and adds arguments, while division subtracts arguments.
3. Prove De Moivre’s theorem for positive and negative integers.
4. Derive all $n$th roots of $Re^{i\Theta}$ and prove that they form a regular polygon.

Numerical Problems

1. Express $(3-4i)/(1+2i)$ in Cartesian form.

   Answer: $-1-2i$.

2. Find the modulus and principal argument of $-!1-i\sqrt3$.

   Answer: modulus $2$; principal argument $-2\pi/3$.

3. Find the area of the triangle in the Argand plane with vertices $z_1=0$, $z_2=2+i$, and $z_3=1+3i$.

   Answer: $5/2$ square units.

4. Solve $z+2\bar z=3+i$.

   Answer: $z=1-i$.

5. Describe the locus $\lvert z-(2-i)\rvert=3$ in the Argand plane.

   Answer: the circle with center $2-i$ and radius $3$.

6. Find both square roots of $5+12i$.

   Answer: $\pm(3+2i)$.

All added algebraic powers, roots, and answer checks are verified in the Unit III Maxima worksheet; every printed residual is zero.

References

1. Complex number — Wikipedia
2. MIT OpenCourseWare 18.04, lecture notes: Topic 1, Complex Algebra and the Complex Plane
3. James Ward Brown and Ruel V. Churchill, Complex Variables and Applications, 9th ed., Chapter 1.