Complex Numbers and the Complex Plane

The imaginary unit was born of embarrassment. Sixteenth-century algebraists solving cubic equations kept passing through square roots of negative numbers on their way to perfectly real answers, and for two hundred years these symbols were tolerated as a bookkeeping trick with no right to exist. What rehabilitated them was not algebra but geometry: the realization, due to Wessel, Argand, and Gauss, that a number $z=x+iy$ is simply a point in a plane, and that the one rule $i^2=-1$ is nothing more mysterious than a quarter-turn performed twice. This single picture converts an algebraic curiosity into the most efficient language physics has for anything that rotates, oscillates, or interferes. Two operations organize everything that follows. Addition is the parallelogram rule — complex numbers add head-to-tail exactly like vectors in the plane. Multiplication is the genuinely new thing, the operation with no counterpart in ordinary vector algebra: it multiplies lengths and adds angles, so multiplying by $z$ is a spiral similarity — a rotation by $\arg z$ together with a scaling by $|z|$ . Writing $z=re^{i\theta}$ makes this transparent, because Euler's formula turns the awkward addition theorems of trigonometry into the single law of exponents $e^{i\alpha}e^{i\beta}=e^{i(\alpha+\beta)}$ . Underneath the geometry sit two structural facts that make $\mathbb{C}$ the natural home of analysis. It is a field — every nonzero number has an inverse, obtained by dividing the conjugate by the squared modulus — yet it admits no order compatible with its arithmetic, so inequalities live only among real quantities like $|z|$ . And it is complete and algebraically closed: every nonconstant polynomial has a root. These are the foundations on which functions of a complex variable, analytic functions, and the whole calculus of residues are built.

Key takeaways

A complex number is $z=x+iy$ with $x,y\in\mathbb{R}$ and $i^2=-1$ ; it is a point in the Argand plane, equivalently a vector from the origin.
Addition is the vector parallelogram rule; multiplication has no vector analogue — it multiplies moduli and adds arguments, a rotation-and-scaling of the plane.
The polar form $z=r(\cos\theta+i\sin\theta)=re^{i\theta}$ with modulus $r=|z|$ and argument $\theta=\arg z$ makes the multiplicative structure obvious.
Conjugation $\bar z=x-iy$ reflects across the real axis and gives $z\bar z=|z|^2$ , hence the inverse $z^{-1}=\bar z/|z|^2$ — this is how a field divides.
Euler's formula $e^{i\theta}=\cos\theta+i\sin\theta$ unifies exponential and trigonometric worlds and yields de Moivre's theorem $(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta$ and the $n$ equally spaced roots of any number.
$\mathbb{C}$ is a complete, algebraically closed field but cannot be ordered; this is why it is the arena for analysis yet forbids inequalities among non-real numbers.
Key idea: multiplying complex numbers multiplies their moduli and adds their arguments, so encoding a number as $z=re^{i\theta}$ turns plane geometry into the algebra of exponents.

The core insight

Multiplication of complex numbers is rotation-and-scaling: $z_1 z_2$ has modulus $|z_1||z_2|$ and argument $\arg z_1+\arg z_2$ . Encoding this as $z=re^{i\theta}$ turns the geometry of the plane into the algebra of exponents.

Where this lives today

The complex plane is not a historical curiosity but the working notation of contemporary physics and engineering. In electrical engineering every steady-state AC network is solved by turning voltages and currents into phasors $V_0e^{i\varphi}$ and impedances into complex numbers, so a differential equation collapses into ordinary algebra in $\mathbb{C}$ — the method behind every power grid and radio circuit. In digital signal processing the fast Fourier transform, the most-used numerical algorithm on Earth, is built directly on the $n$ -th roots of unity: its speed comes from the symmetry of those equally spaced points on the unit circle. In quantum mechanics the state of a system is a complex amplitude whose modulus-squared is a probability and whose argument — the phase — produces every interference effect, from the double slit to the Aharonov-Bohm shift; complex numbers are not a convenience there but structurally essential, as recent experiments testing "real-valued quantum mechanics" have confirmed. And in quantum computing the elementary phase and rotation gates are literally multiplications by unit-modulus complex numbers $e^{i\theta}$ , so that the group $U(1)$ met in this article is the smallest brick of a quantum circuit. Wherever something rotates, oscillates, or interferes, the plane that Wessel drew is doing the bookkeeping.

