What is algebra? Variables, equations and why letters help

Algebra is the part of mathematics that works with numbers you don't know yet, or that can change, by writing them as letters and following rules that are true for every number. Arithmetic answers one sum at a time. Algebra describes the whole pattern, such as "the cost is 40 dollars plus 25 a month", and then lets you rearrange that pattern step by step until the unknown number is found.

In short

• A variable is a letter standing for a number; an equation says two expressions are equal; solving it means finding the numbers that make it true.
• The one move behind most solving is balance: do the same thing to both sides and the equation stays true.
• Algebra works because the rules of arithmetic, such as a × (b + c) = a × b + a × c, hold for every number, so a letter can stand in for any of them.

What does algebra actually do?

Algebra turns a question about a number into a sentence you can reason with. "I'm thinking of a number; double it and add 3 and you get 17" becomes 2x + 3 = 17. Once the question is written that way, it can be untangled mechanically: take 3 from both sides to get 2x = 14, then halve both sides to get x = 7. The letter didn't make the problem harder. It made it something you can hold still and work on.

The second job of algebra is to state things that are true for all numbers at once. Arithmetic can tell you that 3 + 5 = 5 + 3 and that 10 + 2 = 2 + 10, one case at a time. Algebra says a + b = b + a, and that single line covers every pair of numbers there is. A letter in algebra is a placeholder that lets one statement speak for infinitely many sums.

What are variables, expressions and equations?

A variable is a letter, often x, y or n, that stands for a number. Sometimes it is an unknown waiting to be found, as in 2x + 3 = 17. Sometimes it is a quantity that genuinely varies, such as the number of months you keep a phone contract. A constant is a number that stays fixed, like the 3 in 2x + 3. A coefficient is the number multiplying a variable, like the 2.

An expression is a recipe built from numbers, variables and operations: 2x + 3, or 40 + 25m. It has a value once you know the variables, but it makes no claim. An equation joins two expressions with an equals sign and claims they are equal: 2x + 3 = 17. That claim is true for some values of x and false for others, and solving the equation means finding the values that make it true.

The rule that does most of the work is balance. Picture the equation as a pair of scales. Adding, subtracting, multiplying or dividing both sides by the same amount (anything except dividing by zero) keeps the scales level. The Arabic word al-jabr, from which "algebra" comes, names one such move: completing an equation by moving a subtracted term to the other side.

How does algebra solve a real problem?

Here is a choice people actually face. Phone plan A charges a 40 dollar activation fee and then 25 dollars a month. Plan B has no fee and costs 35 dollars a month. Which is cheaper?

Step 1, name the unknown. Let m be the number of months you keep the plan.

Step 2, write each cost as an expression. Plan A costs 40 + 25m. Plan B costs 35m.

Step 3, ask when they cost the same, which gives an equation: 40 + 25m = 35m.

Step 4, balance. Subtract 25m from both sides: 40 = 10m. Divide both sides by 10: m = 4.

Step 5, check and read the answer. At 4 months, plan A costs 40 + 100 = 140 dollars and plan B costs 140 dollars. Before that, B is cheaper; after it, A is.

Months	Plan A: 40 + 25m	Plan B: 35m	Cheaper
1	65	35	B
4	140	140	the same
12	340	420	A
24	640	840	A

The same five steps work whether the fee is 40 dollars or 400, because nothing in them depended on the particular numbers. That is the real gain: a method rather than an answer.

Where did algebra come from?

People solved algebra problems long before anyone wrote letters for them. Babylonian scribes, around 1800 BCE, worked out lengths and areas from clues on clay tablets, problems that today would be quadratic equations. The Greek mathematician Diophantus, around the 3rd century CE, wrote abbreviations for unknowns in his Arithmetica. In India, Brahmagupta set out rules for negative numbers and for zero in 628.

The name comes from Baghdad. Muhammad ibn Musa al-Khwarizmi, who lived from about 780 to about 850 and worked at the House of Wisdom, wrote Hisab al-jabr w'al-muqabala, which, the MacTutor History of Mathematics archive says, can be considered the first book written on algebra. Al-jabr means completion and al-muqabala balancing, the two moves he used to bring equations to standard forms. His own name, Latinized from his book on Indian numerals, gave English the word algorithm. Al-Khwarizmi wrote everything out in words, with no symbols at all.

Symbols came much later. In 1591 the French mathematician François Viète, in his In artem analyticam isagoge, used vowels for unknowns and consonants for known quantities, MacTutor notes. René Descartes, in La Géométrie in 1637, settled the habit still taught today: letters from the start of the alphabet for known amounts, and x, y and z for unknowns.

Why do the same rules work for any number?

Algebra is only trustworthy because a handful of rules hold for every number without exception. Order doesn't matter when adding or multiplying: a + b = b + a and a × b = b × a. Grouping doesn't matter either: (a + b) + c = a + (b + c). And multiplication spreads over addition: a × (b + c) = a × b + a × c. That last one, the distributive law, is why 7 × 13 can be done in your head as 7 × 10 + 7 × 3 = 70 + 21 = 91.

Everything else is built from those rules. Multiply out (a + b) × (a + b) with the distributive law and you get a² + 2ab + b². Check it with a = 3 and b = 4: the left side is 7 × 7 = 49, and the right side is 9 + 24 + 16 = 49. Because each step used only rules true for all numbers, the result is true for all numbers, not just the ones you tested.

The most common misconception is that algebra is arithmetic made harder by letters. It is closer to the opposite: the letters remove detail so the structure shows. A second is that x always hides one secret value. In 40 + 25m, m can be any number of months, and the expression describes how cost changes with it. That second reading, a rule linking one quantity to another, is the step from algebra to a function.

Where does algebra lead?

Nearly all later mathematics is written in algebra. The Pythagorean theorem is an equation, a² + b² = c², that you rearrange to find a missing side. Logarithms are what you need to solve an equation when the unknown sits in an exponent, as in 2^x = 64. Calculus is algebra applied to quantities that change continuously. Outside mathematics, every spreadsheet formula such as =B2*12 + C2 is an algebraic expression, with cells playing the part of variables.

Mathematicians also found that the same rules apply to things other than ordinary numbers: clock arithmetic, rotations of a shape, the matrices used in computer graphics. Studying those rules for their own sake is called abstract algebra, and it is where the subject goes after school.

Check yourself

• Without looking back, explain the difference between an expression and an equation, with an example of each.
• Why does doing the same thing to both sides of an equation keep it true, and which one operation must you never do to both sides?
• In the phone-plan example, what changes and what stays the same if plan A's fee rises to 80 dollars?

Sources

• Al-Khwarizmi, biography — MacTutor History of Mathematics, University of St Andrews
• François Viète, biography — MacTutor History of Mathematics, University of St Andrews
• Solve equations using the subtraction and addition properties of equality — OpenStax, Elementary Algebra 2e, Rice University, 2020
• Algebra — Wikipedia