. This exponential form inherits all the advantages of trigonometric representation while invoking the familiar rules of exponents — multiplication adds exponents, division subtracts them, and powers simply multiply.

Euler's Formula

In 1748, Leonhard Euler published a relationship that would become one of mathematics' most celebrated results:

e^{i\theta} = \cos\theta + i\sin\theta

The formula asserts an equality between an exponential expression and a trigonometric one. On the left, the base

e

— the fundamental constant of calculus, approximately

2.71828

— is raised to an imaginary power

i\theta

. On the right, ordinary cosine and sine combine with the imaginary unit to form a complex number.

The equation holds for every real value of

\theta

. When

\theta = 0

, both sides equal

1

. When

\theta = \pi/2

, the left side is

e^{i\pi/2}

and the right side is

\cos(\pi/2) + i\sin(\pi/2) = 0 + i = i

. The formula claims these are equal, and they are.

Euler's formula bridges two mathematical worlds. Exponential functions describe growth and decay — populations, radioactive isotopes, compound interest. Trigonometric functions describe oscillation and rotation — waves, circles, periodic phenomena. The formula reveals these as two faces of the same underlying structure, connected through the imaginary unit.

The significance extends beyond elegance. Euler's formula transforms complex arithmetic into exponential arithmetic, where the rules are simpler and more intuitive. It provides the foundation for the exponential form of complex numbers and enables computational techniques impossible with purely algebraic methods.

Understanding Euler's Formula

Euler's formula is not an arbitrary definition but a theorem derivable from fundamental principles. The proof emerges from Taylor series — infinite polynomial expansions that represent functions as sums of powers.

The exponential function has the Taylor series:

e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots

Substituting

x = i\theta

e^{i\theta} = 1 + i\theta + \frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!} + \frac{(i\theta)^4}{4!} + \cdots

The powers of

i

cycle:

i^2 = -1

i^3 = -i

i^4 = 1

, and so on. Substituting:

= 1 + i\theta - \frac{\theta^2}{2!} - \frac{i\theta^3}{3!} + \frac{\theta^4}{4!} + \frac{i\theta^5}{5!} - \cdots

Separating real and imaginary parts:

= \left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots\right) + i\left(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots\right)

The real part is the Taylor series for

\cos\theta

. The imaginary part is the Taylor series for

\sin\theta

. Therefore

e^{i\theta} = \cos\theta + i\sin\theta

.

Geometrically, as

\theta

increases from

0

, the point

e^{i\theta}

traces the unit circle in the complex plane. At

\theta = 0

, we start at

1

. At

\theta = \pi/2

, we reach

i

. At

\theta = \pi

, we arrive at

-1

. At

\theta = 2\pi

, we complete the circle and return to

1

. The exponential

e^{i\theta}

parametrizes circular motion.

Euler's Identity

A special case of Euler's formula achieves legendary status. Substituting

\theta = \pi

e^{i\pi} = \cos\pi + i\sin\pi = -1 + 0 = -1

Rearranging:

e^{i\pi} + 1 = 0

This equation — Euler's identity — connects five of mathematics' most fundamental constants in a single statement. The number

e

, base of natural logarithms and central to calculus. The number

i

, foundation of complex arithmetic. The number

\pi

, ratio of circumference to diameter and key to circular geometry. The number

1

, the multiplicative identity. The number

0

, the additive identity.

Each constant emerged from different mathematical needs:

e

from growth and limits,

i

from algebraic completeness,

\pi

from measurement of circles,

1

and

0

from the structure of arithmetic itself. That a single equation links all five suggests deep unity beneath apparently disparate branches of mathematics.

Mathematicians and physicists routinely cite Euler's identity as the most beautiful formula ever discovered. Richard Feynman called it "the most remarkable formula in mathematics." The equation appears on countless posters, t-shirts, and tattoos — a rare piece of abstract mathematics that captures popular imagination.

Beyond aesthetics, the identity encodes practical information. It confirms that

e^{i\pi} = -1

, useful whenever exponential and trigonometric expressions interact. It demonstrates that raising a positive real number to an imaginary power can produce a negative real result — a counterintuitive fact with computational consequences.

The Exponential Form of a Complex Number

Euler's formula transforms the trigonometric form into something more compact. Since

\text{cis}\theta = \cos\theta + i\sin\theta = e^{i\theta}

, any complex number

z = r\text{cis}\theta

can be written:

z = re^{i\theta}

Here

r = |z|

is the modulus and

\theta = \arg(z)

is the argument. The exponential form packages both pieces of information into a single expression resembling a power.

The three representations of a complex number are equivalent:

z = a + bi = r(\cos\theta + i\sin\theta) = r\text{cis}\theta = re^{i\theta}

Algebraic form displays the real and imaginary components directly. Trigonometric form emphasizes modulus and argument through explicit trigonometric functions. Exponential form achieves the same emphasis with more economical notation.

