Complex Numbers

Numbers of the form $a + b i$ with $i^{2} = - 1$ , forming a field in which every polynomial splits. Arithmetic, conjugation and modulus in rectangular form; rotation and scaling in polar form; and the conjugate-pair structure that governs the eigenvalues of a real matrix.

Definition

A complex number is $z = a + b i$ with $a, b \in R$ and $i^{2} = - 1$ . Its real part is $Re z = a$ and its imaginary part $Im z = b$ . A real number, not $b i$ .

Arithmetic. Addition is componentwise. Multiplication expands and uses $i^{2} = - 1$ :

(a + b i) (c + d i) = (a c - b d) + (a d + b c) i .

Conjugate and modulus. $\bar{z} = a - b i$ and $| z | = \sqrt{a^{2} + b^{2}}$ , related by

z \bar{z} = a^{2} + b^{2} = | z |^{2},

which is always a nonnegative real. This is what makes division possible: multiply above and below by the conjugate of the denominator,

\frac{z}{w} = \frac{z \bar{w}}{| w |^{2}} .

Conjugation respects the arithmetic: $\overset{―}{z + w} = \bar{z} + \bar{w}$ and $\overset{―}{z w} = \bar{z} \bar{w}$ . The modulus is multiplicative: $| z w | = | z | | w |$ .

Polar form. Writing $r = | z |$ and $θ = \arg z$ ,

z = r (\cos θ + i \sin θ) = r e^{i θ} .

Multiplication multiplies moduli and adds arguments, so De Moivre's theorem follows: $z^{n} = r^{n} e^{i n θ}$ . Multiplying by $e^{i θ}$ is rotation by $θ$ ; multiplying by a positive real is scaling.

Roots. Every nonzero $z$ has exactly $n$ distinct $n$ th roots, at modulus $r^{1 / n}$ and arguments $(θ + 2 π k) / n$ for $k = 0, \dots, n - 1$ , equally spaced around a circle. The $n$ th roots of unity are the case $z = 1$ , and they sum to zero for $n \geq 2$ .

Why the field matters here. $C$ is algebraically closed: every non-constant polynomial with complex coefficients factors completely. So a characteristic polynomial always splits over $C$ , and every $n \times n$ complex matrix has $n$ eigenvalues counted with multiplicity, which is what a real matrix lacks.

Conjugate pairs. A polynomial with real coefficients satisfies $\overset{―}{p (z)} = p (\bar{z})$ , so its non-real roots occur in conjugate pairs. For a real matrix this means non-real eigenvalues come in pairs $λ, \bar{λ}$ with conjugate eigenvectors.

Assumptions and scope

$C$ is not ordered. Inequalities such as $z < w$ are undefined for non-real numbers, and only moduli, which are real, may be compared.
The imaginary part is the real coefficient $b$ , not $b i$ . Reporting $Im (3 + 4 i)$ as $4 i$ is a type error that propagates.
Conjugate pairing of roots requires the coefficients to be real. A polynomial with genuinely complex coefficients has no such symmetry.
The argument is defined only up to multiples of $2 π$ , and the principal value convention must be stated when it matters. Every $n$ th root computation depends on which representatives are taken.
Viewing $C$ as a real vector space gives dimension 2 with basis ${1, i}$ ; as a complex vector space it has dimension 1. Dimension is relative to the field, so both statements are correct.
A real matrix with complex eigenvalues is not defective on that account. The eigenvalues exist over $C$ and the matrix may be diagonalizable there, which is a different question from having too few eigenvectors.

Forms this is expressed in

The same content in several forms. Each makes something visible that the others leave implicit, so moving between them is part of understanding the topic rather than a presentation choice.

symbolic

A complex number written as $z = a + b i$ , with $a$ and $b$ real. The pair $(a, b)$ places the number in the plane: $a$ along the real axis, $b$ along the imaginary. Equality is componentwise, $a + b i = c + d i$ exactly when $a = c$ and $b = d$ , which is why a single complex equation carries two real ones.

This form makes addition transparent: $(a + b i) + (c + d i) = (a + c) + (b + d) i$ , the same componentwise rule as vector addition in $R^{2}$ . Multiplication is where it becomes laborious: expanding $(a + b i) (c + d i)$ gives four terms that must be collected using $i^{2} = - 1$ into $(a c - b d) + (a d + b c) i$ , a formula that reveals nothing about what the operation does geometrically.