The physical picture

A complex number is best read as an instruction rather than a quantity: "scale by $r$ , then turn by $\theta$ ." The imaginary unit $i=e^{i\pi/2}$ is the instruction "turn a quarter-circle and do not scale." Apply it twice and you have turned a half-circle, which points every vector backward — precisely the statement $i^2=-1$ . Nothing about $i$ is unreal; it is the most concrete object in the plane, the ninety-degree rotation. The first figure fixes the vocabulary — real part, imaginary part, modulus, argument, and conjugate — on a single concrete point.

Figure

The Argand plane: the point z = 3 + 4i sits 3 units along the real axis and 4 up the imaginary axis, with modulus r = √(3² + 4²) = 5 and argument θ = arctan(4/3) ≈ 53.13°. Its conjugate z̄ = 3 − 4i is the mirror image across the real axis, and z times z̄ = 3² + 4² = 25 = r²

Addition of two complex numbers lives in the flat, translational world: lay the vectors tip-to-tail and the sum is the diagonal of the parallelogram — the very same thing you do with forces or displacements. Multiplication lives in a different, rotational world, and it is what makes $\mathbb{C}$ more than a two-dimensional vector space. When two numbers multiply, their lengths multiply and their angles add. The exponential $e^{i\theta}$ is the dictionary between the two worlds, and the second figure shows the multiplicative law in action with concrete numbers.

Figure

Multiplication is rotation-and-scaling: with z₁ = 1 + i (modulus √2, argument 45°) and z₂ = √3 + i (modulus 2, argument 30°), the product z₁z₂ has modulus √2 · 2 = 2√2 ≈ 2.83 and argument 45° + 30° = 75°, landing at 0.73 + 2.73i

The conjugate $\bar z$ , the reflection across the real axis, is the workhorse of division: the product $z\bar z=|z|^2$ is always a nonnegative real number, so dividing by $z$ is arranged by multiplying top and bottom by $\bar z$ and dividing by the real number $|z|^2$ . And because an angle only means anything modulo a full turn, the argument is irreducibly multivalued — a fact that looks like a nuisance now but is the seed of branch cuts, Riemann surfaces, and the winding numbers that later power contour integration.

The core equations

Everything begins with the definition. A complex number is an ordered pair of reals written as a sum, with the single new rule that the square of the imaginary unit is $-1$ ; all of algebra follows by treating $i$ as an ordinary symbol subject to that one relation.

z=x+iy,\qquad i^2=-1,\qquad x,y\in\mathbb{R} \qquad (1)

where: $x=\operatorname{Re}z$ — real part; $y=\operatorname{Im}z$ — imaginary part; $i$ — imaginary unit. Reading: the real numbers sit inside $\mathbb{C}$ as the points with $y=0$ , so $\mathbb{C}$ extends $\mathbb{R}$ without discarding any of it. Multiplication is forced by distributivity together with $i^2=-1$ ; there is no freedom in it. Expanding the product of two numbers and collecting real and imaginary parts gives the rule that every other multiplicative fact rests on.

(x_1+iy_1)(x_2+iy_2)=(x_1x_2-y_1y_2)+i(x_1y_2+x_2y_1) \qquad (2)

where: $x_k,y_k$ — real and imaginary parts of $z_k$ . Reading: unlike vector dot or cross products, this multiplication keeps you inside the plane and is commutative, associative, and invertible — the hallmarks of a field. Conjugation reflects a number across the real axis, and its product with the original is the squared length. This is the identity that lets a field with two-dimensional elements nonetheless divide.