The exponential form excels in computation. Standard rules for exponents — which students learn long before complex numbers — apply directly. Multiplying

e^{i\theta_1}

e^{i\theta_2}

yields

e^{i(\theta_1 + \theta_2)}

. Dividing subtracts exponents. Raising to a power multiplies the exponent. These rules, familiar from real exponentials, now govern complex arithmetic.

Every nonzero complex number has an exponential form. The number

3 + 4i

has modulus

5

and argument

\arctan(4/3) \approx 53.13°

, so

3 + 4i = 5e^{i \cdot 0.927}

(in radians). The number

-2

has modulus

2

and argument

\pi

, so

-2 = 2e^{i\pi}

. Even pure imaginaries fit:

i = 1 \cdot e^{i\pi/2} = e^{i\pi/2}

Converting Between Forms

Moving between algebraic, trigonometric, and exponential forms requires computing modulus and argument or recovering real and imaginary parts.

Algebraic to Exponential:

Given

z = a + bi

, find

r

and

\theta

r = \sqrt{a^2 + b^2}

\theta = \begin{cases} \arctan(b/a) & \text{if } a > 0 \\ \arctan(b/a) + \pi & \text{if } a < 0, b \geq 0 \\ \arctan(b/a) - \pi & \text{if } a < 0, b < 0 \\ \pi/2 & \text{if } a = 0, b > 0 \\ -\pi/2 & \text{if } a = 0, b < 0 \end{cases}

Then write

z = re^{i\theta}

.

Example: Convert

z = 1 + i

. The modulus is

r = \sqrt{1 + 1} = \sqrt{2}

. With

a = 1 > 0

and

b = 1 > 0

, the argument is

\theta = \arctan(1) = \pi/4

. Thus

1 + i = \sqrt{2}e^{i\pi/4}

.

Exponential to Algebraic:

Given

z = re^{i\theta}

, compute:

a = r\cos\theta \qquad b = r\sin\theta

Then write

z = a + bi

.

Example: Convert

z = 4e^{i(2\pi/3)}

. The real part is

a = 4\cos(2\pi/3) = 4(-1/2) = -2

. The imaginary part is

b = 4\sin(2\pi/3) = 4(\sqrt{3}/2) = 2\sqrt{3}

. Thus

4e^{i(2\pi/3)} = -2 + 2\sqrt{3}i

.

Both conversions are routine once the formulas are memorized. The exponential form serves computation; the algebraic form displays components explicitly.

Multiplication in Exponential Form

Multiplying complex numbers in exponential form reduces to elementary exponent rules. Given

z_1 = r_1e^{i\theta_1}

and

z_2 = r_2e^{i\theta_2}

z_1 \cdot z_2 = r_1e^{i\theta_1} \cdot r_2e^{i\theta_2} = r_1r_2 \cdot e^{i\theta_1} \cdot e^{i\theta_2} = r_1r_2 \cdot e^{i(\theta_1 + \theta_2)}

The moduli multiply:

|z_1 \cdot z_2| = r_1r_2

. The arguments add:

\arg(z_1 \cdot z_2) = \theta_1 + \theta_2

.

This rule follows from the standard law of exponents:

e^a \cdot e^b = e^{a+b}

. The complex setting changes nothing about how exponents combine.

Geometrically, multiplication scales and rotates. Multiplying by

z_2 = r_2e^{i\theta_2}

stretches lengths by factor

r_2

and rotates counterclockwise by angle

\theta_2

. Multiplying by

e^{i\pi/2} = i

rotates by

90°

without changing length. Multiplying by

2

doubles length without rotation. Multiplying by

2e^{i\pi/4}

doubles length and rotates

45°

.

Example: Compute

(3e^{i\pi/6})(2e^{i\pi/3})

.

Multiply moduli:

3 \cdot 2 = 6

.
Add arguments:

\pi/6 + \pi/3 = \pi/6 + 2\pi/6 = 3\pi/6 = \pi/2

.

Result:

6e^{i\pi/2} = 6i

.

Compare this to multiplying the equivalent algebraic forms and simplifying — the exponential approach is faster and less error-prone.

Division in Exponential Form

Division mirrors multiplication with subtraction replacing addition. Given

z_1 = r_1e^{i\theta_1}

and

z_2 = r_2e^{i\theta_2}

with

z_2 \neq 0

\frac{z_1}{z_2} = \frac{r_1e^{i\theta_1}}{r_2e^{i\theta_2}} = \frac{r_1}{r_2} \cdot \frac{e^{i\theta_1}}{e^{i\theta_2}} = \frac{r_1}{r_2} \cdot e^{i(\theta_1 - \theta_2)}

The moduli divide:

|z_1/z_2| = r_1/r_2

. The arguments subtract:

\arg(z_1/z_2) = \theta_1 - \theta_2

.

The exponent law

e^a / e^b = e^{a-b}

governs the calculation, just as with real exponentials.

Geometrically, dividing by

z_2

shrinks by factor

1/r_2

and rotates clockwise by angle

\theta_2

(equivalently, counterclockwise by

-\theta_2

). Division undoes multiplication: if

w = z_1 \cdot z_2

, then

z_1 = w/z_2

recovers the original factor.

