Algebra uses letters to stand for numbers, with some shorthand. The × sign is left out and the number is written first, so 5 × n is 5n. Repeated multiplication is written as a power, so a × a = a², and division is written as a fraction. An expression is made of terms; the number in front of a letter is its coefficient, and like terms have exactly the same letters and powers.
Notation rules
- a × b = ab; 5 × n = 5n (number first)
- a × a = a²; a × a × a = a³
- a ÷ b = a/b
- a + a + a = 3a
- Like terms: same letters, same powers (3x² and 5x² are like terms; 3x² and 5x are not)
Writing multiplication
In algebra the × sign is left out. a × b is written ab, and 5 × n is written 5n, with the number first. Adding a letter to itself repeatedly gives a multiple: n + n + n = 3n. Letters are usually written in alphabetical order, so 3 × b × a is 3ab.
Powers and division
Multiplying a letter by itself is written as a power: y × y = y² (“y squared”) and y × y × y = y³. Be careful to tell 2y (y + y) apart from y² (y × y). Division is written as a fraction: a ÷ 4 = a/4, and (n − 3) ÷ 2 = (n − 3)/2, with brackets showing that the whole of n − 3 is divided.
Terms, coefficients and expressions
An expression is a collection of terms joined by + or −, such as 4x + 3y − 7. In the term 4x, the number 4 is the coefficient and x is the variable. A term with no letter, such as −7, is a constant. An expression has no equals sign; an equation does.
Like terms
Like terms have exactly the same letters raised to the same powers, so only the coefficients differ: 5y and −2y are like terms, as are 3ab and ab. 4x and 4x² are not like terms, and neither are 2a and 2b. Only like terms can be added or subtracted.
Writing expressions from words
Translate words one operation at a time. “Three more than n” is n + 3; “n multiplied by itself, then by 4” is 4n²; “subtract 3 from n, then halve the result” is (n − 3)/2. Read the order carefully: “double n then add 5” is 2n + 5, but “add 5 then double” is 2(n + 5).
Worked examples
Example 1. Write “a multiplied by 9” using algebra.
- Leave out the × sign and put the number first.
Answer: 9a.
Example 2. Write y × y using a power.
- y multiplied by itself is y squared.
Answer: y².
Example 3. Write 4 × p × p × q using algebra.
- The number goes first: 4.
- p × p = p², then q.
Answer: 4p²q.
Example 4. Write “subtract 3 from n, then halve the result” as an expression.
- Subtract first: n − 3.
- Then halve the whole result.
Answer: (n − 3)/2.
Example 5. From the list 7x, 2y, x², 7, 3x, write down the like terms.
- Like terms have the same letter and the same power.
- 7x and 3x both contain x to the power 1.
Answer: 7x and 3x.
Common mistakes
- Writing the number after the letter. Write 5n, not n5.
- Confusing 2y and y². 2y = y + y, but y² = y × y.
- Treating x and x² as like terms. The powers must match as well as the letters.
- Missing brackets in division. (n − 3) ÷ 2 is (n − 3)/2, not n − 3/2.
- Getting the order of operations wrong in words. “Double then add 5” is 2n + 5; “add 5 then double” is 2(n + 5).
How algebraic notation is examined
Notation questions are among the first on Foundation papers for Edexcel GCSE (1MA1) and the 4MA1 International GCSE, and every later algebra topic depends on them. Expect to:
- write products, powers and divisions using correct notation
- identify the coefficient of a term or a like term
- write an expression from a description in words
- explain the difference between an expression, an equation and a formula
Practise with the notation and substitution worksheet, which has answers, or use the GCSE algebra revision engine.
Next, see simplifying expressions and substitution.
Frequently asked questions
Why is the × sign left out in algebra?
To avoid confusing it with the letter x, and because it is shorter. ab always means a × b.
What is a coefficient?
The number multiplying a letter in a term. In 7x, the coefficient is 7.
What are like terms?
Terms with exactly the same letters and powers, such as 4a² and −a².