Complex Numbers — Arithmetic and the Complex Plane

A complex number is an expression $z = a + bi$ where $a, b$ are real numbers and $i^{2} = - 1$ . They obey the ordinary rules of algebra (with $i^{2}$ replaced by $- 1$ ) and can be pictured as points or arrows in a two-dimensional complex plane. Used everywhere from solving polynomials to electrical engineering, quantum mechanics, and signal processing. Formalized by Gauss, Argand, and Hamilton in the early 19th century.

A complex number combines a real part and an imaginary part into a single object $z = a + bi$ , where the imaginary unit $i$ satisfies the defining relation $i^{2} = - 1$ . They matter because they make every polynomial equation solvable (the Fundamental Theorem of Algebra) and because they encode rotation and scaling in the plane in one compact multiplication. The surprising fact: although $i$ was introduced as a "fictitious" square root of $- 1$ , complex numbers turn out to be more geometrically natural than the reals — multiplying by $i$ is simply a 90° rotation. They form a complete, consistent number system in which addition, subtraction, multiplication, and division all behave as expected.

The Core Insight

Adjoining a single new symbol $i$ with the rule $i^{2} = - 1$ to the real numbers produces a system where arithmetic stays consistent and geometry becomes algebra. Points in a plane become numbers you can add and multiply; addition is vector translation, and multiplication is simultaneous rotation-and-scaling. Two real dimensions are tamed by one algebraic rule.

Formal Statement

Complex Number

A complex number is an ordered expression

z = a + bi, a, b \in R, i^{2} = - 1.

⟳this is a definition;

i

is the postulated symbol with

i^{2} = - 1

, and

a, b

are arbitrary reals.

$a = Re (z)$ is the real part.
$b = Im (z)$ is the imaginary part (note: $b$ is itself real).
$i$ is the imaginary unit.
The set of all such $z$ is denoted $C$ .

A number with $a = 0$ (so $z = bi$ ) is called purely imaginary; one with $b = 0$ is just a real number, so $R \subset C$ .

Arithmetic Operations

For $z_{1} = a + bi$ and $z_{2} = c + d i$ :

z_{1} + z_{2} = (a + c) + (b + d) i .

⟳add real parts to real parts and imaginary parts to imaginary parts, treating

i

as a literal symbol.

z_{1} - z_{2} = (a - c) + (b - d) i .

⟳subtract componentwise, exactly as for addition.

z_{1} \cdot z_{2} = (a c - b d) + (a d + b c) i .

⟳expand

(a + bi) (c + d i)

by distribution, collect terms, then replace

i^{2}

with

- 1

in the

b d i^{2}

term.

Complex Conjugate and Modulus

The complex conjugate of $z = a + bi$ is

\overset{z}{ˉ} = a - bi .

⟳flip the sign of the imaginary part only.

The modulus (absolute value, magnitude) is

∣ z ∣ = a^{2} + b^{2} = z \overset{z}{ˉ} .

⟳

z \overset{z}{ˉ} = (a + bi) (a - bi) = a^{2} - (bi)^{2} = a^{2} + b^{2}

using

i^{2} = - 1

; take the square root.

Division is then defined by multiplying numerator and denominator by the conjugate of the denominator:

\frac{z _{1}}{z _{2}} = \frac{z _{1} z _{2} ˉ}{∣ z _{2} ∣ ^{2}} = \frac{( a c + b d ) + ( b c - a d ) i}{c ^{2} + d ^{2}}, z_{2} \neq = 0.

⟳multiply top and bottom by

\overset{z_{2}}{ˉ} = c - d i

; the denominator

z_{2} \overset{z_{2}}{ˉ} = c^{2} + d^{2}

becomes real, making the quotient a standard complex number.

The Complex Numbers Form a Field

(C, +, \cdot)

is a field: addition and multiplication are commutative, associative, and distributive;

0 = 0 + 0 i

is the additive identity,

1 = 1 + 0 i

the multiplicative identity; every

z

has an additive inverse

- z

, and every

z \neq = 0

has a multiplicative inverse

z^{- 1} = \frac{z ˉ}{∣ z ∣ ^{2}} = \frac{a - bi}{a ^{2} + b ^{2}} .

⟳solve

z \cdot z^{- 1} = 1

by setting

z^{- 1} = \overset{z}{ˉ} /∣ z ∣^{2}

and checking

z \overset{z}{ˉ} /∣ z ∣^{2} = ∣ z ∣^{2} /∣ z ∣^{2} = 1

Why It Holds / Derivation

Consistency of Complex Multiplication

We show the product rule [4] follows purely from algebra plus $i^{2} = - 1$ .

Start with $z_{1} z_{2} = (a + bi) (c + d i)$ .
Apply the distributive law (FOIL): $= a c + a d i + b c i + b d i^{2}$ .
Substitute $i^{2} = - 1$ : $b d i^{2} = - b d$ .
Collect real terms $a c - b d$ and imaginary terms $(a d + b c) i$ .
Result: $z_{1} z_{2} = (a c - b d) + (a d + b c) i$ , which is again of the form $a^{'} + b^{'} i$ — so $C$ is closed under multiplication.