Example: Compute

\frac{8e^{i(5\pi/6)}}{2e^{i(\pi/3)}}

.

Divide moduli:

8/2 = 4

.
Subtract arguments:

5\pi/6 - \pi/3 = 5\pi/6 - 2\pi/6 = 3\pi/6 = \pi/2

.

Result:

4e^{i\pi/2} = 4i

.

In algebraic form, this division would require multiplying by the conjugate of the denominator and simplifying — a lengthier process. Exponential form bypasses that machinery entirely.

Powers in Exponential Form

Raising a complex number to a power becomes trivial in exponential form. For

z = re^{i\theta}

and any integer

n

z^n = (re^{i\theta})^n = r^n \cdot (e^{i\theta})^n = r^n e^{in\theta}

The modulus raises to the

n

-th power. The argument multiplies by

n

. Both operations are elementary.

This is De Moivre's theorem expressed in exponential notation. The trigonometric statement

(\text{cis}\theta)^n = \text{cis}(n\theta)

becomes the exponential statement

(e^{i\theta})^n = e^{in\theta}

, a direct consequence of the power rule for exponents.

Example: Compute

(1 + i)^{12}

.

Convert to exponential form:

1 + i = \sqrt{2}e^{i\pi/4}

.

Apply the power rule:

(\sqrt{2}e^{i\pi/4})^{12} = (\sqrt{2})^{12} \cdot e^{i \cdot 12 \cdot \pi/4} = 2^6 \cdot e^{i \cdot 3\pi} = 64e^{i \cdot 3\pi}

.

Simplify the argument:

3\pi = 2\pi + \pi

, so

e^{i \cdot 3\pi} = e^{i\pi} = -1

.

Result:

64 \cdot (-1) = -64

.

The computation required no binomial expansion, no tracking of

i

powers, no collection of terms. Converting to exponential form, applying the power rule, and simplifying completed the calculation in a few lines.

Roots in Exponential Form

Finding

n

-th roots inverts the power operation. Given

w = Re^{i\phi}

, the solutions to

z^n = w

are:

z_k = R^{1/n} e^{i(\phi + 2\pi k)/n} \quad \text{for } k = 0, 1, 2, \ldots, n-1

Each value of

k

produces a distinct root. Beyond

k = n-1

, the arguments differ by full rotations of

2\pi

, repeating previous roots.

All

n

roots share the same modulus

R^{1/n}

, placing them on a circle of that radius in the complex plane. Their arguments differ by

2\pi/n

, spacing them evenly around the circle. The roots form vertices of a regular

n

-gon.

Example: Find the cube roots of

8

.

Write

8 = 8e^{i \cdot 0}

(modulus

8

, argument

0

).

Apply the formula:

z_k = 8^{1/3} e^{i(0 + 2\pi k)/3} = 2e^{i \cdot 2\pi k/3}

for

k = 0, 1, 2

z_0 = 2e^{i \cdot 0} = 2

z_1 = 2e^{i \cdot 2\pi/3} = 2(-\frac{1}{2} + \frac{\sqrt{3}}{2}i) = -1 + \sqrt{3}i

z_2 = 2e^{i \cdot 4\pi/3} = 2(-\frac{1}{2} - \frac{\sqrt{3}}{2}i) = -1 - \sqrt{3}i

Three roots forming an equilateral triangle of radius

2

, one vertex at

2

on the positive real axis.

The exponential formula makes root extraction systematic: convert to exponential form, apply the root formula, enumerate distinct values of

k

Why Exponential Form Matters

The exponential representation of complex numbers is not merely a notational convenience — it fundamentally simplifies how we think about and compute with complex quantities.

Computational Efficiency: Multiplication becomes addition of exponents. Division becomes subtraction. Powers become multiplication by integers. Roots become division by integers. Every operation reduces to arithmetic on the modulus and argument, bypassing the algebraic expansion and simplification that rectangular form demands.

Conceptual Unification: Exponential form reveals that circular motion and exponential growth are manifestations of the same mathematical structure. A point rotating around the origin traces

e^{i\theta}

\theta

increases — circular motion encoded as an exponential. Conversely, the exponential function, when extended to imaginary inputs, produces rotation rather than growth.

Foundation for Advanced Mathematics: Fourier analysis decomposes signals into sums of complex exponentials

e^{i\omega t}

, each representing a pure oscillation at frequency

\omega

. Signal processing relies on this decomposition to filter, compress, and analyze audio, images, and communications. Quantum mechanics writes wave functions using complex exponentials, encoding the probability amplitudes that govern particle behavior.

The exponential form appears wherever oscillation, rotation, or periodic phenomena arise. Electrical engineers use it to analyze AC circuits. Physicists use it in wave equations. Mathematicians use it throughout complex analysis. Learning to think in exponential form equips you with a tool that extends far beyond the algebra of complex numbers into the mathematical description of the physical world.