Conjugation is a sign flip, $\bar{z} = a - b i$ , and division is engineered from it: multiplying above and below by $\bar{w}$ turns the denominator into the real number $| w |^{2} = c^{2} + d^{2}$ .

What this form cannot show is that multiplication rotates. The modulus $\sqrt{a^{2} + b^{2}}$ and the direction of the point are both implicit, recoverable only by further computation, so questions about powers, roots, or repeated rotation are answered badly here, $(1 + i)^{8}$ by repeated expansion is eight rounds of collecting terms. Those belong to the polar form.

Translates into: symbolic

symbolic

The same number written as $z = r e^{i θ}$ , equivalently $r (\cos θ + i \sin θ)$ , where $r = | z | \geq 0$ is the distance from the origin and $θ$ is the direction. The two descriptions are connected by $a = r \cos θ$ , $b = r \sin θ$ and, in reverse, $r = \sqrt{a^{2} + b^{2}}$ with $θ$ the angle of the point $(a, b)$ .

This form makes multiplication a single statement: moduli multiply and arguments add, $r_{1} e^{i θ_{1}} \cdot r_{2} e^{i θ_{2}} = r_{1} r_{2} e^{i (θ_{1} + θ_{2})}$ . Multiplying by $z$ therefore scales by $r$ and rotates by $θ$ , and the four-term expansion of the Cartesian form is revealed as bookkeeping for those two facts. Powers follow immediately as $z^{n} = r^{n} e^{i n θ}$ , which is De Moivre's theorem. The $n$ $n$ th roots appear as $n$ points at modulus $r^{1 / n}$ , spaced evenly by $2 π / n$ around a circle.

Two hazards attach to the conversion. The argument is defined only up to multiples of $2 π$ , so a principal range must be fixed, conventionally $(- π, π]$ . And $\arctan (b / a)$ cannot determine it alone: the ratio discards the individual signs, so it returns the same angle for $(a, b)$ and $(- a, - b)$ , placing third-quadrant numbers in the first and second-quadrant numbers in the fourth. The quadrant must come from the signs of $a$ and $b$ themselves.

Addition is what this form loses. There is no useful rule combining $r_{1} e^{i θ_{1}}$ and $r_{2} e^{i θ_{2}}$ into a sum, so anything additive, including reading off real and imaginary parts, returns to the Cartesian form.

Translates into: symbolic

geometric

Modulus is distance, argument is angle, multiplication is rotation

A complex number drawn as a point of the plane: $z = a + b i$ at $(a, b)$ , with modulus $r = | z |$ its distance from the origin and argument $θ = \arg z$ the angle from the positive real axis.

This form makes multiplication intelligible. Multiplying by $e^{i φ}$ rotates by $φ$ and changes nothing else, so the point travels along the circle of radius $r$ ; multiplying by a real $ρ$ scales the distance and leaves the angle. A general product does both, which is why moduli multiply and arguments add.

Conjugation is reflection in the real axis, and $z \bar{z} = | z |^{2}$ is Pythagoras. The $n$ -th roots of unity are $n$ equally spaced points on the unit circle, which the algebra states and the picture makes obvious.

What the plane cannot supply is exact arithmetic: the Cartesian form adds easily and the polar form multiplies easily, and the drawing settles neither.

Translates into: symbolic, symbolic

Worked material

Example

Numbers whose structure is visible in the plane

The unit imaginary. $i = e^{i π / 2}$ has modulus 1 and argument a quarter turn. Multiplying by it rotates without scaling: $1 \mapsto i \mapsto - 1 \mapsto - i \mapsto 1$ , returning after four steps. That cycle is $i^{4} = 1$ , and it is why powers of $i$ repeat with period four.

A real number. $- 3 = 3 e^{i π}$ : modulus 3, argument $π$ . Real numbers are the points with argument $0$ or $π$ , and multiplying by a negative real scales and reflects through the origin. Conjugation fixes them, which is the defining property of $R$ inside $C$ .