Because every step uses only field axioms of $R$ plus one substitution, no contradiction can arise; the system is consistent. A rigorous foundation defines $z$ as the ordered pair $(a, b)$ with multiplication $(a, b) (c, d) = (a c - b d, a d + b c)$ , avoiding any appeal to a "mysterious" $- 1$ . $■$

Modulus is Multiplicative

Claim: $∣ z_{1} z_{2} ∣ = ∣ z_{1} ∣ ∣ z_{2} ∣$ .

By [6], $∣ z ∣^{2} = z \overset{z}{ˉ}$ .
Note conjugation distributes over products: $\overline{z_{1} z_{2}} = \overset{z_{1}}{ˉ} \overset{z_{2}}{ˉ}$ (check by expanding both sides).
Then $∣ z_{1} z_{2} ∣^{2} = (z_{1} z_{2}) \overline{(z_{1} z_{2})} = z_{1} z_{2} \overset{z_{1}}{ˉ} \overset{z_{2}}{ˉ}$ .
Rearrange (commutativity): $= (z_{1} \overset{z_{1}}{ˉ}) (z_{2} \overset{z_{2}}{ˉ}) = ∣ z_{1} ∣^{2} ∣ z_{2} ∣^{2}$ .
Take positive square roots: $∣ z_{1} z_{2} ∣ = ∣ z_{1} ∣∣ z_{2} ∣$ . $■$

This is the algebraic shadow of the geometric fact that multiplying complex numbers multiplies their lengths.

Carl Friedrich Gauss | 1777–1855

Gauss gave the term "complex number" its modern usage and championed the geometric picture of $C$ as a plane (the "Gaussian plane"). In his 1799 doctoral thesis he provided one of the first essentially complete proofs of the Fundamental Theorem of Algebra, establishing that complex numbers are exactly what is needed to factor every real polynomial. He treated $i$ not as something paradoxical but as a legitimate, well-defined quantity, helping dissolve centuries of suspicion around "imaginary" roots.

Building Intuition

Think of $a + bi$ as the point $(a, b)$ or the arrow from the origin to that point.

Addition = stacking arrows tip-to-tail (vector addition). The real and imaginary axes behave like ordinary $x$ and $y$ axes.
Multiplication by $i$ = rotate the arrow 90° counterclockwise. Apply it twice and you point backwards: $i \cdot i = - 1$ . That single picture is the meaning of $i^{2} = - 1$ .
The modulus $∣ z ∣$ is the arrow's length; the argument $ar g z$ is its angle from the positive real axis.
General multiplication = "rotate by the angles, multiply the lengths." Multiplying by $z$ rotates the whole plane by $ar g z$ and scales it by $∣ z ∣$ .

So complex numbers are not "extra" numbers tacked on — they are the natural arithmetic of the plane itself.

The complex plane: the point $z=3+2i$, its conjugate, modulus, and argument

Worked Examples

Multiplying and Dividing Complex Numbers

Given:

z_{1} = 3 + 2 i

and

z_{2} = 1 - 4 i

. Compute

z_{1} z_{2}

and

z_{1} / z_{2}

. Product:

Expand: $(3 + 2 i) (1 - 4 i) = 3 - 12 i + 2 i - 8 i^{2}$ .
Use $i^{2} = - 1$ : $- 8 i^{2} = + 8$ .
Combine: $(3 + 8) + (- 12 + 2) i = 11 - 10 i$ .

z_{1} z_{2} = 11 - 10 i

Quotient: multiply by the conjugate

\overset{z_{2}}{ˉ} = 1 + 4 i

$\frac{( 3 + 2 i ) ( 1 + 4 i )}{( 1 - 4 i ) ( 1 + 4 i )}$ .
Numerator: $3 + 12 i + 2 i + 8 i^{2} = 3 + 14 i - 8 = - 5 + 14 i$ .
Denominator: $1^{2} + 4^{2} = 17$ .

\frac{z _{1}}{z _{2}} = \frac{- 5}{17} + \frac{14}{17} i \approx - 0.294 + 0.824 i

(Quantities here are pure numbers — complex numbers are dimensionless unless a physical context assigns units, e.g. ohms for impedance.)

Modulus, Conjugate, and a Verification

Given:

z = 5 - 12 i

. Find

∣ z ∣

\overset{z}{ˉ}

z \overset{z}{ˉ}

, and

z^{- 1}

Modulus: $∣ z ∣ = 5^{2} + (- 12)^{2} = 25 + 144 = 169 = 13$ .
Conjugate: $\overset{z}{ˉ} = 5 + 12 i$ .
Product check: $z \overset{z}{ˉ} = (5 - 12 i) (5 + 12 i) = 25 - (12 i)^{2} = 25 + 144 = 169 = ∣ z ∣^{2}$ . ✓
Inverse via [8]: $z^{- 1} = \frac{z ˉ}{∣ z ∣ ^{2}} = \frac{5 + 12 i}{169}$ .

