Exponential Form: Euler's Formula & Identity

Euler's Formula

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

Euler's Formula

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

Learn more about this formula: Euler's Formula →

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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.

Fact:

i^i

is a real number

Raising both sides of

e^{i\pi/2} = i

to the power

i

gives

i^i = e^{i \cdot i\pi/2} = e^{-\pi/2} \approx 0.2079

. The other angles of

i

,

\frac{\pi}{2} + 2\pi k

, give infinitely many values

e^{-\pi/2 - 2\pi k}

, and every one of them is real.

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.

e^(iθ) at θ = 45°: the point (cos θ, sin θ)

The point e^(iθ) sits on the unit circle at angle θ, with horizontal leg cos θ and vertical leg sin θ drawn in: Euler’s formula is the statement that the exponential of an imaginary number is a point on this circle. At 45° both legs are √2/2. Sweep θ round the circle on the Euler formula visualizer.

The formula turns the trigonometric form into a single exponential, which is what makes it useful.

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:

Understanding Euler's Formula

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

Substituting

x = i\theta

:

Understanding Euler's Formula

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:

Understanding Euler's Formula

= 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:

Understanding Euler's Formula

= \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.

θ	eⁱᶿ via Euler's formula	Cartesian a + bi	Location on unit circle
0	cos 0 + i sin 0	1	positive real axis
π⁄6	cos(π⁄6) + i sin(π⁄6)	√3⁄2 + (1⁄2)i	30° from positive real axis
π⁄4	cos(π⁄4) + i sin(π⁄4)	√2⁄2 + (√2⁄2)i	45° (first quadrant diagonal)
π⁄3	cos(π⁄3) + i sin(π⁄3)	1⁄2 + (√3⁄2)i	60°
π⁄2	cos(π⁄2) + i sin(π⁄2)	i	positive imaginary axis
π	cos π + i sin π	−1	negative real axis (Euler's identity)
3π⁄2	cos(3π⁄2) + i sin(3π⁄2)	−i	negative imaginary axis
2π	cos 2π + i sin 2π	1	back to start; full rotation

Feed an imaginary number into the series for

e^x

and it splits exactly into the series for cosine and sine.

Euler's Identity

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

\theta = \pi

:

Euler's Identity

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

Rearranging:

Euler's Identity

e^{i\pi} + 1 = 0

Learn more about this formula: Euler's Identity →

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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.

Fact: The logarithm of

-1

was once disputed

Leibniz and Johann Bernoulli argued in letters over

\ln(-1)

, and Bernoulli held that it was

0

, like

\ln 1

. Euler settled it with this identity:

e^{i\pi} = -1

makes

\ln(-1) = i\pi

, and adding full turns gives infinitely many values

i\pi + 2\pi k i

.

Mathematicians and physicists routinely cite Euler's identity as the most beautiful formula ever discovered. Richard Feynman called its parent, Euler's formula, "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.

e^(iπ) = −1

At θ = π the point has travelled half way round the unit circle and sits at −1 on the real axis: the sine leg has vanished and the cosine leg points left. That single position is Euler’s identity, e^(iπ) + 1 = 0. Park the angle at π on the Euler formula visualizer.

The identity is Euler’s formula evaluated at the one angle where both legs are integers.

The Exponential Form of a Complex Number

The trigonometric form already describes every complex number by distance and angle.

2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)

is the point at distance

2

and angle

\frac{\pi}{3}

, which is

1 + \sqrt{3}i

. Nothing is wrong with it, so any new way of writing the same number has to earn its place.

Squaring it in trigonometric form takes five steps; written as an exponential it takes one.

The trigonometric form writes the angle twice and keeps cosine and sine apart, so every product has to be expanded and then put back together with trigonometric identities. The exponential form carries the angle once, in an exponent, and the ordinary exponent rules do the work.

step	squaring 2(cos π/3 + i sin π/3)	squaring 2e^(iπ/3)
1	4(cos π/3 + i sin π/3)²	4e^(2iπ/3) — done
2	4(cos² π/3 + 2i sin π/3 cos π/3 − sin² π/3)
3	4((cos² π/3 − sin² π/3) + i · 2 sin π/3 cos π/3)
4	recognise two double-angle identities
5	4(cos 2π/3 + i sin 2π/3)

It is the difference between writing

2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

and writing

2^5

: the same number, but one of them can be worked with directly.