\bar z=x-iy,\qquad z\bar z=x^2+y^2=|z|^2,\qquad z^{-1}=\frac{\bar z}{|z|^2}\ \ (z\neq 0) \qquad (3)

where: $\bar z$ — complex conjugate; $|z|=\sqrt{x^2+y^2}$ — modulus; $z^{-1}$ — multiplicative inverse. Reading: to divide by $z$ , multiply by $\bar z$ and rescale by the real number $|z|^2$ — the complex analogue of rationalizing a denominator. Cartesian form hides the multiplicative structure; polar form reveals it. Every nonzero number is a length times a direction, and Euler's formula packages that direction as an exponential.

z=r(\cos\theta+i\sin\theta)=re^{i\theta},\qquad e^{i\theta}=\cos\theta+i\sin\theta \qquad (4)

where: $r=|z|\ge 0$ — modulus; $\theta=\arg z$ — argument, defined only modulo $2\pi$ . Reading: the exponential converts a sum of angles into a product of numbers, which is exactly why it linearizes rotation. With the exponential in hand the product law and de Moivre's theorem become one-line consequences of the exponent rule $e^{a}e^{b}=e^{a+b}$ .

z_1z_2=r_1r_2\,e^{i(\theta_1+\theta_2)},\qquad (\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta \qquad (5)

where: $r_k,\theta_k$ — modulus and argument of $z_k$ ; $n$ — integer power. Reading: multiplying rotates and scales; raising to a power multiplies the angle, which is how trigonometric multiple-angle identities fall out for free. Two further relations complete the toolkit: the extraction of roots, where multivaluedness first pays off by producing several answers, and the triangle inequality, the metric backbone of all later analysis.

z^{1/n}=r^{1/n}e^{i(\theta+2\pi k)/n}\ (k=0,\dots,n-1),\qquad \big||z_1|-|z_2|\big|\le|z_1+z_2|\le|z_1|+|z_2| \qquad (6)

where: $k$ — index labelling the $n$ distinct roots. Reading: a nonzero number has exactly $n$ $n$ -th roots, spaced $2\pi/n$ apart on a circle; and the length of a sum is caught between the sum and difference of the lengths.

Derivation AEuler's formula and de Moivre's theorem

Euler's formula is the hinge on which this whole subject turns: it identifies the exponential of an imaginary angle with a point on the unit circle, and from that one identity de Moivre's theorem, the trigonometric addition formulae, and the entire multiplicative geometry of the plane follow as immediate corollaries. The goal here is to build it from the single tool that survives the passage from real to complex argument — the absolutely convergent power series of the exponential — then separate the even and odd powers and recognize the cosine and sine series hiding inside, and finally read off de Moivre by raising both sides to an integer power.

Step 1. Write the exponential as its power series $e^{i\theta}=\sum_{n\ge 0}(i\theta)^n/n!$ . This series converges absolutely for every complex argument, so its terms may be reordered without changing the sum. Why: the series is the only definition of the exponential that does not secretly assume a real argument, so it is the safe bridge from $\mathbb{R}$ into $\mathbb{C}$ .

Step 2. The powers of $i$ cycle through $1,i,-1,-i$ with period four. Collecting the even- $n$ terms (which are real) apart from the odd- $n$ terms (which carry a single factor of $i$ ) splits the sum into $\big(1-\tfrac{\theta^2}{2!}+\tfrac{\theta^4}{4!}-\cdots\big)+i\big(\theta-\tfrac{\theta^3}{3!}+\cdots\big)$ . Why: the four-fold cycle of $i$ is exactly the mechanism that sorts one series into a real channel and an imaginary channel.

Step 3. The two bracketed series are precisely the Maclaurin expansions of $\cos\theta$ and $\sin\theta$ , so $e^{i\theta}=\cos\theta+i\sin\theta$ . Why: recognizing the standard series is the entire content of Euler's identity — no geometry is assumed, it is forced by the algebra of the series.

