Algebra: What It Is and How Symbolic Notation Works

Where the word «algebra» comes from

The name comes from the Arabic al-jabr, first used in a mathematical sense by Muhammad ibn Musa al-Khwarizmi, a mathematician active in Baghdad in the 9th century, at the House of Wisdom, under the patronage of the caliph Al-Ma’mun (who became caliph in 813). In his treatise, devoted to symbolic notation and the theory of equations, al-jabr named a precise technical operation: “completion,” meaning the move of a term from one side of an equation to the other, along with the sign change that move requires. It’s the same move still made today solving x + 3 = 7, taking the 3 to the other side as −3.

The title of “father of algebra” remains a debated question among historians. According to the historian Gandz, al-Khwarizmi has a stronger claim to it than the Greek mathematician Diophantus, because he was among the first to treat algebra as a discipline in its own right, independent of the number theory in which Diophantus framed it.

Al-Khwarizmi’s own name, garbled through medieval Latin, also gave us the word algorithm: the step-by-step procedures that make computers work.

Symbolic notation, or letters instead of numbers

Symbolic notation is the set of rules and procedures that use letters of the alphabet to stand for general numbers. Because a letter like a or b can equal any number in a given set, the properties and rules derived through symbolic notation carry general validity: they stay true for every numerical value the variables can take.

Worked example: the distributive property is written a · (b + c) = a · b + a · c. It doesn’t matter which numbers you pick for a, b and c — trying 2, 3 and 4 gives 2 · (3 + 4) = 14 and 2 · 3 + 2 · 4 = 14 — the rule still holds. That’s the whole point of symbolic notation: one piece of writing that works for infinite calculations, not a result that holds only once.

Symbolic notation is what defines algebra and is used across every field of mathematics: it’s the language algebra uses to study the theory of equations in particular, the set of rules for solving equations or systems of equations in one or more unknowns.

Arithmetic and algebra: where the difference starts

Arithmetic works with fixed, already-known numbers: 3 + 5 has only one possible result, and it stays that way. Algebra introduces, through symbolic notation, a more general way of representing numbers: the basic operations of arithmetic are handled using letters instead of numbers, which makes it possible to reach solutions that hold regardless of the specific case at hand.

The line between them is simple: arithmetic stays with numbers already known, algebra starts once a number is still to be found.

Equations, analytic geometry and trigonometry: after symbolic notation

Symbolic notation is the foundation several branches of mathematics build on, each deserving a Recap of its own. The study of equations uses symbolic notation to find the value of one or more unknowns. Analytic geometry applies the same letters and the same rules to points, lines and shapes, describing them with numerical coordinates. Trigonometry, finally, uses symbolic formulas to relate the angles and sides of triangles. These are three different applications of the same tool, not three separate branches of mathematics.

Beyond pure mathematics, symbolic notation is also the language physics laws are written in: Newton’s three laws of motion, for instance, are expressed with algebraic formulas like F = m · a, where the letters stand for any force, mass and acceleration, not one specific case. The same language is also at work inside machine learning: a model learns by updating letters that represent its parameters, not numbers fixed in advance.

Elementary algebra and abstract algebra

What’s been described so far is elementary algebra, the same kind taught in school: numbers, letters, equations. A development and generalization of symbolic notation is instead abstract algebra, which examines in more detail the settings — sets of numbers or other objects — where the general laws of operations are laid down as axioms. That’s a later stage of study, typically at university level, and falls outside the scope of this Recap.