The definition

Writing a complex number as

z = re^{i\theta}

, with

r

its modulus and

\theta

its argument in radians, is its exponential form.

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:

Exponential Form

z = re^{i\theta}

Learn more about this formula: Exponential Form →

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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.

So

1 + \sqrt{3}i = 2e^{i\pi/3}

.

It is the same number as before, in a third notation.

The three representations of a complex number are equivalent:

The Exponential Form of a Complex Number

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.

RULE

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}

.

z = 2e^(iπ/3)

The exponential form reads as directions for reaching z: turn to the angle θ = π/3, then go out a distance r = 2. The factor e^(iθ) supplies the direction on the unit circle and r stretches it, so the point lands on the circle of radius 2, with coordinates r cos θ = 1 and r sin θ = √3 — the algebraic form 1 + √3 i. Change r and θ on the Euler's formula visualizer.

With r = 1 the same point sits on the unit circle: e^(iθ) alone is a pure direction.

Multiplication, powers, reciprocals and division then follow without being asked for.

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

e^{i\theta_1}

by

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.

with z = 2e^(iπ/3)	exponent rule	result
z · z	add the exponents	4e^(2iπ/3) = −2 + 2√3 i
z³	multiply the exponent by 3	8e^(iπ) = −8
1 / z	negate the exponent	(1/2)e^(−iπ/3)
z / z	subtract the exponents	e⁰ = 1

All of this has one cause — Euler's formula makes e^(iθ) obey the exponent rules — and it has a price.

The exponent rules only work with the angle in radians, because Euler's formula is stated in radians. And the form helps only with multiplying: adding two numbers in exponential form still means going back to

a + bi

, since there is no rule for

e^{i\alpha} + e^{i\beta}

. Euler's formula itself is at the top of the page.

Pitfall: Putting degrees in the exponent

Writing

3 + 4i = 5e^{i \cdot 53.13}

feeds degrees into a formula that reads radians:

5e^{i \cdot 53.13} \approx -4.81 + 1.37i

, a different point. With the angle in radians,

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

.

The rest of the page covers the notation, converting between the forms, and multiplication, division, powers and roots in exponential form.

Exponential Form Notation

Notation

Exponential Form Notation

One mark does the work here: the imaginary exponent. This section owns its reading, the full form that carries a modulus in front, and the periodicity that makes exponents non-unique — the price of importing power rules into rotation.

z

,

a + bi

and

\mathbb{C}

come from complex numbers basics,

i

from the imaginary unit,

r

,

\theta

,

\arg

and

\operatorname{cis}

from the trigonometric form; the letter

e

itself is a reserved constant.

e^{i\theta}

e to the i theta

A power with an imaginary exponent — defined, not computed: Euler's Formula above assigns it the value

\cos\theta + i\sin\theta

, and the notation is chosen precisely so the exponent rules keep working.

CasesAlways modulus

1

:

e^{i\theta}

walks the unit circle,

\theta

the arc walked. The exponent writes

i

first —

e^{i\theta}

— echoing the i-first order used whenever the coefficient is an expression.

Also written

\exp(i\theta)

— the functional spelling of analysis and programming, obligatory once the exponent outgrows a superscript.

Do not confuseA growth process. Real

e^x

grows; an imaginary exponent rotates — larger

\theta

means more turning, never more magnitude.

re^{i\theta}

r e to the i theta

The full form: size out front, direction in the exponent —

r = |z|

from the absolute value,

\theta = \arg(z)

from the trigonometric form. The chain

a + bi = r\operatorname{cis}\theta = re^{i\theta}

of The Exponential Form of a Complex Number above closes the three-form family.

CasesThe sign lives in the exponent, never on

r

:

-2 = 2e^{i\pi}

, not

(-2)e^{i0}

; pure imaginaries drop the coefficient entirely —

i = e^{i\pi/2}

.

Also written

Ae^{i\omega t}

— the physics and engineering costume: amplitude

A

, angular frequency

\omega

, time

t

; the same mark running oscillations and waves.

Do not confuseA merged exponent. Only the

e^{i\theta}

factor rotates; sliding

r

into the exponent —

e^{ri\theta}

or

e^{r + i\theta}

— names a different number.

e^{i(\theta + 2\pi k)}

e to the i, theta plus two pi k

The exponent is determined only up to full turns:

e^{i(\theta + 2\pi)} = e^{i\theta}

. Equal values under visibly different exponents — the multivalued argument wearing exponential clothing.

