The Central Idea
A nonzero complex number
\[ z=a+bi \]determines a point \((a,b)\) in the complex plane. The same point can be described by its distance \(r\) from the origin and an angle \(\theta\) from the positive real axis.
\[ a=r\cos\theta, \qquad b=r\sin\theta. \]Therefore,
\[ \boxed{z=r(\cos\theta+i\sin\theta)}. \]This is the polar, or trigonometric, form of a complex number.
The Modulus
For
\[ z=a+bi, \]the distance from the origin is
\[ \boxed{|z|=\sqrt{a^2+b^2}}. \]In polar notation this distance is written \(r\), so
\[ \boxed{r=|z|}. \]The Argument
The angle from the positive real axis to the vector representing \(z\) is called an argument of \(z\).
\[ \theta=\arg z. \]When \(a\neq0\), the angle satisfies
\[ \tan\theta=\frac{b}{a}. \]The quadrant of \((a,b)\) must be considered when selecting the correct angle.
The Argument Is Not Unique
A complete rotation changes an angle by \(2\pi\) without changing its direction. Therefore, if \(\theta\) is one argument of \(z\), then
\[ \boxed{\theta+2\pi k,\qquad k\in\mathbb{Z}} \]is also an argument.
The Principal Argument
A common convention chooses one standard argument called the principal argument, written \(\operatorname{Arg}(z)\), with
\[ \boxed{-\pi<\operatorname{Arg}(z)\le\pi}. \]Other conventions are possible, so the interval should always be stated or understood from context.
The Special Case \(z=0\)
The complex number \(0\) has modulus \(0\), but it has no defined argument. The zero vector has no direction, so formulas involving \(\arg z\) normally assume \(z\neq0\).
Converting Rectangular Form to Polar Form
For \(z=a+bi\), first calculate
\[ r=\sqrt{a^2+b^2}. \]Then determine an angle \(\theta\) satisfying
\[ \cos\theta=\frac{a}{r}, \qquad \sin\theta=\frac{b}{r}. \]Finally write
\[ \boxed{z=r(\cos\theta+i\sin\theta)}. \]Example: Convert \(1+i\) to Polar Form
The modulus is
\[ r=\sqrt{1^2+1^2}=\sqrt2. \]The point \((1,1)\) lies at angle
\[ \theta=\frac{\pi}{4}. \]Therefore,
\[ \boxed{ 1+i=\sqrt2\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right) }. \]Example: Quadrant Matters
Convert
\[ z=-1+\sqrt3\,i \]to polar form. The modulus is
\[ r=\sqrt{(-1)^2+(\sqrt3)^2}=2. \]The reference angle is \(\pi/3\), but the point lies in Quadrant II, so the principal argument is
\[ \theta=\frac{2\pi}{3}. \]Thus,
\[ \boxed{ -1+\sqrt3\,i = 2\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right) }. \]Converting Polar Form to Rectangular Form
Starting with
\[ z=r(\cos\theta+i\sin\theta), \]distribute \(r\):
\[ z=r\cos\theta+i(r\sin\theta). \]Therefore,
\[ \boxed{a=r\cos\theta,\qquad b=r\sin\theta}. \]Example: Convert Polar Form to Rectangular Form
Convert
\[ 6\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right). \]Since
\[ \cos\frac{\pi}{6}=\frac{\sqrt3}{2}, \qquad \sin\frac{\pi}{6}=\frac12, \]we get
\[ z=6\left(\frac{\sqrt3}{2}\right)+6i\left(\frac12\right). \]Therefore,
\[ \boxed{z=3\sqrt3+3i}. \]Multiplying Complex Numbers in Polar Form
Let
\[ z_1=r_1(\cos\theta_1+i\sin\theta_1) \]and
\[ z_2=r_2(\cos\theta_2+i\sin\theta_2). \]Using the trigonometric angle-addition identities,
\[ \boxed{ z_1z_2 = r_1r_2 \left[ \cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2) \right] }. \]The Geometry of Complex Multiplication
The multiplication rule says:
- multiply the moduli;
- add the arguments.
and
\[ \boxed{\arg(z_1z_2)=\arg z_1+\arg z_2\pmod{2\pi}}. \]Complex multiplication is therefore a combination of scaling and rotation.
