Complex Numbers

⚡ Key Takeaways

Complex numbers are numbers of the form $a + bi$ where $a, b$ are real and $i^2 = -1$ . They extend the real numbers so that every polynomial equation has a solution. They are essential in algebra, geometry, signal processing, quantum mechanics, and electrical engineering. The concept matured through Cardano (1545), Euler (1748), Argand (1806), and Gauss (1831).

💡 The Core Insight

The single move that creates the entire theory is declaring a new number $i$ with $i^2 = -1$ . Once you accept this, nothing else needs to be invented: addition, subtraction, multiplication, and division all follow from ordinary algebra plus this one rule. The payoff is enormous — the plane becomes a number system where rotation is just multiplication, and the long-standing impossibility "you cannot take the square root of a negative number" simply dissolves.

DefinitionThe Imaginary Unit and Complex Numbers

The imaginary unit $i$ is defined by the property

1

\quad i^2 = -1.

Derivation hint: This is a definition, not a derived fact — $i$ is introduced precisely as a root of the equation $x^2 + 1 = 0$ . A complex number is any expression

2

\quad z = a + bi, \qquad a, b \in \mathbb{R}.

Derivation hint: $a$ and $b$ are arbitrary real numbers; the set of all such $z$ is denoted $\mathbb{C}$ . Here $a = \operatorname{Re}(z)$ is the real part and $b = \operatorname{Im}(z)$ is the imaginary part (note: $\operatorname{Im}(z)$ is the real number $b$ , not $bi$ ).

DefinitionArithmetic Operations

For $z_1 = a + bi$ and $z_2 = c + di$ :

Addition / Subtraction:

3

\quad z_1 \pm z_2 = (a \pm c) + (b \pm d)i.

Derivation hint: Combine like terms — real with real, $i$ -terms with $i$ -terms.

Multiplication:

4

\quad z_1 z_2 = (ac - bd) + (ad + bc)i.

Derivation hint: Expand $(a+bi)(c+di)$ by distribution, then replace $i^2$ with $-1$ .

Complex conjugate:

5

\quad \bar{z} = a - bi.

Derivation hint: Flip the sign of the imaginary part; this reflects $z$ across the real axis.

Modulus (absolute value):

6

\quad |z| = \sqrt{a^2 + b^2} = \sqrt{z\bar{z}}.

Derivation hint: Compute $z\bar{z} = (a+bi)(a-bi) = a^2 + b^2$ using equation [4]; the modulus is its square root.

Division:

7

\quad \frac{z_1}{z_2} = \frac{z_1 \bar{z_2}}{|z_2|^2} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}.

Derivation hint: Multiply numerator and denominator by $\bar{z_2}$ to make the denominator real.

DefinitionPolar Form

Every complex number can be written using its modulus $r = |z|$ and argument $\theta$ (the angle from the positive real axis):

8

\quad z = r(\cos\theta + i\sin\theta), \qquad r = \sqrt{a^2+b^2}, \quad \tan\theta = \frac{b}{a}.

Derivation hint: Read off $a = r\cos\theta$ and $b = r\sin\theta$ from the right triangle formed by $z$ in the plane. Using Euler's formula this compresses to:

9

\quad z = r e^{i\theta}, \qquad e^{i\theta} = \cos\theta + i\sin\theta.

Derivation hint: Compare the Taylor series of $e^{i\theta}$ , $\cos\theta$ , and $\sin\theta$ ; the series of $e^{i\theta}$ splits exactly into the cosine (real) and sine (imaginary) series.

TheoremFundamental Theorem of Algebra

Every polynomial of degree $n \geq 1$ with complex coefficients has exactly $n$ roots in $\mathbb{C}$ (counted with multiplicity). Equivalently, $\mathbb{C}$ is algebraically closed.

10

\quad p(z) = a_n z^n + \cdots + a_1 z + a_0 = a_n (z - r_1)(z - r_2)\cdots(z - r_n).

Derivation hint: Existence of one root follows from analytic or topological arguments (e.g., Liouville's theorem); dividing out $(z - r_1)$ and inducting gives the full factorization.