A number on the unit circle. $\frac{1}{\sqrt{2}} (1 + i) = e^{i π / 4}$ has modulus 1. Multiplying by it is pure rotation by $45 °$ , and its eighth power is $e^{2 π i} = 1$ . Every modulus-1 number is a pure rotation, which is exactly the set whose powers stay bounded.

The cube roots of unity. $1$ , $ω = - \frac{1}{2} + \frac{\sqrt{3}}{2} i$ and $\bar{ω} = - \frac{1}{2} - \frac{\sqrt{3}}{2} i$ sit at $0 °$ , $120 °$ and $240 °$ on the unit circle. Each cubes to 1, and they sum to zero. The two real parts of $- \frac{1}{2}$ cancel the 1 exactly.

They illustrate two things at once: that $n$ th roots are equally spaced, and that the non-real ones come in a conjugate pair because $z^{3} - 1$ has real coefficients.

A conjugate pair from a real quadratic. $x^{2} - 2 x + 5$ has roots $1 \pm 2 i$ , at modulus $\sqrt{5}$ and arguments $\pm \arctan 2$ . Their sum is $2$ and their product is $5$ . Both real, which is forced: a conjugate pair always sums and multiplies to real numbers, which is why a real quadratic can have complex roots without any complex coefficient appearing.

Eigenvalues of a rotation. $(\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix})$ has eigenvalues $\pm i$ , on the unit circle at $\pm 90 °$ . The modulus 1 says the map preserves length; the argument $π / 2$ says it turns by a quarter. For a real matrix a complex eigenvalue $r e^{i θ}$ always reads this way, rotate by $θ$ , scale by $r$ , in some plane.

What the plane picture supplies. Modulus is distance, argument is angle, conjugation is reflection in the real axis, multiplication is rotate-and-scale. Every fact in this unit is one of those four read off a diagram, which is why sketching the point is the first move when an arctan or a root count is in doubt.

Non-example

Five errors with complex numbers

Reporting the imaginary part with its $i$ . Writing $Im (3 + 4 i) = 4 i$ . The imaginary part is the real number 4. A coordinate of the point, like the 3. The error makes $Im$ return complex values, and identities such as $z = Re z + i Im z$ then double-count the $i$ .

Losing the sign flip in a product. Computing $(3 + 4 i) (1 - 2 i)$ as $3 - 2 i - 8 = - 5 - 2 i$ by treating $i^{2}$ as $+ 1$ , or as $3 - 2 i$ by dropping the term entirely. The correct value is $11 - 2 i$ , the $+ 8$ arising because $- 8 i^{2} = + 8$ . The modulus check catches it: $| 11 - 2 i | = \sqrt{125}$ matches $| z | | w | = 5 \sqrt{5}$ , while the wrong answers do not.

Comparing complex numbers. Writing $i > 0$ or "the larger root". $C$ carries no order compatible with its arithmetic, if $i > 0$ then $i^{2} > 0$ , so $- 1 > 0$ ; if $i < 0$ then $(- i) > 0$ and squaring gives the same contradiction. Only moduli may be compared, and those are real.

Taking one root instead of $n$ . Solving $z^{3} = 1$ and answering $z = 1$ . There are three cube roots, at angles $0$ , $2 π / 3$ and $4 π / 3$ ; the other two are $- \frac{1}{2} \pm \frac{\sqrt{3}}{2} i$ . Every nonzero complex number has exactly $n$ distinct $n$ th roots, and stopping at the first one found loses the rest.

Expecting an unpaired complex eigenvalue from a real matrix. Reporting that a real matrix has eigenvalues $2$ and $3 + i$ and nothing else. The characteristic polynomial has real coefficients, so $3 - i$ must also be a root. The pairing is forced. A stated spectrum that omits a conjugate is evidence of an arithmetic error rather than of an unusual matrix.

What unites them. The first two are arithmetic with a conceptual cause: not tracking what $i$ contributes. The middle one imports a property $R$ has and $C$ does not. The last two stop at the first answer when the structure guarantees more. One root of $n$ , one half of a pair.

Common errors

Common misconception

The imaginary part of $a + b i$ is $b i$ , so $Im (3 + 4 i) = 4 i$ and the imaginary part is itself a complex number.

Complex Numbers

Assumptions and scope

Forms this is expressed in

Worked material

Numbers whose structure is visible in the plane

Five errors with complex numbers

Common errors

Related units

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