Example: Multiplication in Polar Form
Let
\[ z_1=2\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right) \]and
\[ z_2=3\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right). \]Multiply the moduli:
\[ 2\cdot3=6. \]Add the angles:
\[ \frac{\pi}{6}+\frac{\pi}{4}=\frac{5\pi}{12}. \]Therefore,
\[ \boxed{ z_1z_2 = 6\left(\cos\frac{5\pi}{12}+i\sin\frac{5\pi}{12}\right) }. \]Dividing Complex Numbers in Polar Form
For \(z_2\neq0\),
\[ \boxed{ \frac{z_1}{z_2} = \frac{r_1}{r_2} \left[ \cos(\theta_1-\theta_2) + i\sin(\theta_1-\theta_2) \right] }. \]Division divides moduli and subtracts arguments.
Example: Division in Polar Form
Compute
\[ \frac{ 8\left(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4}\right) }{ 2\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right) }. \]Divide the moduli:
\[ \frac82=4. \]Subtract the arguments:
\[ \frac{3\pi}{4}-\frac{\pi}{6}=\frac{7\pi}{12}. \]Therefore,
\[ \boxed{ 4\left(\cos\frac{7\pi}{12}+i\sin\frac{7\pi}{12}\right) }. \]De Moivre's Formula
Repeated multiplication leads to a convenient rule for integer powers. If
\[ z=r(\cos\theta+i\sin\theta), \]then for an integer \(n\),
\[ \boxed{ z^n = r^n(\cos n\theta+i\sin n\theta) }. \]This is De Moivre's formula.
Example: Using De Moivre's Formula
Compute
\[ \left[ 2\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right) \right]^4. \]Raise the modulus to the fourth power:
\[ 2^4=16. \]Multiply the argument by \(4\):
\[ 4\cdot\frac{\pi}{3}=\frac{4\pi}{3}. \]Therefore,
\[ \boxed{ 16\left(\cos\frac{4\pi}{3}+i\sin\frac{4\pi}{3}\right) }. \]Roots of a Complex Number
Suppose
\[ w^n=z \]and write
\[ z=r(\cos\theta+i\sin\theta). \]The \(n\) distinct roots are
\[ \boxed{ w_k = r^{1/n} \left[ \cos\left(\frac{\theta+2\pi k}{n}\right) + i\sin\left(\frac{\theta+2\pi k}{n}\right) \right] } \]for
\[ k=0,1,\ldots,n-1. \]The Geometry of Complex Roots
All \(n\)th roots of a nonzero complex number have modulus
\[ r^{1/n}. \]Their arguments differ by
\[ \frac{2\pi}{n}. \]Therefore, the roots are equally spaced around a circle centered at the origin.
The \(n\)th Roots of Unity
The equation
\[ z^n=1 \]has the roots
\[ \boxed{ z_k = \cos\left(\frac{2\pi k}{n}\right) + i\sin\left(\frac{2\pi k}{n}\right) } \]for \(k=0,1,\ldots,n-1\). They lie at the vertices of a regular \(n\)-gon on the unit circle.
Example: Cube Roots of Unity
The equation
\[ z^3=1 \]gives the angles
\[ 0, \qquad \frac{2\pi}{3}, \qquad \frac{4\pi}{3}. \]Therefore, the roots are
\[ \boxed{1}, \] \[ \boxed{-\frac12+\frac{\sqrt3}{2}i}, \] \[ \boxed{-\frac12-\frac{\sqrt3}{2}i}. \]The Complex Exponential
The exponential function extends naturally to complex inputs. The central identity is
\[ \boxed{e^{i\theta}=\cos\theta+i\sin\theta}. \]This is Euler's formula. It links exponential functions, trigonometry, and rotation in the complex plane.
Special Values from Euler's Formula
At \(\theta=0\),
\[ e^{0i}=1. \]At \(\theta=\frac{\pi}{2}\),
\[ e^{i\pi/2}=i. \]At \(\theta=\pi\),
\[ e^{i\pi}=-1. \]Euler's Identity
Since
\[ e^{i\pi}=-1, \]we obtain
\[ \boxed{e^{i\pi}+1=0}. \]This single equation connects \(e\), \(i\), \(\pi\), \(1\), and \(0\).
Exponential Form of a Complex Number
Combining polar form with Euler's formula gives
\[ z=r(\cos\theta+i\sin\theta) =re^{i\theta}. \]Therefore,
\[ \boxed{z=re^{i\theta}}. \]This is the exponential form of a nonzero complex number.