ProofDe Moivre's Theorem from Polar Form

We show that $\big(r(\cos\theta + i\sin\theta)\big)^n = r^n(\cos n\theta + i\sin n\theta)$ . 1. Multiplication in polar form. Take $z_1 = r_1 e^{i\theta_1}$ , $z_2 = r_2 e^{i\theta_2}$ . Then

11

\quad z_1 z_2 = r_1 r_2\, e^{i(\theta_1 + \theta_2)}.

Derivation hint: Multiply equation [4] out in trig form and apply the angle-addition identities for $\cos$ and $\sin$ . 2. Moduli multiply, arguments add. From step 1, $|z_1 z_2| = r_1 r_2$ and $\arg(z_1 z_2) = \theta_1 + \theta_2$ . 3. Apply repeatedly. Setting $z_1 = z_2 = \cdots = z = re^{i\theta}$ and multiplying $n$ copies:

12

\quad z^n = r^n e^{i n\theta} = r^n(\cos n\theta + i\sin n\theta).

Derivation hint: Iterate equation [11] $n$ times; the modulus becomes $r^n$ and the angle accumulates to $n\theta$ . 4. Conclusion. Equation [12] is exactly De Moivre's theorem, proven for all positive integers $n$ (and, by inversion, for negative integers). $\blacksquare$

∎

mathematician · Leonhard Euler | 1707–1783

Euler did more than anyone to make complex numbers respectable and useful. He introduced the symbol $i$ for $\sqrt{-1}$ , systematically used complex exponentials, and discovered the relation $e^{i\theta} = \cos\theta + i\sin\theta$ (1748) — from which the celebrated identity $e^{i\pi} + 1 = 0$ follows by setting $\theta = \pi$ . Euler treated complex numbers as legitimate computational objects decades before their geometric meaning was fully understood, computing infinite series and integrals with them and arriving at correct results that would only later be rigorously justified.

🧠 Intuition

Think of a complex number as a point in a plane (the Argand plane), with the real part along the horizontal axis and the imaginary part along the vertical axis. Addition then behaves like vector addition — you slide one arrow to the tip of the other. The deepest intuition is about multiplication. When you multiply by $z = re^{i\theta}$ , you do two things at once: scale every length by $r$ , and rotate the whole plane by angle $\theta$ . So multiplying by $i$ (which has $r=1$ , $\theta = 90°$ ) rotates the plane a quarter turn. That is why $i^2 = -1$ : rotating by $90°$ twice is a $180°$ rotation, which sends $1$ to $-1$ . The "impossible" rule $i^2 = -1$ is just the geometry of turning around.

Figure

The Argand plane: a complex number as a point, with modulus and argument

ExampleMultiplication and Division

Given: $z_1 = 3 + 2i$ and $z_2 = 1 - 4i$ . Find $z_1 z_2$ and $z_1 / z_2$ .

Multiplication (using equation [4] with $a=3, b=2, c=1, d=-4$ ):

z_1 z_2 = (3\cdot 1 - 2\cdot(-4)) + (3\cdot(-4) + 2\cdot 1)i = (3 + 8) + (-12 + 2)i.

\boxed{z_1 z_2 = 11 - 10i}

Division (using equation [7]; first $|z_2|^2 = 1^2 + (-4)^2 = 17$ ):

\frac{z_1}{z_2} = \frac{(3 + 2i)(1 + 4i)}{17} = \frac{(3 - 8) + (12 + 2)i}{17} = \frac{-5 + 14i}{17}.

\boxed{\frac{z_1}{z_2} = -\tfrac{5}{17} + \tfrac{14}{17}i \approx -0.294 + 0.824i}

ExamplePowers via De Moivre's Theorem

Given: Compute $(1 + i)^8$ .

Step 1 — convert to polar form. Modulus: $r = \sqrt{1^2 + 1^2} = \sqrt{2}$ . Argument: $\theta = \arctan(1/1) = 45° = \pi/4$ . So $1 + i = \sqrt{2}\,e^{i\pi/4}$ .

Step 2 — apply De Moivre (equation [12]) with $n = 8$ :

(1+i)^8 = (\sqrt{2})^8\, e^{i\cdot 8\pi/4} = 2^4\, e^{i 2\pi} = 16(\cos 2\pi + i\sin 2\pi).