CasesHarmless in a single computation, load-bearing for roots: Roots in Exponential Form below finds all

n

answers precisely by writing

z = re^{i(\theta + 2\pi k)}

before dividing the exponent by

n

.

Do not confuseReal-exponential injectivity. For real exponents,

e^{x_1} = e^{x_2}

forces

x_1 = x_2

; with imaginary exponents that cancellation is invalid — silently "equating exponents" is how root solutions get lost.

All complex number symbols →↑ Complex Numbers

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Notation · Cheat sheet

Complex Number Notation at a Glance

The marks from the notation cards across the complex-number pages: how each is read and what it means. Every group links to the full card on its page.

Basics

z, \; w

z, w: complex numbers

The usual letters for complex numbers, as

x

and

y

are for reals.

a + bi

a plus b i

Real part

a

, imaginary part

b

; both are real numbers.

\mathbb{C}

C, the complex numbers

The set of all

a + bi

; it contains

\mathbb{R}

.

Imaginary numbers

i

i, the imaginary unit

The number with

i^2 = -1

.

\sqrt{-1}

the square root of minus one

Another name for

i

; root rules for reals fail with it.

bi

b i

A pure imaginary number: real part

0

.

Algebraic form

\operatorname{Re}(z), \; \operatorname{Im}(z)

real part, imaginary part

For

z = a + bi

:

a

and

b

.

\operatorname{Im}(z)

is

b

, not

bi

.

a + (-b)i

a plus negative b, i

Written

a - bi

; the imaginary part is

-b

.

z = x + yi

z equals x plus y i

Same form with plane letters: the point

(x, y)

.

Complex conjugate

\bar{z}

z bar, the conjugate

For

z = a + bi

,

\bar{z} = a - bi

: the mirror image across the real axis.

\overline{z_1 + z_2}

the conjugate of the sum

The bar covers the whole expression; it equals

\bar{z}_1 + \bar{z}_2

.

z^*

z star

The same conjugate, as written in physics and engineering.

Absolute value

|z|

the modulus of z

Distance from

0

:

\sqrt{a^2 + b^2}

.

|z|^2 = z\bar{z}

mod z squared

Equals

a^2 + b^2

, with no root to take.

|z - c| = R

mod z minus c equals R

The circle of radius

R

around

c

.

Additive inverse

-z

negative z

Both parts change sign:

-a - bi

.

-(a + bi)

minus, quantity a plus b i

The brackets carry the minus to both parts.

z - w

z minus w

Adding the inverse:

z + (-w)

.

Multiplicative inverse

z^{-1}, \; \frac{1}{z}

z inverse; one over z

The number with

z \cdot z^{-1} = 1

; it equals

\frac{\bar{z}}{|z|^2}

.

(3 + 2i)^{-1}

the inverse of three plus two i

Found by multiplying top and bottom by the conjugate:

\frac{3 - 2i}{13}

.

Trigonometric form

r, \; \theta

r and theta, the polar pair

Distance from

0

and angle from the positive real axis.

\arg(z)

the argument of z

Any angle of

z

; there are infinitely many,

2\pi

apart.

\operatorname{Arg}(z)

the principal argument

One chosen angle, usually in

(-\pi, \pi]

; some texts use

[0, 2\pi)

.

\operatorname{cis}\theta

sis theta

Short for

\cos\theta + i\sin\theta

.

Exponential formThis page

e^{i\theta}

e to the i theta

Equals

\operatorname{cis}\theta

(Euler's formula): a point on the unit circle.

re^{i\theta}

r e to the i theta

The whole number

z

: length

r

, angle

\theta

.

e^{i(\theta + 2\pi k)}

theta plus two pi k

The same number for every integer

k

.

De Moivre's theorem

z_k

z sub k, the k-th root

The roots of

z^n = w

, numbered

k = 0, \ldots, n - 1

.

\omega

omega

A primitive root of unity: its powers give all

n

roots of

1

.

R^{1/n}

R to the one over n

The real root of the length; the complex root

w^{1/n}

has

n

values.

Same marks, different job

|z|

The modulus here; for a real number, the absolute value; for a matrix, the determinant.