Step 4. Raise both sides to the $n$ -th power. The exponent law gives $(e^{i\theta})^n=e^{in\theta}$ , and reading each side through Euler's formula yields $(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta$ . Why: de Moivre's theorem is just the law of exponents wearing trigonometric clothing; setting $n=2$ or $n=3$ and expanding the left side hands you the double- and triple-angle formulae with no extra work. *As a check, setting $n=2$ and expanding the left side of de Moivre's theorem immediately reproduces the double-angle identity $\cos 2\theta=\cos^2\theta-\sin^2\theta$ .*

Go deeper · Derivation B · 1 minThe n distinct roots of a complex number

Derivation BThe n distinct roots of a complex number

The equation $w^n=z$ is the first place where the multivaluedness of the argument stops being a nuisance and starts doing useful work: instead of one root it produces exactly $n$ , spread evenly around a circle. The goal is to solve $w^n=z$ completely by writing both numbers in polar form, matching moduli and arguments separately, and then counting how many genuinely distinct answers the periodicity of the argument allows before the list starts repeating. The payoff is a formula and a picture — a regular $n$ -gon — in a single stroke.

Step 1. Put both numbers in polar form, $z=re^{i\theta}$ and $w=\rho e^{i\phi}$ , and demand $w^n=z$ , i.e. $\rho^n e^{in\phi}=re^{i\theta}$ , treating $\rho>0$ and real $\phi$ as the unknowns. Why: polar form separates the problem into a length equation and an angle equation, which decouple cleanly.

Step 2. Match the moduli: $\rho^n=r$ forces $\rho=r^{1/n}$ , the ordinary positive real $n$ -th root. Why: moduli are nonnegative reals, where the $n$ -th root is unique, so no ambiguity enters through the length.

Step 3. Match the arguments, but only modulo a full turn: $n\phi=\theta+2\pi k$ for any integer $k$ , hence $\phi=(\theta+2\pi k)/n$ . Why: equality of two directions holds only up to whole revolutions, and this is precisely where the several roots are born.

Step 4. Count the distinct answers. The values $k=0,1,\dots,n-1$ give $n$ different angles; $k=n$ returns to the $k=0$ direction, so the list closes. The roots therefore read $w_k=r^{1/n}e^{i(\theta+2\pi k)/n}$ , and successive roots differ in argument by $2\pi/n$ , placing them at the vertices of a regular $n$ -gon on the circle of radius $r^{1/n}$ . Why: the fixed angular step $2\pi/n$ is what turns an algebra problem into a statement of symmetry. *As a check, setting $z=1$ gives the $n$ -th roots of unity $e^{2\pi ik/n}$ , whose sum vanishes for $n\ge 2$ because it is a finite geometric series with ratio $e^{2\pi i/n}\neq 1$ .*

Go deeper · Derivation C · 1 minThe triangle inequality from the modulus

Derivation CThe triangle inequality from the modulus

The triangle inequality is the statement that a straight path is the shortest, transcribed into complex arithmetic, and it is the single fact that makes $|z_1-z_2|$ a genuine distance and $\mathbb{C}$ a metric space on which limits, continuity, and convergence make sense. The goal is to prove it purely algebraically — no diagrams — by expanding the squared modulus of a sum with the identity $|w|^2=w\bar w$ , bounding the one cross term that appears by the elementary fact that a real part never exceeds a modulus, and then taking a square root.

Step 1. Expand the squared modulus of the sum: $|z_1+z_2|^2=(z_1+z_2)\overline{(z_1+z_2)}=|z_1|^2+|z_2|^2+2\operatorname{Re}(z_1\bar z_2)$ , using $z_1\bar z_2+\bar z_1 z_2=2\operatorname{Re}(z_1\bar z_2)$ . Why: writing length-squared as $w\bar w$ converts a geometric quantity into an algebraic one that multiplies out cleanly.

