Complex Numbers: Choose the Angle, Find Every Root

1. Read the point before choosing the angle

Write $z=a+bi$, where $a$ and $b$ are real numbers and $i^2=-1$. This is Cartesian form. The real part is $a$; the imaginary part is the real number $b$, not $bi$. Plot $z$ at $(a,b)$: horizontal real axis, vertical imaginary axis.

The modulus $r=|z|=\sqrt{a^2+b^2}$ is the distance from the origin. For $z\ne0$, an argument $\theta$ is an angle from the positive real axis to the point, positive counterclockwise. Then

$$\begin{gathered}z=r(\cos\theta+i\sin\theta),\\ a=r\cos\theta,\quad b=r\sin\theta.\end{gathered}$$

This is polar form. We sometimes write $\operatorname{cis}\theta=\cos\theta+i\sin\theta$ to save space. Euler’s notation $e^{i\theta}$ means the same expression for real $\theta$; you do not need it for the calculations below.

Adding any whole number of turns gives the same point. All arguments are $\theta+2\pi k$, where $k$ is any integer. In this lesson the principal argument $\operatorname{Arg}z$ is the one in $(-\pi,\pi]$. A course may use another interval; follow the stated convention.

For $z=0$, the modulus is zero and there is no defined argument. Although $0(\cos\theta+i\sin\theta)=0$ for any angle, none is a direction of the origin.

2. Worked example: why the quadrant matters

Convert $z=-1+i$ to polar form.

1. Find the length

   $a=-1$, $b=1$, so $r=\sqrt{(-1)^2+1^2}=\sqrt2$.

2. Locate the point

   The point $(-1,1)$ is in quadrant II. Its reference angle, the acute angle to the nearest real-axis direction, is $\pi/4$. Measured counterclockwise from the positive real axis, the angle is $\pi-\pi/4=3\pi/4$.

3. Write and check

   $$z=\sqrt2\operatorname{cis}(3\pi/4).$$

   Reconstruct the coordinates: $\sqrt2\cos(3\pi/4)=-1$ and $\sqrt2\sin(3\pi/4)=1$. Both match.

A calculator may give $\arctan(1/(-1))=-\pi/4$. But that angle has positive cosine and negative sine: it points to quadrant IV and produces $1-i$. It is not an argument of $-1+i$. In contrast, $-5\pi/4$ is an argument because it differs from $3\pi/4$ by one full turn.

Quick check: What is the principal argument of $-2i$? Its point lies straight down the imaginary axis, so the answer is $-\pi/2$. The angle $3\pi/2$ describes the same direction, but is outside our principal interval.

3. Multiply lengths and add angles

For nonzero numbers $z=r\operatorname{cis}\theta$ and $u=s\operatorname{cis}\phi$, with positive moduli $r,s$ and real arguments $\theta,\phi$, the sine and cosine addition formulas give

$$zu=rs\operatorname{cis}(\theta+\phi).$$

Multiplication scales the length and turns the direction. For a positive integer $n$, repeating that multiplication gives De Moivre’s formula:

$$z^n=r^n\operatorname{cis}(n\theta).$$

You multiply the angle by $n$, and raise the modulus to the power $n$. You may subtract whole turns from the resulting angle.

Check the same square in two forms

For $z=-1+i$, Cartesian expansion gives

$$(-1+i)^2=1-2i+i^2=-2i.$$

Polar form gives

$$\begin{gathered}(\sqrt2\operatorname{cis}(3\pi/4))^2\\ =2\operatorname{cis}(3\pi/2)\\ =-2i.\end{gathered}$$

The new modulus is $2$, and the new point is directly below the origin. Both methods agree. Squaring the two terms separately would miss the middle term $-2i$.

For addition and subtraction, Cartesian form is usually simpler: combine real parts and imaginary parts separately. The add-the-angles rule is a multiplication rule.

4. Find every root

Suppose you want every complex number $w$ satisfying $w^n=z$, where $n$ is a positive integer and $z=r\operatorname{cis}\theta\ne0$. Write $w=\rho\operatorname{cis}\alpha$, with positive modulus $\rho$ and argument $\alpha$. Comparing lengths gives $\rho^n=r$, so $\rho=r^{1/n}$, the positive real nth root of $r$. Comparing angles gives $n\alpha=\theta+2\pi k$. Thus

$$\begin{gathered}w_k=r^{1/n}\operatorname{cis}\left(\frac{\theta+2\pi k}{n}\right),\\ k=0,1,\ldots,n-1.\end{gathered}$$

The subscript $k$ labels the different roots. They lie on a circle of radius $r^{1/n}$, with neighbouring angles separated by $2\pi/n$. These $n$ choices are distinct; increasing $k$ by $n$ adds a full turn and repeats a root. Dividing only one argument by $n$ usually finds just one root. If $z=0$, the only distinct root is $w=0$.

Worked example: solve $w^3=-8$

Use $-8=8\operatorname{cis}\pi$. Each root has modulus $2$. The three angles are $\pi/3$, $\pi$ and $5\pi/3$, from $k=0,1,2$. Therefore

$$w\in\{1+i\sqrt3,\ -2,\ 1-i\sqrt3\}.$$

Each cube has modulus $2^3=8$; its angle is an odd multiple of $\pi$, so each cube is $-8$. An independent Cartesian check for the first root gives $(1+i\sqrt3)^2=-2+2i\sqrt3$, then $(-2+2i\sqrt3)(1+i\sqrt3)=-8$. The other nonreal root is its conjugate, obtained by reversing the imaginary sign; its cube is also $-8$. Finally $(-2)^3=-8$.

5. Try it, check it, try again

Write your attempt before opening an answer. If a result looks wrong, first check the quadrant, then the modulus, then the number of distinct roots.

1. Convert a new point

Find the modulus and principal argument of $-\sqrt3-i$, and write its polar form.

2. Use a power, then check it

Compute $(1+i)^4$ using polar form. Check by squaring twice.

3. Recover both square roots

Solve $w^2=-2i$. Give both answers in Cartesian form.

4. Find four distinct roots

Solve $w^4=-16$. List the roots and explain why adding $k=4$ gives no new answer.

5. Retry with a rotated target

Without copying the previous angles, solve $w^3=8i$. Predict the root spacing before calculating, then check one root in Cartesian form.

6. Choose your next step

If angles are the sticking point, sketch the real and imaginary signs before calculating. If roots are the sticking point, write the full family $\theta+2\pi k$ before dividing. A good final check asks: correct modulus, correct direction, and enough distinct answers?

For course context, see MATH 202 support or ENGR 213 support. For a different algebra topic, try finite sums and induction; for algebra preparation, revisit algebra skills before calculus.

If you would like help, bring an attempt and your current course outline to a tutoring session. We can work through where the quadrant or root count became unclear, then use a fresh practice question to check your understanding. Request tutoring.