The Babylonian quadratic equations

Quadratic equations are among the oldest problems in mathematics: they were already being solved by the scribes of Mesopotamia almost four thousand years ago, many centuries before algebraic symbols existed. On clay tablets from the Old Babylonian period (roughly 1800–1600 BC), written in sexagesimal (base-60) numeration, we find problems such as: “I added the area and the side of a square and obtained $\tfrac{3}{4}$; what is the side?” In modern notation this is the equation $x^2 + x = \tfrac{3}{4}$.

A recipe without formulae

The scribes stated neither a general formula nor used letters: they gave a procedure, a list of operations to carry out on the numbers of the problem — precisely what we would today call an algorithm. For the equation $x^2 + x = \tfrac{3}{4}$ the recipe ran like this: take half of $1$ (the coefficient of the side), that is $\tfrac{1}{2}$; square it, obtaining $\tfrac{1}{4}$; add it to $\tfrac{3}{4}$, obtaining $1$; take its square root, that is $1$; finally subtract the half from before, $\tfrac{1}{2}$. The result is $x = 1 - \tfrac{1}{2} = \tfrac{1}{2}$.

If we retrace those steps with today’s symbols we see that it is exactly completing the square: $x^2 + x = \tfrac{3}{4} \;\Rightarrow\; \left(x + \tfrac{1}{2}\right)^2 = \tfrac{3}{4} + \tfrac{1}{4} = 1 \;\Rightarrow\; x + \tfrac{1}{2} = 1.$

Moreover, many Babylonian problems are posed in the form “I know the sum and the product of two numbers, find them”: a system that, by substitution, reduces to a quadratic. It is the same scheme we shall meet again in symmetric systems and in the formulae for the sum and product of the roots.

From the Babylonians to al-Khwārizmī

The Babylonians thus knew the method, but applied it case by case, always on concrete numbers and taking only one root (never the negative one, which for them would have made no sense, since these were lengths). The step towards a general theory of quadratic equations — with a classification of types and a geometric justification of each recipe — comes much later, around the year 820, with the work of al-Khwārizmī, the founding text of algebra. The quadratic formula with the square root of the discriminant, which we learn today in a single stroke, is the modern symbolic version of a technique that humankind had been using, step by step, for almost four millennia.

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Topics: Quadratic equations
Concepts: Completing the square · Quadratic equation
People: al-Khwārizmī