Example: Write \(1+i\) in Exponential Form
We know
\[ |1+i|=\sqrt2 \]and
\[ \arg(1+i)=\frac{\pi}{4}. \]Therefore,
\[ \boxed{1+i=\sqrt2\,e^{i\pi/4}}. \]Multiplication in Exponential Form
If
\[ z_1=r_1e^{i\theta_1} \]and
\[ z_2=r_2e^{i\theta_2}, \]then ordinary exponent rules give
\[ \boxed{z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}}. \]Division in Exponential Form
For \(z_2\neq0\),
\[ \boxed{ \frac{z_1}{z_2} = \frac{r_1}{r_2}e^{i(\theta_1-\theta_2)} }. \]Powers in Exponential Form
If
\[ z=re^{i\theta}, \]then
\[ \boxed{z^n=r^ne^{in\theta}}. \]Applying Euler's formula recovers De Moivre's formula immediately.
The Unit Circle in Exponential Form
Every complex number of modulus \(1\) can be written as
\[ \boxed{e^{i\theta}}. \]As \(\theta\) varies, \(e^{i\theta}\) traces the unit circle.
Rotation by Multiplication with \(e^{i\phi}\)
Let
\[ z=re^{i\theta}. \]Multiplication by \(e^{i\phi}\) gives
\[ e^{i\phi}z=re^{i(\theta+\phi)}. \]The modulus remains unchanged while the argument increases by \(\phi\). Therefore,
\[ \boxed{z\mapsto e^{i\phi}z} \]rotates the plane counterclockwise by angle \(\phi\) about the origin.
Scaling and Rotation Together
Let
\[ w=\rho e^{i\phi}. \]Multiplying
\[ z=re^{i\theta} \]by \(w\) gives
\[ wz=\rho r e^{i(\phi+\theta)}. \]Thus multiplication by \(w\):
- scales distances from the origin by \(\rho=|w|\);
- rotates directions by \(\phi=\arg w\).
Conjugation in Exponential Form
If
\[ z=re^{i\theta}, \]then
\[ \boxed{\overline{z}=re^{-i\theta}}. \]Reversing the sign of the angle reflects the point across the real axis.
Reciprocals in Exponential Form
For \(z=re^{i\theta}\neq0\),
\[ \boxed{\frac1z=\frac1r e^{-i\theta}}. \]The reciprocal inverts the modulus and reverses the argument.
The Negative Real Axis
If \(r>0\), then the negative real number \(-r\) has principal argument \(\pi\). Therefore,
\[ \boxed{-r=re^{i\pi}}. \]All equivalent arguments are
\[ \pi+2\pi k, \qquad k\in\mathbb{Z}. \]Exponential Form Is Not Unique
Since
\[ e^{i(\theta+2\pi k)}=e^{i\theta}, \]a nonzero complex number has infinitely many exponential representations:
\[ \boxed{z=re^{i(\theta+2\pi k)},\qquad k\in\mathbb{Z}}. \]A Preview of the Complex Logarithm
For a nonzero complex number
\[ z=re^{i\theta}, \]expressions of the form
\[ \ln r+i\theta \]naturally arise when solving for an exponent. Because \(\theta\) can differ by multiples of \(2\pi\), the complex logarithm is naturally multivalued. Later lessons will treat branches and principal values carefully.
Why Exponential Form Is Useful
Rectangular form is usually most efficient for addition and subtraction. Polar or exponential form is usually most efficient for:
- multiplication;
- division;
- powers;
- roots;
- rotations;
- oscillatory models;
- frequency-domain methods;
- later analysis of complex functions.
Connection to Oscillation and Signals
Euler's formula gives
\[ e^{i\omega t}=\cos(\omega t)+i\sin(\omega t). \]A complex exponential therefore packages sine and cosine into one expression. This representation appears throughout differential equations, Fourier analysis, electrical engineering, wave models, communications, and signal processing.
A First Look at Phasors
A sinusoidal expression such as
\[ A\cos(\omega t+\phi) \]can be viewed as the real part of
\[ Ae^{i(\omega t+\phi)}. \]This rotating complex representation separates amplitude and phase from repetitive trigonometric calculations. It is the basic idea behind phasor methods.