Step 2. Bound the cross term. For any complex number its real part cannot exceed its modulus, so $\operatorname{Re}(z_1\bar z_2)\le|z_1\bar z_2|=|z_1||z_2|$ , where $|\bar z_2|=|z_2|$ . Why: this one inequality, $\operatorname{Re}w\le|w|$ , is the entire source of the bound.

Step 3. Combine and take roots: $|z_1+z_2|^2\le|z_1|^2+2|z_1||z_2|+|z_2|^2=(|z_1|+|z_2|)^2$ , and both sides are nonnegative, so $|z_1+z_2|\le|z_1|+|z_2|$ . Replacing $z_2$ by $-z_2$ and relabelling gives the lower bound $\big||z_1|-|z_2|\big|\le|z_1+z_2|$ . Why: the square-root step is legitimate only because moduli are nonnegative, the same property that blocks an order on $\mathbb{C}$ yet grants it a metric. *Equality holds if and only if $z_1$ and $z_2$ share the same argument — that is, they point in the same direction from the origin (are collinear and co-directed).*

Limiting cases

Special positions in the plane strip the algebra down to its geometric meaning:

Case	Result	Meaning
Pure real	$z=x,\ y=0$	on the real axis; $\bar z=z$ , $\arg z=0$ or $\pi$ . The copy of $\mathbb{R}$ inside $\mathbb{C}$ .
Pure imaginary	$z=iy,\ x=0$	on the imaginary axis; $\bar z=-z$ , $\arg z=\pm\pi/2$ . Multiplying by $i$ is a $90^\circ$ turn.
Unit modulus	$\|z\|=1,\ z=e^{i\theta}$	on the unit circle; multiplication is pure rotation. The circle is the group $U(1)$ .
Roots of unity	$z_k=e^{2\pi ik/n}$	vertices of a regular $n$ -gon on the unit circle, summing to $0$ for $n\ge 2$ .
Small argument	$e^{i\theta}\approx 1+i\theta$	for $\theta\ll 1$ a tiny rotation is an infinitesimal shift up the imaginary axis — the generator of $U(1)$ .

Common trapThe square-root and logarithm laws carry over from the reals unchanged

They do not, and forgetting it is the classic complex-number blunder. The argument is defined only modulo $2\pi$ , so identities such as $\arg(z_1z_2)=\arg z_1+\arg z_2$ and $\sqrt{z_1z_2}=\sqrt{z_1}\sqrt{z_2}$ hold only after a branch of the argument has been fixed, and not for every branch. Treat $\sqrt{\ }$ or $\log$ as if they were single-valued and you can "prove" nonsense: $1=\sqrt{1}=\sqrt{(-1)(-1)}=\sqrt{-1}\,\sqrt{-1}=i\cdot i=-1$ . The error is not in the arithmetic but in pretending a multivalued function has one value; every root and logarithm on $\mathbb{C}$ demands an explicit choice of branch before its algebraic laws can be trusted.

Worked examples

1 · Dividing complex numbers is subtracting angles (confidence builder)

Evaluate $\dfrac{1+i}{1-i}$ two ways, algebraically and geometrically, and say what the answer means.

Step 1. Rationalize by the conjugate of the denominator. Multiply top and bottom by $1+i$ : the denominator becomes $(1-i)(1+i)=1-i^2=2$ , and the numerator becomes $(1+i)^2=1+2i+i^2=2i$ . Hence $\dfrac{1+i}{1-i}=\dfrac{2i}{2}=i$ .

Step 2. Now read it in polar form. Both numbers have modulus $\sqrt{1^2+1^2}=\sqrt2$ , while $\arg(1+i)=+\pi/4$ and $\arg(1-i)=-\pi/4$ . Division divides the moduli, $\sqrt2/\sqrt2=1$ , and subtracts the arguments, $\pi/4-(-\pi/4)=\pi/2$ . So the quotient is $1\cdot e^{i\pi/2}=i$ , agreeing with Step 1.