Step 3 — evaluate. Since $\cos 2\pi = 1$ and $\sin 2\pi = 0$ :

\boxed{(1+i)^8 = 16}

(A real answer, as a sanity check: $(1+i)^2 = 2i$ , so $(1+i)^8 = (2i)^4 = 16 i^4 = 16$ . ✓)

⚠️ Common mistake

" $\sqrt{-4} = 2i$ means $\sqrt{a}\sqrt{b} = \sqrt{ab}$ always." This identity fails for negative numbers. For example,

\sqrt{-1}\cdot\sqrt{-1} = i \cdot i = -1, \quad\text{but}\quad \sqrt{(-1)(-1)} = \sqrt{1} = 1.

The rule $\sqrt{a}\sqrt{b}=\sqrt{ab}$ is only valid when at least one of $a, b$ is non-negative. Always reduce to $i$ first, then multiply.

⚠️ Common mistake

"The imaginary part of $z = a + bi$ is $bi$ ." No — by definition $\operatorname{Im}(z) = b$ , a real number, not $bi$ . This matters for formulas like $|z|^2 = (\operatorname{Re} z)^2 + (\operatorname{Im} z)^2$ , which would be dimensionally wrong if $\operatorname{Im}(z)$ carried an $i$ .

Counterexample

Claim: "Complex numbers can be ordered like the reals." This is false. Suppose an ordering exists compatible with arithmetic, with $i > 0$ or $i < 0$ .

If $i > 0$ : then $i \cdot i = -1 > 0$ , contradiction (since $-1 < 0$ ).
If $i < 0$ : then $-i > 0$ , and $(-i)(-i) = i^2 = -1 > 0$ , again contradiction.

So $\mathbb{C}$ cannot be an ordered field. There is no consistent way to say one genuinely complex number is "greater than" another.

connection

Complex numbers tie together many areas. They generalize the Real Numbers by adjoining a root of $x^2+1$ . Their polar form rests entirely on Trigonometry, and Euler's formula links them to the Exponential Function. Geometrically they are Vectors in the Plane with an extra multiplication that encodes rotation, connecting to Rotation Matrices (multiplication by $e^{i\theta}$ corresponds to a $2\times 2$ rotation matrix). They are the natural setting for Polynomial Equations via the Fundamental Theorem of Algebra, and they open the door to Complex Analysis, where differentiable complex functions exhibit astonishingly rigid behavior.

🔬 Where this appears

⏳ History

- ~50 AD — Heron of Alexandria encounters

\sqrt{81 - 144}

in a pyramid volume and quietly drops the negative sign, sidestepping the issue.

- 1545 — Gerolamo Cardano publishes Ars Magna, manipulating "sophistic" square roots of negatives (e.g.,

5 \pm \sqrt{-15}

) while distrusting them.

- 1572 — Rafael Bombelli gives the first systematic rules for arithmetic with these quantities, crucially showing they appear in real solutions of cubics (the casus irreducibilis).

- 1637 — René Descartes coins the dismissive term "imaginary."

- 1748 — Leonhard Euler establishes

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

and uses

i

extensively.

- 1799 / 1806 — Caspar Wessel and Jean-Robert Argand independently give the geometric (plane) interpretation, making the numbers concrete.

- 1831 — Carl Friedrich Gauss introduces the term "complex number," firmly legitimizes them, and gives a rigorous proof of the Fundamental Theorem of Algebra.

- 1833 — William Rowan Hamilton defines complex numbers rigorously as ordered pairs

(a, b)

of reals with specific addition and multiplication rules, removing the mystery of

i

entirely.

open-problem

Complex numbers are completely understood as an algebraic structure — yet they open onto some of the deepest unsolved questions in mathematics. The Riemann Hypothesis, perhaps the most famous open problem, concerns the location of the complex zeros of the Riemann zeta function $\zeta(s)$ and asserts they all lie on the line $\operatorname{Re}(s) = \tfrac12$ ; its truth would pin down the distribution of prime numbers. More broadly, while $\mathbb{C}$ is algebraically closed and "complete," extending the rich theory of complex analysis to several complex variables introduces phenomena with no one-variable analogue (e.g., Hartogs' extension), and the geometry of such functions remains an active research frontier.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also