Algebraic Notation and Vocabulary

In algebra, letters stand for numbers. Algebra has its own short way of writing calculations, and its own words for the different parts of what you write. This page covers both.

In the exam you are usually asked to simplify something like 6 \times r \times s, or to write y \times y \times y using a power, for 1 mark. You may also be asked to write an expression or a formula from words, for 1 to 3 marks. Now and then a question gives you a box of words, such as expression, equation, formula, identity and term, and asks you to choose the right word.

The words matter in other questions too. If a question asks for an expression and you write a formula such as T = 5n, you can lose a mark.

In algebra, leave out the times sign. Write the number first, then the letter: 3 \times t is written 3t, and t \times 3 is also 3t. A letter on its own needs no 1 in front: 1 \times t is just t

Letters multiplied together are written side by side, usually in alphabetical order: m \times n = mn. If there is more than one number, multiply the numbers together first: 2 \times m \times 4 = 8m

Division is written as a fraction: k \div 6 is written \frac{k}{6}. The order matters, because 6 \div k is \frac{6}{k}, which is different. A fraction line keeps the whole top together, so (x + 5) \div 3 is \frac{x + 5}{3}

To divide a term such as 20t by a number, divide the number part: 20t \div 5 = 4t, because 20 \div 5 = 4

When a letter is multiplied by itself, use a power. y \times y = y^2, said y squared, and y \times y \times y = y^3, said y cubed. The power shows how many copies of y are multiplied together.

Don't mix this up with adding. y + y + y = 3y, but y \times y \times y = y^3

With numbers and different letters, multiply the numbers, then write the letters with their powers: 2x \times 3y = 6xy, and p \times p \times q = p^2q

A power belongs only to the letter or number just before it. 3n^2 means 3 \times n \times n: only the n is squared. To square all of 3n you need a bracket: (3n)^2 means 3n \times 3n

A term is one part of an expression. Terms are separated by + and - signs, and each sign belongs to the term after it. So 8p - 3q + 6 has three terms: 8p, -3q and +6

Like terms have exactly the same letters, with the same powers, such as 5x and 2x. To add or subtract like terms, add or subtract the numbers in front of the letters, called the coefficients: 5x + 2x = 7x, and 4k - k + 2k = 5k, because k on its own means 1k. Terms with different letters, such as 3x and 4y, are not like terms, so 3x + 4y can't be simplified. It is not 7xy

Things that are multiplied together are called factors. In 5(x + 2) the factors are 5 and (x + 2), and in 3pq the factors include 3, p and q.

An expression is one or more terms with no equals sign, such as 4n + 1. It doesn't have one answer: its value changes as n changes.

An equation has an equals sign, and it is true only for certain values of the letter. 4n + 1 = 13 is true only when n = 3. Finding that value is called solving the equation.

A formula is a rule that links two or more quantities, and each letter stands for something. In P = 4s, P is the perimeter of a square and s is its side length. The letter on its own, here P, is the subject of the formula.

An identity is true for every value of the letter. 3(y + 2) = 3y + 6 is an identity, because expanding the bracket on the left gives exactly the right side.

If you are not sure whether something is an identity, try a number for the letter. For 3(y + 2) = 3y + 2, try y = 1. The left side is 3 \times 3 = 9 but the right side is 3 + 2 = 5. The two sides are different, so it is not an identity.

An inequality uses one of these signs instead of an equals sign.

• t > 7 means t is greater than 7.
• t \geq 7 means t is greater than or equal to 7, so t is at least 7.
• t < 7 means t is less than 7.
• t \leq 7 means t is less than or equal to 7, so t is at most 7.

• Leaving a times sign in the answer. 4 \times pq and 2r \times 3s are not finished: write 4pq and 6rs.
• Writing 3y or y3 for y \times y \times y. Multiplying a letter by itself gives a power, y^3, with a small 3 written up high. 3y means y + y + y.
• Squaring the number when only the letter is squared. 3n^2 means 3 \times n^2, so when n = 5 it is 75, not 225.
• Simplifying terms that are not like terms. 6x + 7y is already finished: it is not 13xy.
• Writing a formula when the question asks for an expression. Give just the algebra, such as 4n + 3, not C = 4n + 3.
• Mixing up an equation and an identity. An equation is true for only some values of the letter; an identity is true for every value.