Step 3. Interpretation: dividing by $1-i$ and multiplying by $1+i$ cancel in length but add a quarter-turn, so the net effect on any number is a pure $90^\circ$ rotation. The quotient is $i$ — unit modulus, argument $\pi/2$ , a pure rotation.

2 · All three cube roots of a real number (fully worked)

Find every solution of $w^3=-8$ and describe them geometrically.

Step 1. Write the target in polar form: $-8$ has modulus $8$ and argument $\pi$ , so $z=8e^{i\pi}$ . The three roots share the modulus $r^{1/3}=8^{1/3}=2$ ; only their arguments differ.

Step 2. The arguments are $\phi_k=\dfrac{\pi+2\pi k}{3}$ for $k=0,1,2$ , giving $\phi_0=\dfrac{\pi}{3}$ , $\phi_1=\pi$ , and $\phi_2=\dfrac{5\pi}{3}$ — three directions spaced $2\pi/3=120^\circ$ apart.

Step 3. Convert to Cartesian form. With $\cos(\pi/3)=\tfrac12$ and $\sin(\pi/3)=\tfrac{\sqrt3}{2}$ : $w_0=2e^{i\pi/3}=1+i\sqrt3$ , $\quad w_1=2e^{i\pi}=-2$ , $\quad w_2=2e^{i5\pi/3}=1-i\sqrt3$ .

Step 4. Check: $(-2)^3=-8$ , and $(1+i\sqrt3)^3=(2e^{i\pi/3})^3=8e^{i\pi}=-8$ ; likewise for $w_2$ . Their sum is $(1+i\sqrt3)+(-2)+(1-i\sqrt3)=0$ . The roots $-2$ and $1\pm i\sqrt3$ sit at the vertices of an equilateral triangle of circumradius $2$ centred at the origin, and they sum to zero.

3 · Multiple-angle formulae from de Moivre

Derive $\cos 3\theta$ and $\sin 3\theta$ in terms of $\cos\theta$ and $\sin\theta$ .

Step 1. By de Moivre, $(\cos\theta+i\sin\theta)^3=\cos 3\theta+i\sin 3\theta$ . Expand the left side with the binomial theorem: $\cos^3\theta+3i\cos^2\theta\sin\theta+3\cos\theta(i\sin\theta)^2+(i\sin\theta)^3$ .

Step 2. Simplify the powers of $i$ ( $i^2=-1$ , $i^3=-i$ ): the expansion becomes $\big(\cos^3\theta-3\cos\theta\sin^2\theta\big)+i\big(3\cos^2\theta\sin\theta-\sin^3\theta\big)$ .

Step 3. Match real and imaginary parts. Using $\sin^2\theta=1-\cos^2\theta$ and $\cos^2\theta=1-\sin^2\theta$ : $\cos 3\theta=4\cos^3\theta-3\cos\theta$ and $\sin 3\theta=3\sin\theta-4\sin^3\theta$ — two trigonometric identities obtained with no trigonometry, only the algebra of $\mathbb{C}$ .

Physics of the portrait

Complex numbers have no single inventor and no single portrait; what they have is a plane, and the story of that plane is the story of how a symbol of impossibility became the coordinate system of rotation. The portrait here is the geometric idea itself.

Origins · 1 minthe complex plane and how this was discovered

the complex plane

geometric representation, Wessel 1799 · Argand 1806 · Gauss 1831 · foundations of complex analysis