\bar{z}

The conjugate here; in statistics a bar means a mean, in sets a complement.

z^{-1}

vs

z^*

The reciprocal, against the conjugate: they agree only on the unit circle.

w^{1/n}

n

values for a complex

w

; a single real root for a positive real number.

Complex number symbols table Complex numbers

Complex Number Notation at a Glance·/complex-numbers/basicsLearn Math Class

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

:

Converting Between Forms

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

Converting Between Forms

\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}

.

Pitfall: Ignoring the quadrant

For

z = -1 - i

, taking

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

gives

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

, the opposite point. Since

a < 0

and

b < 0

,

\theta = \pi/4 - \pi = -3\pi/4

, and

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

.

In this example both parts are positive, so the point lies in the first quadrant and the first line of the case formula applies. The remaining lines handle points in the second and third quadrants and on the imaginary axis, adding or subtracting

\pi

or replacing the arctangent outright. The modulus formula needs no such cases: it is the same in every quadrant.

Exponential to Algebraic:

Given

z = re^{i\theta}

, compute:

Polar to Rectangular Conversion

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

Learn more about this formula: Polar to Rectangular Conversion →

Browse all complex numbers formulas →

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.

Conversion	Formulas	Worked example
Algebraic → Exponential (a + bi → reⁱᶿ)	r = √(a² + b²) θ = arctan(b ⁄ a), adjusted for quadrant z = reⁱᶿ	z = 1 + i r = √(1² + 1²) = √2 θ = arctan(1) = π⁄4 (first quadrant) → 1 + i = √2 · eⁱ⁽π/4⁾
Exponential → Algebraic (reⁱᶿ → a + bi)	a = r · cos θ b = r · sin θ z = a + bi	z = 4eⁱ⁽²π/3⁾ a = 4 · cos(2π⁄3) = 4 · (−1⁄2) = −2 b = 4 · sin(2π⁄3) = 4 · (√3⁄2) = 2√3 → 4eⁱ⁽²π/3⁾ = −2 + 2√3 i

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}

:

Multiplication in Exponential Form

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.

Pitfall: Adding the moduli

Adding the moduli along with the arguments turns

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

into

5e^{i\pi/2} = 5i

. Only the arguments add; the moduli multiply, so the product is

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

.

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.

z₁z₂: arguments add, moduli multiply

The product sits at the angle θ₁ + θ₂ marked in the picture and at distance |z₁||z₂| from the origin: multiplying by a complex number rotates by its argument and scales by its modulus. Nothing in the algebraic expansion shows this; the picture does. Turn one factor and watch the product swing round on the complex multiplication tool.

Adding exponents and adding angles are the same operation.

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

:

Division in Exponential Form

\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.

Pitfall: Subtracting in the wrong order

Computing

\frac{6e^{i\pi/2}}{3e^{i\pi/6}}

with

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

gives

2e^{-i\pi/3} = 1 - \sqrt{3}i

. The denominator's argument is the one subtracted:

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

, so the quotient is

2e^{i\pi/3} = 1 + \sqrt{3}i

.

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.

Dividing subtracts the angles

z₁ = 4 + 2i divided by z₂ = 1 − i. In exponential form the quotient is (r₁/r₂) e^(i(θ₁ − θ₂)): its angle is θ₁ − θ₂, marked on the figure, and its length is √20/√2 = √10. The result is 1 + 3i. Try other pairs on the complex division visualizer.

Here θ₂ is negative, so subtracting it turns the quotient further counterclockwise than z₁.

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

:

Power in Polar Form

z^n = r^n e^{in\theta}

Learn more about this formula: Power in Polar Form →

Browse all complex numbers formulas →

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

.

Pitfall: Leaving the modulus unpowered

Multiplying only the angle by

12

turns

(\sqrt{2}e^{i\pi/4})^{12}

into

\sqrt{2}e^{i \cdot 3\pi} = -\sqrt{2}

. The modulus is raised too,

(\sqrt{2})^{12} = 64

, so the result is

64e^{i \cdot 3\pi} = -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.

z and z²: (re^(iθ))² = r²e^(2iθ)

Squaring doubles the angle and squares the length, exactly what the exponent rule (e^(iθ))² = e^(2iθ) predicts; the picture and the exponential form say the same thing in two languages. Raise the power and watch the exponent rule turn into a spiral on the De Moivre visualizer.