Common Error: Ignoring the Quadrant
Using
\[ \theta=\arctan\left(\frac{b}{a}\right) \]without checking the signs of \(a\) and \(b\) can produce the wrong angle. For example, \(-1+i\) lies in Quadrant II, so its principal argument is
\[ \boxed{\frac{3\pi}{4}}, \]not \(-\frac{\pi}{4}\).
Common Error: Mixing Degrees and Radians
Complex analysis normally uses radians. For example,
\[ 90^\circ=\frac{\pi}{2}. \]When using a calculator, confirm the angle mode before interpreting an inverse-trigonometric result.
Common Error: Treating the Argument as Unique
If \(\theta\) is an argument, then
\[ \theta+2\pi k \]is also an argument for every integer \(k\). Only the principal argument is chosen as one conventional representative.
Common Error: Finding Only One Root
An equation such as
\[ w^n=z \]has \(n\) distinct roots when \(z\neq0\). Omitting the \(2\pi k\) term from the root formula usually causes the remaining roots to be missed.
Common Error: Applying the Wrong Operations
In polar form, multiplication means:
- multiply moduli;
- add arguments.
Division means:
- divide moduli;
- subtract arguments.
A Reliable Procedure: Rectangular to Polar Form
- Identify \(a=\operatorname{Re}(z)\) and \(b=\operatorname{Im}(z)\).
- Compute \(r=\sqrt{a^2+b^2}\).
- Find a reference angle.
- Use the quadrant to determine the correct argument.
- Write \(z=r(\cos\theta+i\sin\theta)\).
- When useful, rewrite it as \(z=re^{i\theta}\).
- Check using \(a=r\cos\theta\) and \(b=r\sin\theta\).
A Reliable Procedure: Polar Operations
- Write each nonzero number as \(re^{i\theta}\).
- For multiplication, multiply moduli and add arguments.
- For division, divide moduli and subtract arguments.
- For powers, raise the modulus and multiply the argument.
- For \(n\)th roots, take the \(n\)th root of the modulus.
- Use \((\theta+2\pi k)/n\) to generate every root argument.
- Convert back to rectangular form only when useful.
Worked Example: Rectangular, Polar, and Exponential Forms
Let
\[ z=-\sqrt3+i. \]The modulus is
\[ r=\sqrt{3+1}=2. \]The point lies in Quadrant II with reference angle \(\pi/6\), so
\[ \theta=\frac{5\pi}{6}. \]Polar form:
\[ \boxed{ z=2\left(\cos\frac{5\pi}{6}+i\sin\frac{5\pi}{6}\right) }. \]Exponential form:
\[ \boxed{z=2e^{i5\pi/6}}. \]Worked Example: Product by Exponential Form
Let
\[ z_1=2e^{i\pi/3} \]and
\[ z_2=5e^{-i\pi/6}. \]Then
\[ z_1z_2 = 10e^{i(\pi/3-\pi/6)} = 10e^{i\pi/6}. \]Therefore,
\[ \boxed{z_1z_2=10e^{i\pi/6}}. \]In rectangular form,
\[ \boxed{z_1z_2=5\sqrt3+5i}. \]Worked Example: Find All Fourth Roots of \(16\)
Write
\[ 16=16e^{i2\pi k}. \]The fourth-root modulus is
\[ 16^{1/4}=2. \]The root arguments are
\[ \frac{2\pi k}{4}=\frac{\pi k}{2}, \qquad k=0,1,2,3. \]Therefore, the roots are
\[ \boxed{2,\quad2i,\quad-2,\quad-2i}. \]Practice 1: Convert to Polar Form
Write
\[ z=\sqrt3+i \]in polar form.
Practice 2: Convert to Exponential Form
Write
\[ z=-2i \]in exponential form using the principal argument.
Practice 3: Convert to Rectangular Form
Convert
\[ 4e^{i\pi/3} \]to rectangular form.
Practice 4: Multiply
Compute
\[ \left(3e^{i\pi/4}\right)\left(2e^{i\pi/6}\right). \]Practice 5: Divide
Compute
\[ \frac{12e^{i5\pi/6}}{3e^{i\pi/3}}. \]Practice 6: Power
Use De Moivre's formula to compute
\[ \left[ 2\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right) \right]^3. \]Practice 7: Roots of Unity
Find all solutions of
\[ z^4=1. \]Practice 8: General Arguments
Give all arguments of
\[ z=-1-i. \]Practice 9: Rotation
If
\[ z=3-2i, \]find the result of rotating \(z\) counterclockwise by \(\pi/2\) using complex multiplication.
