Algebra underlies arithmetic classes, engineering formulas and computer code alike, yet many people encounter the word only as a school subject without a clear sense of what it does. At its core, algebra is a way of writing general rules about numbers using symbols instead of writing out every specific case.
What algebra actually is
According to the educational site Mathsisfun, algebra is a branch of mathematics that uses letters such as x or y as variables standing in for numbers that are unknown or that can take many values, together with rules for how expressions and equations built from those symbols can be combined and changed.
Mathsisfun's glossary of basic terms defines the building blocks that make up any algebraic statement. A variable is a symbol, usually a letter such as x or y, for a number that is not yet known. A constant is a fixed number on its own, with no variable attached, such as the −7 in the expression 4x − 7. A coefficient is a number used to multiply a variable — the 4 in 4x, for instance — and a term is a single number or variable, or a product of numbers and variables, while an expression is a group of terms joined by plus or minus signs.
An equation adds one more ingredient: an equals sign. Per Mathsisfun, an equation states that two expressions are equal, meaning the value on the left side of the sign must match the value on the right side, as in x + 2 = 6. Solving an equation means finding which value of the variable makes that statement true.
Where the word 'algebra' came from
The word itself traces to 9th-century Baghdad. According to the mathematics education site Mathigon, the Persian scholar Muhammad al-Khwarizmi (780–850) worked at the House of Wisdom and wrote a book known in English as The Compendious Book on Calculation by Completion and Balancing. That book showed how to solve linear and quadratic equations and became the main mathematics textbook used in European universities for centuries, per Mathigon.
Mathigon states that the term 'algebra' derives directly from the Arabic title of al-Khwarizmi's book, and that the word 'algorithm' likewise derives from his name. For this reason he is often described as the father of algebra. The University of St Andrews' MacTutor History of Mathematics archive, a resource maintained by mathematicians John O'Connor and Edmund Robertson, catalogs related historical topics including the solving of quadratic, cubic and quartic equations and the later 16th-century dispute between mathematicians Niccolò Tartaglia and Gerolamo Cardano over cubic-equation methods, without providing further narrative detail on its index page.
Types of equations and what distinguishes them
Not every equation is solved the same way. A linear equation involves a variable raised only to the first power (such as x + 2 = 6), while a quadratic equation includes a variable raised to the second power (such as x² in the equation). Al-Khwarizmi's book, according to Mathigon, showed how to solve both linear and quadratic equations using systematic rules.
Basic algebra vocabulary| Term | Definition (per Mathsisfun) | Example |
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| Variable | A symbol for a number not yet known | x, y |
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| Constant | A fixed number with no symbol attached | −7 in 4x − 7 |
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| Coefficient | A number that multiplies a variable | 4 in 4x |
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| Term | A number, a variable, or their product | 4x, or 7 |
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| Expression | Terms joined by + or − signs, no equals sign | 4x − 7 |
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| Equation | A statement that two expressions are equal | x + 2 = 6 |
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A four-step method for solving problems
Beyond vocabulary, mathematics education also relies on general strategies for approaching problems, not just equations. The most widely cited framework was set out by Hungarian mathematician George Pólya in his book How to Solve It. A review of the book published in The Mathematical Gazette in 1946, accessed via Cambridge University Press, records that the first edition ran to 204 pages, cost 16 shillings and 6 pence, and was published in 1945 by Princeton University Press together with Humphrey Milford.
Pólya's method, as laid out on a University of Kentucky mathematics course page and on a Mathematics LibreTexts resource, consists of four steps. The first is to understand the problem: reading it carefully, restating it in one's own words, and identifying what is given and what is being asked. The second is to devise a plan, choosing from strategies such as drawing a picture, making a list, looking for a pattern, introducing a variable, or breaking the problem into smaller parts.
The third step is to carry out the plan, working through the chosen method and reconsidering the plan if it does not lead anywhere. The fourth step, looking back, means checking whether the result is reasonable, verifying the answer, and considering whether a simpler method existed or whether the problem could be generalized.
- Understand the problem — identify what is given and what is being asked.
- Devise a plan — pick a strategy such as using a variable, drawing a diagram, or finding a pattern.
- Carry out the plan — apply the chosen method step by step.
- Look back — check the answer for reasonableness and consider simpler or more general solutions.
The Kentucky course page and LibreTexts resource both present this sequence as a general-purpose approach rather than a formula limited to any one type of equation, which is why it continues to appear in mathematics teacher training material today.
Both historical threads describe systematic procedures for determining an unknown value: al-Khwarizmi's 9th-century rules for completing and balancing equations, and Pólya's 20th-century four-step process, each offer a repeatable method rather than a one-off trick for turning an unknown quantity into a known one.
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