Roots divide the exponent, and therefore the angle, in the same way.

Roots in Exponential Form

Finding

n

-th roots inverts the power operation. Given

w = Re^{i\phi}

, the solutions to

z^n = w

are:

Roots in Exponential Form

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.

Pitfall: Adding the full turn after dividing

Writing the square roots of

-9 = 9e^{i\pi}

as

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

gives

3i

for every

k

, so

-3i

is lost. The full turn is divided too:

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

gives

3i

for

k = 0

and

-3i

for

k = 1

.

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

.

k	Argument formula (φ + 2πk) ⁄ n	Argument	z_k exponential	z_k algebraic
0	(0 + 0) ⁄ 3	0	2 eⁱ⁽⁰⁾	2
1	(0 + 2π) ⁄ 3	2π⁄3	2 eⁱ⁽²π/3⁾	−1 + √3 i
2	(0 + 4π) ⁄ 3	4π⁄3	2 eⁱ⁽⁴π/3⁾	−1 − √3 i

The full turns hidden in a number's angle are where its

n

different

n

-th roots come from.

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}

as

\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.

Summary: Operations in Algebraic vs Exponential Form

The advantage of exponential form lies in computational simplicity — every operation that requires careful expansion in algebraic form reduces to elementary exponent arithmetic in exponential form. The table below collects each operation in both representations side by side, making the contrast explicit: where the algebraic column needs distribution, conjugates, or term-collection, the exponential column performs the same work via addition, subtraction, or multiplication of the argument

\theta

.

Operation on z, w	Algebraic form z = a + bi, w = c + di	Exponential form z = r₁eⁱ⁽ᶿ¹⁾, w = r₂eⁱ⁽ᶿ²⁾
Multiplication z · w	(ac − bd) + (ad + bc)i (FOIL + apply i² = −1)	r₁r₂ · eⁱ⁽ᶿ¹ ⁺ ᶿ²⁾ (moduli multiply, args add)
Division z ⁄ w	multiply num and denom by w̄, then expand	(r₁ ⁄ r₂) · eⁱ⁽ᶿ¹ ⁻ ᶿ²⁾ (moduli divide, args subtract)
Power zⁿ (integer n)	binomial expansion + repeated i²-collapse	r₁ⁿ · eⁱ⁽ⁿᶿ¹⁾ (modulus to nth power, arg × n; De Moivre)
n-th root (zⁿ = w)	no clean closed-form method	r₂¹/ⁿ · eⁱ⁽ᶿ² ⁺ ²πᵏ⁾/ⁿ for k = 0, 1, …, n − 1
Modulus \|z\|	√(a² + b²)	r₁ (read directly)
Argument arg(z)	arctan(b ⁄ a) with quadrant adjustment	θ₁ (read directly)
Conjugate z̄	a − bi	r₁ · e⁻ⁱᶿ¹ (modulus same, argument negates)

Common Pitfalls

Six mistakes account for many of the wrong answers in work with the exponential form. Each is marked where it happens in the sections above; here each is worked through in full, with the wrong and the right working side by side.

Putting degrees in the exponent

The argument of a number is found in degrees, and the degree value is written straight into the exponent. Euler's formula comes from Taylor series in which

\theta

is measured in radians, so

e^{i\theta}

reads any number in the exponent as radians.

Wrong working: the argument of

3 + 4i

is

\arctan(4/3) \approx 53.13°

, so

3 + 4i = 5e^{i \cdot 53.13}

. Read in radians,

5e^{i \cdot 53.13} = 5(\cos 53.13 + i\sin 53.13) \approx -4.81 + 1.37i

.

Right working: convert first,

53.13° \cdot \frac{\pi}{180°} \approx 0.927

, so

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

. Check:

5(\cos 0.927 + i\sin 0.927) \approx 3 + 4i

.

Why it happens: the trigonometric form is often written in degrees, as in

\text{cis}(45°)

, and calculators are often left in degree mode. The habit carries over to the exponent.

How to check: convert back with

a = r\cos\theta

and

b = r\sin\theta

in radian mode; the result must be the original number. An exponent larger than

2\pi \approx 6.28

in a plain conversion is a warning sign.

Ignoring the quadrant

The argument is taken as

\arctan(b/a)

with no adjustment. The ratio

b/a

is the same for

z

and

-z

, and the arctangent only returns angles between

-\pi/2

and

\pi/2

, so a point with

a < 0

is given the angle of the opposite point.

