Algebraic Techniques: The Language of Generalisation

Algebra replaces specific numbers with letters — called variables — so that a single expression or equation captures a general truth rather than one particular case. Instead of saying '3 + 5 = 8 and 7 + 9 = 16 and …', algebra says a + b = b + a, which is true for every pair of numbers at once. Algebraic techniques include writing and simplifying expressions, expanding brackets, factorising, and substituting values.

Named after a book

The word 'algebra' comes from the title of a 9th-century Arabic text: al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala, written by Persian mathematician Muhammad ibn Musa al-Khwārizmī around 820 AD. Al-jabr — the restoration of balance — gave us the word algebra, and al-Khwārizmī's name gave us the word 'algorithm'. For centuries, algebra was written out in words — no symbols at all. French mathematician François Viète introduced the first systematic use of letters for unknowns in the late 1500s, and Descartes refined the notation into something close to what we use today.

Algebra in the real world

Every formula in science and engineering is an algebraic expression. The formula for speed (v = d/t) lets you calculate time, distance, or speed depending on which you know. Computer programming is built on algebraic thinking — variables store values, and expressions compute new ones. Financial modellers use algebraic equations to project revenues. Engineers write algebraic expressions for stress, load, and strain. Even a simple spreadsheet formula like =A1*B1+C1 is algebra. Understanding how to manipulate expressions and solve equations unlocks the ability to work with quantitative information in almost any field.