∣ z ∣ = 13, \overset{z}{ˉ} = 5 + 12 i, z^{- 1} = \frac{5}{169} + \frac{12}{169} i

Verify: $z \cdot z^{- 1} = \frac{( 5 - 12 i ) ( 5 + 12 i )}{169} = \frac{169}{169} = 1$ . ✓

Common Misconceptions

" $i = - 1$ , so $- 1 \cdot - 1 = (- 1) (- 1) = 1 = 1$ ." This is FALSE. The identity

x y = x y

holds only for non-negative reals. The correct, unambiguous defining property is

i^{2} = - 1

, not a careless square-root manipulation. Always reduce using

i^{2} = - 1

directly. Complex numbers are not ordered. There is no consistent way to say

i > 0

i < 0

that respects arithmetic. If

i > 0

then

i \cdot i = - 1 > 0

, a contradiction; if

i < 0

then

(- i) > 0

and

(- i)^{2} = - 1 > 0

, again a contradiction. Inequalities like

z_{1} < z_{2}

are meaningless in

C

— only their moduli

∣ z_{1} ∣, ∣ z_{2} ∣

(real numbers) can be compared.

One might guess that squaring preserves "size" of the real and imaginary parts independently — it does not. Take $z = 1 + i$ . Then $z^{2} = (1 + i)^{2} = 1 + 2 i + i^{2} = 2 i$ , which has zero real part. The parts mix completely under multiplication; only the modulus behaves predictably: $∣ z ∣^{2} = (2)^{2} = 2 = ∣ z^{2} ∣$ . ✓

Connections

Complex arithmetic sits at a crossroads of algebra and geometry. The geometric multiplication rule "rotate and scale" is made precise by Polar Form and Euler's Formula, where $z = r (cos θ + i sin θ) = r e^{i θ}$ , turning multiplication into addition of angles. The fact that every polynomial factors completely over $C$ is the Fundamental Theorem of Algebra. Addition mirrors Vectors in the Plane exactly, while the field structure connects to Abstract Algebra: Fields. The conjugate and modulus generalize the dot product and length from Euclidean Geometry.

Real-World Applications

Electrical engineering (AC circuits): Impedance is written $Z = R + i X$ (resistance plus reactance); using $j$ for $i$ , engineers compute currents and voltages with complex arithmetic instead of differential equations.
Signal processing: The Fourier transform decomposes signals into complex exponentials $e^{iω t}$ ; phase and amplitude are encoded in a single complex number.
Quantum mechanics: Wavefunctions are complex-valued; probabilities are $∣ ψ ∣^{2}$ , and interference arises from adding complex amplitudes.
Control theory: System stability is read from the location of complex poles in the plane (left vs. right half-plane).
Fluid dynamics & aerodynamics: Two-dimensional potential flow is modeled with complex potentials; the Joukowsky transform maps circles to airfoil shapes.
Computer graphics: Multiplication by a unit complex number performs efficient 2D rotation (the precursor to quaternions in 3D).
Fractals: The Mandelbrot set is defined by iterating $z \mapsto z^{2} + c$ over complex $c$ .

Historical Development

1545 — Gerolamo Cardano, in Ars Magna, encounters square roots of negatives while solving cubics; he calls them "sophistic" but manipulates them anyway.
1572 — Rafael Bombelli formulates explicit rules for arithmetic with $- 1$ , showing they yield correct real answers for cubic equations.
1637 — René Descartes coins the dismissive term "imaginary" for these numbers in La Géométrie.
1748 — Leonhard Euler systematizes the notation, introduces $i$ for $- 1$ (used widely after 1777), and discovers $e^{i θ} = cos θ + i sin θ$ .
1797–1806 — Caspar Wessel (1797) and Jean-Robert Argand (1806) independently describe the geometric plane representation — the "Argand diagram."
1799 — Gauss proves the Fundamental Theorem of Algebra in his doctoral dissertation.
1831 — Gauss publishes the geometric interpretation and popularizes the name "complex number," lending the subject full respectability.
1833–1837 — William Rowan Hamilton defines complex numbers rigorously as ordered pairs $(a, b)$ of reals, removing any need for a "mysterious" $i$ .
1843 — Hamilton extends the idea to four dimensions, discovering the quaternions.

Subtleties & Limits

Complex numbers are algebraically closed and complete, but they sacrifice ordering — there is no field ordering on $C$ compatible with arithmetic. A deeper structural question is how far one can push the pattern: the reals ( $R$ ), complex numbers ( $C$ ), quaternions ( $H$ ), and octonions ( $O$ ) form a tower of dimensions $1, 2, 4, 8$ , each losing a property (order, then commutativity, then associativity). Hurwitz's theorem (1898) proves these four are the only normed division algebras over $R$ — there is no consistent "complex-number-like" multiplication in dimension 3, 5, or beyond 8. Why the universe of well-behaved number systems stops at eight dimensions remains a subject of deep interest connecting algebra, topology, and theoretical physics.