Wrong working: for

z = -1 - i

,

\theta = \arctan\frac{-1}{-1} = \arctan(1) = \pi/4

, so

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

. But

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

.

Right working:

r = \sqrt{2}

. Since

a < 0

and

b < 0

, the third line of the case formula applies:

\theta = \arctan(1) - \pi = -3\pi/4

. So

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

, and indeed

\sqrt{2}\left(\cos(-3\pi/4) + i\sin(-3\pi/4)\right) = \sqrt{2}\left(-\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i\right) = -1 - i

.

Why it happens: in first-quadrant examples such as

1 + i

the plain arctangent is correct, and the case formula looks like fine print.

How to check: sketch the point.

-1 - i

lies in the third quadrant, so its argument must lie between

-\pi

and

-\pi/2

;

\pi/4

points into the first quadrant.

Adding the moduli

When two numbers are multiplied, the arguments add, and the moduli are added as well instead of multiplied.

Wrong working:

(2e^{i\pi/4})(3e^{i\pi/4}) = (2 + 3)e^{i(\pi/4 + \pi/4)} = 5e^{i\pi/2} = 5i

.

Right working: the moduli multiply,

2 \cdot 3 = 6

, and the arguments add,

\pi/4 + \pi/4 = \pi/2

. The product is

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

.

Why it happens: the arguments sit in the exponent, where multiplication turns into addition; the moduli are ordinary factors in front, where multiplication stays multiplication. The two parts look alike and get the same treatment.

How to check: try a real case. With both arguments

0

,

(2e^{i \cdot 0})(3e^{i \cdot 0}) = 2 \cdot 3 = 6

, not

5

.

Subtracting in the wrong order

In a quotient the arguments are subtracted the wrong way round, the numerator's argument from the denominator's.

Wrong working:

\frac{6e^{i\pi/2}}{3e^{i\pi/6}}

: moduli

6/3 = 2

, arguments

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

, so the quotient is

2e^{-i\pi/3} = 1 - \sqrt{3}i

.

Right working: the argument of the quotient is

\theta_1 - \theta_2 = \pi/2 - \pi/6 = \pi/3

, so the quotient is

2e^{i\pi/3} = 1 + \sqrt{3}i

.

Why it happens: subtraction is the one step where order matters, and the wrong answer differs only in the sign of the angle, so it looks just as plausible.

How to check: multiply the answer by the denominator; it must give the numerator back.

2e^{i\pi/3} \cdot 3e^{i\pi/6} = 6e^{i\pi/2}

, while

2e^{-i\pi/3} \cdot 3e^{i\pi/6} = 6e^{-i\pi/6}

.

Leaving the modulus unpowered

In a power the angle is multiplied by

n

, but the modulus is left as it was.

Wrong working:

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

.

Right working:

(\sqrt{2}e^{i\pi/4})^{12} = (\sqrt{2})^{12} \cdot e^{i \cdot 3\pi} = 64 \cdot (-1) = -64

.

Why it happens: in De Moivre's theorem for the unit circle,

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

, there is no modulus at all, and the angle step is the memorable part.

How to check:

|z^n| = |z|^n

. Here

|1 + i| = \sqrt{2}

, so

|(1 + i)^{12}|

must be

(\sqrt{2})^{12} = 64

. A second route agrees:

(1 + i)^2 = 2i

, so

(1 + i)^{12} = (2i)^6 = 64i^6 = -64

.

Adding the full turn after dividing

In the root formula, the argument

\phi

is divided by

n

and the full turn

2\pi k

is added afterwards, instead of dividing

\phi + 2\pi k

as a whole. The full turns then survive undivided, and every

k

gives the same point.

Wrong working:

z^2 = -9

with

-9 = 9e^{i\pi}

. Then

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

:

k = 0

gives

3e^{i\pi/2} = 3i

, and

k = 1

gives

3e^{i \cdot 5\pi/2} = 3i

again. One root is found twice.

Right working:

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

. For

k = 0

,

3e^{i\pi/2} = 3i

; for

k = 1

,

3e^{i \cdot 3\pi/2} = -3i

. Both satisfy

(\pm 3i)^2 = -9

.

Why it happens: adding

2\pi k

at the end looks like the familiar remark that an argument repeats every full turn, which is true for a number but not for its roots.

