Atlas / Mathematics / The Algebra Thread

Field · Emerged c. 820 – 1799

Theory of Equations

How can an unknown quantity be found from an equation, and is there always a formula for it?

5 chapters3 min read5 turning points 1 open problem

Branched from: Root of the thread
Branched into: Complex Analysis + Galois Theory
Figures: Muḥammad ibn Mūsā al-Khwārizmī, Gerolamo Cardano, Scipione del Ferro, Niccolò Tartaglia, Lodovico Ferrari, François Viète, Carl Friedrich Gauss

In brief

The theory of equations is algebra's original question: given a relation like $x^2 + 10x = 39$ , find the unknown. For quadratic equations there is a formula, known in essence to the Babylonians. For cubics and quartics, formulas were found in sixteenth-century Italy amid secrecy, public contests and a broken oath. For the quintic, a formula was sought for another two and a half centuries.

Along the way the subject built the tools of modern algebra. It gave a name ("al-jabr"), symbolic notation with letters for unknowns, and complex numbers, forced into existence by the cubic formula. It ended with a theorem guaranteeing that every polynomial equation has as many roots as its degree.

Key ideas

Equation and unknownEnters c. 820: A statement of equality involving a quantity to be found. Solving means isolating the unknown by operations that keep both sides equal.
Al-jabrEnters c. 820: "Restoration": moving a subtracted term to the other side of an equation to make it positive. Al-Khwārizmī's name for the operation became the name of the subject, algebra.
Solution by radicalsEnters 1515 – 1545: A formula for the roots built from the coefficients using only arithmetic and $n$ th roots, like the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
Symbolic notationEnters 1591: Letters standing for both unknowns and known coefficients, so that whole families of equations can be solved at once. It came from Viète, and in modern form from Descartes.
Fundamental theorem of algebraEnters 1799: Every polynomial of degree $n$ has exactly $n$ roots in the complex numbers, counted with multiplicity. Solutions always exist, even when no formula finds them.

Chapter I

Recipes on Clay

The oldest algebra is nearly four thousand years old. Old Babylonian scribes posed problems like adding the area and the side of a square to get three-quarters, and solved them with fixed procedures: halve this, square that, add, take the square root. These are the steps of completing the square, stated as recipes without symbols. Greek mathematicians treated such problems geometrically, as rectangles and squares of unknown size.

Chapter II

Al-Jabr

Around 820, at the House of Wisdom in Baghdad, Muḥammad ibn Mūsā al-Khwārizmī wrote a short book that made equation-solving a discipline. He sorted linear and quadratic equations into six standard types and gave a general method, with a geometric proof, for each. His operation al-jabr, "restoring" a subtracted quantity by moving it to the other side, gave the subject its name. His own name, Latinised, gave us "algorithm". The Persian poet-mathematician Omar Khayyam later solved cubic equations geometrically, by intersecting conic sections, and said an algebraic formula might be found by others.

Chapter III

Duels in Italy

It was found in sixteenth-century Italy, where mathematicians made their reputations in public problem-solving contests and guarded their methods. Around 1515 Scipione del Ferro found a formula for one kind of cubic and told only a few people, among them a student. In 1535 Niccolò Tartaglia rediscovered it and won a contest with it. Gerolamo Cardano persuaded Tartaglia to reveal it under a solemn oath of secrecy. Then, having seen del Ferro's older notes, he published it in Ars Magna (1545), with his student Lodovico Ferrari's solution of the quartic. Tartaglia never forgave him.

The formula had a strange feature. For some cubics with three real roots it passes through square roots of negative numbers. Working with those "impossible" numbers was the start of complex analysis.

Chapter IV

A Closer Look: Bombelli's Impossible Numbers

Cardano's formula solves the cubic $x^3 = px + q$ by

x = \sqrt[3]{\frac{q}{2} + \sqrt{\frac{q^2}{4} - \frac{p^3}{27}}} \;+\; \sqrt[3]{\frac{q}{2} - \sqrt{\frac{q^2}{4} - \frac{p^3}{27}}} .

Apply it, as Rafael Bombelli did in 1572, to $x^3 = 15x + 4$ . The obvious solution is $x = 4$ , since $64 = 60 + 4$ . But the formula gives

\frac{q^2}{4} - \frac{p^3}{27} = 4 - 125 = -121, \qquad x = \sqrt[3]{2 + \sqrt{-121}} + \sqrt[3]{2 - \sqrt{-121}} .

A perfectly real equation with a perfectly real answer leads straight through the square root of a negative number. Such cases came to be called "irreducible", and Cardano set them aside. Bombelli decided to calculate anyway, treating $\sqrt{-1}$ as a new kind of number with $(\sqrt{-1})^2 = -1$ . Writing $i$ for it, $\sqrt{-121} = 11i$ . He guessed that the cube roots might have the form $2 \pm i$ , and checked:

(2 + i)^3 = 8 + 12i + 6i^2 + i^3 = 8 + 12i - 6 - i = 2 + 11i .

So $\sqrt[3]{2 + 11i} = 2 + i$ and $\sqrt[3]{2 - 11i} = 2 - i$ , and the formula gives $x = (2 + i) + (2 - i) = 4$ . The imaginary parts cancel and the true answer appears.

It was one of the first times anyone had computed with complex numbers and got a meaningful result. Bombelli could not say what these numbers were, only that the rules worked. It took another two centuries, and the picture of numbers as points in a plane, before complex analysis could say.