For most of their history complex numbers were an algebraic embarrassment. Gerolamo Cardano's Ars Magna of 1545 was forced to write square roots of negative numbers while solving cubics, and Rafael Bombelli in 1572 gave working rules for calculating with them, but neither could say what they were. Descartes coined "imaginary" as a term of contempt, and even Leibniz called $i$ "an amphibian between being and not-being." The rehabilitation was geometric and came, remarkably, from outside the mathematical mainstream. The Norwegian-Danish surveyor Caspar Wessel presented the plane picture to the Royal Danish Academy in 1797 (published 1799), representing $x+iy$ as a directed line segment and multiplication as the addition of angles — the whole modern picture, from a man remembered otherwise only for maps. His paper sank without trace. The Swiss bookkeeper Jean-Robert Argand rediscovered the same diagram in 1806, and the plane now carries his name. What made the idea unignorable was Carl Friedrich Gauss, who had used the geometry privately since the 1790s, made it the engine of his 1799 proof that every polynomial has a root, and in 1831 gave the numbers their lasting name — complex — and full public authority. Once the plane was accepted, the subject exploded: William Rowan Hamilton recast $\mathbb{C}$ in 1837 as ordered pairs of reals with a twisted multiplication, removing the last whiff of mysticism, and Cauchy and Riemann built upon it the calculus of analytic functions that dominates mathematical physics to this day.

History

1545

— Cardano's Ars Magna manipulates square roots of negatives to solve cubic equations

1572

— Bombelli lays down consistent rules of arithmetic for the symbol √−1

1748

— Euler's Introductio publishes the identity e^{iθ} = cos θ + i sin θ

1799

— Wessel presents the first geometric representation of complex numbers as directed segments

1831

— Gauss coins the term complex number and gives the plane its full authority

Going further

Texts. Stone & Goldbart, Mathematics for Physics, ch. 17 (Complex Analysis) · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., ch. 11 · Hassani, Mathematical Physics, 2nd ed., Part II · Byron & Fuller, Mathematics of Classical and Quantum Physics, ch. 6 · Nakahara, Geometry, Topology and Physics, 2nd ed., §1 (for the $U(1)$ and manifold viewpoint) · Needham, Visual Complex Analysis, ch. 1 · Ahlfors, Complex Analysis, ch. 1. Key results to keep in mind: the multiplication rule (2) follows from expanding with $i^2=-1$ ; Euler's formula (4) from comparing the $\exp$ , $\cos$ , $\sin$ Maclaurin series; de Moivre (5) from the exponent law; the $n$ -th root formula (6) and the fact that the roots of unity sum to zero; the triangle inequality via $|w|^2=w\bar w$ ; and the worked results $\tfrac{1+i}{1-i}=i$ , the cube roots of $-8$ being $-2$ and $1\pm i\sqrt3$ (each cubing to $-8$ , summing to $0$ ), and $\cos 3\theta=4\cos^3\theta-3\cos\theta$ . Anchor numbers: $|3+4i|=5$ , $\arg(3+4i)\approx 53.13^\circ$ , $|1+i|=\sqrt2$ , $(\sqrt2\,\angle 45^\circ)(2\,\angle 30^\circ)=2\sqrt2\,\angle 75^\circ$ .

Prerequisites. Real numbers and the field axioms · Vectors in the Euclidean plane · Trigonometry and the exponential function. Leads to. Functions of a Complex Variable and Limits · Elementary Complex Functions · Analytic Functions and the Cauchy-Riemann Equations.

Ask the tutor: Ask me to walk through any root, power, or division in the plane — draw the vectors, add the angles, and watch the algebra become geometry.

i

a real thing or just a convenient fiction?

It is exactly as real as $-1$ , and no more fictional. Every number is a human construction meant to model something: the negatives model debt and direction, the reals model continuous magnitude, and the complex numbers model rotation in a plane. Once you identify $i$ with the ninety-degree turn, there is nothing imaginary about it — it is the most concrete operation in the plane, and the relation $i^2=-1$ is the plain statement that two quarter-turns reverse a direction. The label "imaginary," coined by Descartes as an insult, has unfortunately stuck. Physicists use $i$ to describe genuinely physical rotations and phases, so it earns its place the same way negative numbers do.