Practice 10: Find All Cube Roots
Find all solutions of
\[ w^3=8. \]Practice Answers
Practice 1
The modulus is
\[ r=2, \]and the argument is \(\pi/6\). Therefore,
\[ \boxed{ \sqrt3+i = 2\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right) }. \]Practice 2
The modulus is \(2\), and the principal argument is \(-\pi/2\). Therefore,
\[ \boxed{-2i=2e^{-i\pi/2}}. \]Practice 3
\[ 4e^{i\pi/3} = 4\left(\frac12+i\frac{\sqrt3}{2}\right). \]Therefore,
\[ \boxed{2+2\sqrt3\,i}. \]Practice 4
\[ \boxed{6e^{i5\pi/12}}. \]Practice 5
\[ \frac{12e^{i5\pi/6}}{3e^{i\pi/3}} = 4e^{i\pi/2}. \]Therefore,
\[ \boxed{4i}. \]Practice 6
\[ 2^3=8 \]and
\[ 3\cdot\frac{\pi}{4}=\frac{3\pi}{4}. \]Therefore,
\[ \boxed{ 8\left(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4}\right) }. \]In rectangular form,
\[ \boxed{-4\sqrt2+4\sqrt2\,i}. \]Practice 7
The fourth roots of unity are
\[ \boxed{1,\quad i,\quad-1,\quad-i}. \]Practice 8
The principal argument is \(-3\pi/4\). Therefore, all arguments are
\[ \boxed{-\frac{3\pi}{4}+2\pi k,\qquad k\in\mathbb{Z}}. \]Practice 9
A counterclockwise rotation by \(\pi/2\) is multiplication by \(i\):
\[ i(3-2i)=3i-2i^2=2+3i. \]Therefore,
\[ \boxed{2+3i}. \]Practice 10
The cube-root modulus is \(2\), and the arguments are
\[ 0, \qquad \frac{2\pi}{3}, \qquad \frac{4\pi}{3}. \]Therefore,
\[ \boxed{2,\quad-1+\sqrt3\,i,\quad-1-\sqrt3\,i}. \]Checking Strategies
- After converting to polar form, verify \(r\cos\theta=a\) and \(r\sin\theta=b\).
- Confirm that every modulus is nonnegative.
- Check the quadrant before accepting an inverse-tangent angle.
- For multiplication, verify that the product modulus equals the product of the original moduli.
- For roots, verify that the solutions are equally spaced in angle.
- Raise each proposed root to the required power when practical.
Connections to Earlier Mathematics
- Coordinate geometry: rectangular and polar coordinates describe the same point.
- Trigonometry: sine and cosine encode direction on the unit circle.
- Exponentiation: exponential laws become rotation laws.
- Vectors: modulus behaves like length and argument behaves like direction.
- Algebra: powers and roots become easier when multiplication is expressed geometrically.
What to Remember
For
\[ z=a+bi\neq0, \]the modulus is
\[ \boxed{r=|z|=\sqrt{a^2+b^2}}. \]Polar form is
\[ \boxed{z=r(\cos\theta+i\sin\theta)}. \]Euler's formula is
\[ \boxed{e^{i\theta}=\cos\theta+i\sin\theta}. \]Exponential form is
\[ \boxed{z=re^{i\theta}}. \]In this form,
\[ \boxed{ (r_1e^{i\theta_1})(r_2e^{i\theta_2}) = r_1r_2e^{i(\theta_1+\theta_2)} } \]and
\[ \boxed{ \frac{r_1e^{i\theta_1}}{r_2e^{i\theta_2}} = \frac{r_1}{r_2}e^{i(\theta_1-\theta_2)} }. \]The central geometric rule is: complex multiplication multiplies scale and adds rotation.
Where We Go Next
We now have three complementary representations of a complex number: rectangular form \(a+bi\), polar form \(r(\cos\theta+i\sin\theta)\), and exponential form \(re^{i\theta}\).
The next lesson, Complex Functions, uses these representations to study mappings from the complex plane to itself. We will examine how functions transform points, regions, magnitude, direction, and geometric structure before moving into complex limits and continuity.