How to check: the

n

roots must be spaced

2\pi/n

apart around the circle. If two values of

k

give the same point, the

2\pi k

was not divided by

n

.

Exponential Form FAQ

Why does Euler's formula work?

+

Because the power series for the exponential, split into its even and odd terms, produces exactly the series for cosine and for sine once i is substituted. Nothing is assumed: the identity falls out of expansions already known for real numbers. Differentiating both sides gives a second, shorter argument.Read more →

What is the relationship between cis and the exponential form?

+

They are the same thing written differently, since cis theta and e to the i theta both denote cosine plus i sine. The exponential spelling is preferred because it obeys the ordinary rules of exponents, making the multiplication and power laws automatic rather than facts to memorise separately.Read more →

Is $i^i$ a real number?

+

Yes. Since

i = e^{i\pi/2}

, raising it to the power

i

gives

i^i = e^{i \cdot i\pi/2} = e^{-\pi/2} \approx 0.2079

. The value is not unique:

i

also equals

e^{i(\pi/2 + 2\pi k)}

for every integer

k

, and these angles give infinitely many values

e^{-\pi/2 - 2\pi k}

. Every one of them is a positive real number;

e^{-\pi/2}

is the one that comes from the principal argument.Read more →

What is $\ln(-1)$ ?

+

In the complex numbers it is

i\pi

, because Euler's identity says

e^{i\pi} = -1

. Adding full turns gives every other value,

i\pi + 2\pi k i

for any integer

k

, so the complex logarithm has infinitely many values. Among the real numbers

\ln(-1)

stays undefined, since

e^x

is positive for every real

x

. Leibniz and Johann Bernoulli argued over it in letters, and Bernoulli held that it was

0

, like

\ln 1

.Read more →

Can the angle in $e^{i\theta}$ be in degrees?

+

No, it must be in radians, because Euler's formula comes from Taylor series in which

\theta

is measured in radians. Writing

3 + 4i = 5e^{i \cdot 53.13}

with the angle in degrees gives a different point, about

-4.81 + 1.37i

. Converting first,

53.13° \cdot \frac{\pi}{180°} \approx 0.927

, gives

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

. Degrees are only safe where the unit is written down, as in

\text{cis}\,53.13°

.Read more →

How do you add two complex numbers in exponential form?

+

By converting them back to

a + bi

first. The exponent rules cover products, quotients and powers, but there is no rule for

e^{i\alpha} + e^{i\beta}

, so a sum is worked out across and up and then turned back into

re^{i\theta}

if needed. For example,

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

. This is why the algebraic form stays in use beside the exponential one.Read more →

Exponential Form

The Elegant Union of Exponentials and Circles

Key Terms

Euler's Formula

Understanding Euler's Formula

Euler's Identity

The Exponential Form of a Complex Number

Exponential Form Notation

Exponential Form Notation

Converting Between Forms

Multiplication in Exponential Form

Division in Exponential Form

Powers in Exponential Form

Roots in Exponential Form

Why Exponential Form Matters

Summary: Operations in Algebraic vs Exponential Form

Common Pitfalls

Exponential Form FAQ

Why does Euler's formula work?

What is the relationship between cis and the exponential form?

Is $i^i$ a real number?

What is $\ln(-1)$ ?

Can the angle in $e^{i\theta}$ be in degrees?

How do you add two complex numbers in exponential form?

Exponential Form

The Elegant Union of Exponentials and Circles

Key Terms

Euler's Formula

Understanding Euler's Formula

Euler's Identity

The Exponential Form of a Complex Number

Exponential Form Notation

Exponential Form Notation

Converting Between Forms

Multiplication in Exponential Form

Division in Exponential Form

Powers in Exponential Form

Roots in Exponential Form

Why Exponential Form Matters

Summary: Operations in Algebraic vs Exponential Form

Common Pitfalls

Exponential Form FAQ

Why does Euler's formula work?

What is the relationship between cis and the exponential form?

Is iii^iii a real number?

What is ln⁡(−1)\ln(-1)ln(−1)?

Can the angle in eiθe^{i\theta}eiθ be in degrees?

How do you add two complex numbers in exponential form?

Is $i^i$ a real number?

What is $\ln(-1)$ ?

Can the angle in $e^{i\theta}$ be in degrees?