Why can't I compare two complex numbers with

<

>

Because no ordering of $\mathbb{C}$ is compatible with its arithmetic. In any ordered field every nonzero square must be positive, so $1=1^2>0$ ; but then $i^2=-1$ would also have to be positive, making $-1>0$ , a contradiction. So there is no way to rank complex numbers by $<$ that respects addition and multiplication. What you can order are real quantities extracted from them — most often the modulus $|z|$ , which measures distance from the origin. When a textbook writes $|z_1|<|z_2|$ it is comparing two nonnegative reals, never the complex numbers themselves.

What does multiplying by a complex number actually do to the plane?

It performs a spiral similarity: a rotation combined with a uniform scaling, with no shearing or reflection. Multiplying every point of the plane by $z=re^{i\theta}$ turns the whole plane by the angle $\theta$ and stretches it by the factor $r$ about the origin. If $r=1$ it is a pure rotation; if $\theta=0$ it is a pure scaling; multiplying by $i$ ( $r=1$ , $\theta=90^\circ$ ) is a clean quarter-turn. This is the single fact that makes complex numbers indispensable in physics: anything built from rotations and oscillations — waves, phasors, quantum phases — is expressed most economically as multiplication in $\mathbb{C}$ .

Why is the argument "multivalued," and does it cause trouble?

The argument is an angle, and angles only mean anything up to whole turns: $\theta$ , $\theta+2\pi$ , and $\theta-2\pi$ all point the same way, so $\arg z$ is really a whole family of values differing by multiples of $2\pi$ . Usually this is harmless — you pick a principal value, say $-\pi<\arg z\le\pi$ , and proceed. It becomes essential, not merely a bother, when you take roots or logarithms: $z^{1/n}$ has $n$ values and $\log z$ has infinitely many precisely because the argument is multivalued. Handled carelessly it produces false identities like $\sqrt{-1}\sqrt{-1}=1$ ; handled properly, through branch cuts and Riemann surfaces, it is the gateway to contour integration.

How can you raise

e

to an imaginary power at all?

You cannot by "repeated multiplication" — that only makes sense for whole-number exponents. The exponential is instead defined by its power series $e^{w}=\sum_{n\ge 0}w^n/n!$ , which converges for every complex $w$ , real or not. Feeding in $w=i\theta$ and sorting the terms by their powers of $i$ produces the cosine series in the real part and the sine series in the imaginary part, which is exactly Euler's formula $e^{i\theta}=\cos\theta+i\sin\theta$ . So $e^{i\theta}$ is not a mysterious new operation but the ordinary exponential series evaluated at an imaginary argument, and it inherits the law $e^{a}e^{b}=e^{a+b}$ that makes it so useful.

If the physical field is real, why do physicists write waves as complex exponentials?

Convenience that never sacrifices correctness. A real oscillation $A\cos(kx-\omega t+\varphi)$ is the real part of the complex wave $Ae^{i\varphi}e^{i(kx-\omega t)}$ , and because differentiation, integration, and superposition are all linear, you may carry the whole complex expression through a calculation and take the real part only at the end. The reward is that phase shifts become multiplications by $e^{i\varphi}$ and interference becomes the addition of complex amplitudes, so a page of trigonometric addition formulae collapses to one line of exponent arithmetic. The one place you must revert to the real part is a nonlinear step — squaring to get an intensity, for instance — where mixing real and imaginary parts would give a wrong answer.

What is the difference between the modulus and the ordinary absolute value?

None in spirit — the modulus is the absolute value, extended from the line to the plane. For a real number $x$ the absolute value $|x|$ is its distance from $0$ on the number line; for a complex number $z=x+iy$ the modulus $|z|=\sqrt{x^2+y^2}$ is its distance from $0$ in the plane, by Pythagoras. When $y=0$ the two coincide, so the notation is deliberately the same. The modulus keeps the key properties of the absolute value — it is nonnegative, vanishes only at $0$ , is multiplicative ( $|z_1z_2|=|z_1||z_2|$ ), and obeys the triangle inequality — which is exactly why $|z_1-z_2|$ serves as a genuine distance and turns $\mathbb{C}$ into a metric space.