Chapter V

Symbols and Existence

Notation turned recipes into theory. François Viète in 1591 used letters for known quantities as well as unknowns, so that one equation could stand for every problem of its type. Descartes, in 1637, gave the modern convention of $x$ and $y$ .

Two questions remained. Does every equation have a solution? Gauss's 1799 thesis proved that every polynomial equation has a root among the complex numbers, the fundamental theorem of algebra. And can every equation be solved by a formula? For degree five, the best mathematicians of the eighteenth century tried and failed. Understanding why is Galois theory.

Turning points5

c. 1800 BCEREFORMULATION
Babylonian scribes solve quadratic problems
Clay tablets from Old Babylonian scribal schools pose and solve problems equivalent to quadratic equations, such as finding a rectangle's sides from its area and the difference of its sides. They use a step-by-step procedure that amounts to completing the square, and state it as a recipe, without symbols.
› Sources (1)
- Høyrup, J. (2002). Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin. Springer.
c. 820REFORMULATION
Al-Khwārizmī's book of al-jabr
At the House of Wisdom in Baghdad, Muḥammad ibn Mūsā al-Khwārizmī writes a manual classifying linear and quadratic equations into standard forms and giving general methods, with geometric proofs, for solving each. It is algebra as a discipline in its own right. "Algebra" comes from its title and "algorithm" from his name.
Muḥammad ibn Mūsā al-Khwārizmī
› Sources (1)
- Rashed, R. (2009). Al-Khwārizmī: The Beginnings of Algebra. Saqi Books.
1515 – 1545PROOFContested
The cubic and quartic are solved
Scipione del Ferro finds a formula for cubic equations around 1515 and keeps it secret. Niccolò Tartaglia rediscovers it in 1535 to win a public contest. Gerolamo Cardano coaxes it from him under an oath of secrecy, then publishes it, with Lodovico Ferrari's solution of the quartic, in his Ars Magna (1545). The formula sometimes requires square roots of negative numbers even when every root is real.
What is disputed
One of the most famous priority quarrels in mathematics. Cardano credited del Ferro and Tartaglia in the book, but he had sworn to Tartaglia never to publish the method. He justified breaking the oath by having seen del Ferro's earlier papers. Tartaglia denounced him, and Ferrari defended Cardano in a series of public challenges. The formula has been called "Cardano's formula" ever since, a naming many regard as unfair.
Gerolamo Cardano, Scipione del Ferro, Niccolò Tartaglia, Lodovico Ferrari
› Sources (2)
- Cardano, G. (1545). Ars Magna. Nuremberg.
- Stedall, J. (2011). From Cardano's Great Art to Lagrange's Reflections: Filling a Gap in the History of Algebra. European Mathematical Society.
1591REFORMULATION
Viète puts letters for the known quantities
François Viète uses vowels for unknowns and consonants for known quantities, so that an equation can stand for an entire family of problems and be solved in general. Descartes' convention of $x, y, z$ for unknowns and $a, b, c$ for constants followed in 1637. Algebra becomes a language.
François Viète
› Sources (1)
- Viète, F. (1591). In artem analyticem isagoge. Tours.
1799PROOFContested
The fundamental theorem of algebra
In his doctoral thesis, Carl Friedrich Gauss gives the first substantial proof that every polynomial with real coefficients factors into linear and quadratic pieces, so every polynomial equation has a complex root. Existence was guaranteed. Finding the roots by formula was another matter.
What is disputed
Priority and rigour are both qualified. d'Alembert (1746), Euler and Lagrange had given attempted proofs, which Gauss criticised for assuming what they needed. Gauss's own 1799 proof relied on an unproved topological fact about curves, only filled in by Ostrowski in 1920. Argand's 1806/1814 proof is sometimes counted as the first essentially complete one.
Carl Friedrich Gauss
› Sources (2)
- Gauss, C. F. (1799). Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse. Helmstedt.
- Fine, B. & Rosenberger, G. (1997). The Fundamental Theorem of Algebra. Springer.

Applications

Where it is used

Engineering
Stability from the roots of a polynomial
Whether a machine, circuit or aircraft control system settles down or oscillates out of control depends on the roots of its characteristic polynomial. Routh (1877) and Hurwitz (1895) found tests that decide from the coefficients alone whether every root lies in the stable half of the complex plane.
› Sources (1)
- Routh, E. J. (1877). A Treatise on the Stability of a Given State of Motion. Macmillan.

Open problems

Where the map runs out

Open

Hilbert's thirteenth problem

The continuous version was solved by Kolmogorov and Arnold (1957); the algebraic version is open as of 2026.

Hilbert asked whether the roots of the general degree-7 equation can be written using only functions of two variables. More broadly: how many variables do you really need to express the solution of an equation? After clever substitutions the general septic depends on three parameters, and the question is whether it can be done with fewer.

Why it is hard: For continuous functions Kolmogorov and Arnold showed, surprisingly, that two variables always suffice, but their functions are wild. For algebraic functions, the natural setting, no degree is yet known to need algebraic functions of more than one variable, even though the septic seems to need three. The modern theory of "resolvent degree" reformulates the question geometrically but has not answered it.
What resolving it unlocks: A precise measure of how complex the solutions of polynomial equations really are, beyond "solvable by radicals or not".

› Sources (1)

Farb, B. & Wolfson, J. (2019). Resolvent degree, Hilbert's 13th problem and geometry. L'Enseignement Mathématique 65: 303–376.