Algebraic Notation & Vocabulary

Algebraic Notation

What is algebraic notation?

When writing expressions in algebra (as opposed to sums in numbers) there are conventions and symbols that are used that take on a particular meaning

This is what we mean by algebraic notation

In number work, for adding and subtracting, we use + and –

In algebra, we still do!

Examples:
a + b
c + d – e

However for multiplication, no symbol is used, and for division, fractions are used

Examples:
ab (means a × b)
$\frac{a}{b}$ (means a ÷ b)
3ab (means 3 × a × b)

We can have combinations of these of course

Examples:
$a b + \frac{c}{3}$ (means a×b + c÷3)
The order of operations still apply in algebra so a×b and c÷d would happen before the addition

Powers (indices) and roots are used the same way as with numbers

Examples:
a² means a × a
4a² means 4 × a²
With the order of operations, a² will happen before multiplying by 4

Brackets also work in the same way as with numbers

Examples
$3 (a + b)$ means $3 \times (a + b)$ but with brackets taking priority, when known, $a + b$ would be worked out first, then we would multiply by 3
$5 x (2 x + 3)$ means $5 \times x \times (2 \times x + 3)$ , again with the brackets worked out first

How will I need to use algebraic notation?

Algebraic expressions are used in many parts of the course
- You need to be able to understand their meaning and work with them
  - e.g. for rearranging a formula
- You may be given a situation in words that you then have to write in algebra
  - Which could lead to an equation you may then have to solve

Worked example

At the start of a competition, Raheem has p conkers and Howard has 2 q conkers.
At the end of the competition Raheem still has the same number of conkers he started with but Howard won 6 and lost none.

(a)

Write down an expression for the number of conkers that Howard has at the end of the competition.

Howard has 2 q + 6

At the end of the competition, Raheem and Howard have a total of 40 conkers.

(b)

Write an equation in terms of p and q that shows the number of conkers Raheem and Howard have in total at the end of the competition.

p + 2 q + 6 = 40

This can be simplified to p + 2 q = 34

Algebraic Vocabulary

You need to know the meanings of the word term and factor, as they are the basic building blocks in algebra.

You need to know the differences between an expression, equation, formula, identity and inequality in order to fully understand algebra and proofs. You may be asked to identify which is which.

What is a term?

A term is either…

…a letter (variable) on its own, e.g. x
..a number on its own, e.g. 20
…a number multiplied by a letter, e.g. 5x

The number in front of a letter is called a coefficient

The coefficient of x in the term 6x is 6
The coefficient of y in the term -5y is -5

Terms that are just numbers (with no letters) are called constants
Terms can include powers and more than one letter

E.g. 6xy, 4x², ab³c, …

What is a factor?

A factor is any number or letter that divides a term exactly (with no remainder)

E.g. all the factors of 4xy are 1, 2, 4, x, 2x, 4x, y, 2y, 4y, xy, 2xy and 4xy

A term can be separated into factors that multiply together to give that term

E.g. two factors of 5x are 5 and x

To factorise means to write something as a multiplication of factors
A common factor is one that divides both terms

E.g. the common factors of 6xy and 4x are 2, x and 2x

The highest (or greatest) common factor is 2x

What is an expression?

An expression is an algebraic statement that does not have an equals sign

There is nothing to solve

An expression is made by adding, subtracting, multiplying or dividing terms

E.g. 2x + 5y, b² – 2cd, $\frac{6 y}{5 t}$ , …
A single term can be an expression

Expressions can be simplified (made easier)

E.g. x + x + x simplifies to 3x

What is an equation?

An equation is an algebraic statement with an equals sign between a left-hand side and a right-hand side

Both sides are equal in value
E.g. if 2x has the same value as 10, then 2x = 10

An equation can be solved by finding the missing values of the letters that make the left-hand side equal to the right-hand side

E.g. the equation 2x = 10 is solved by x = 5

x = 5 is called the solution

What is a formula?

A formula is a worded rule, definition or relationship between different quantities, written in shorthand using letters

E.g. weight, w, is mass, m, multiplied by gravitational acceleration, g

The formula is w = mg

It is common to substitute numbers into a formula, but a formula on its own cannot be solved
To turn a formula into an equation, more information is needed

E.g. In the formula w = mg, if w = 50 and m = 5 then the equation 50 = 5g can be formed

What is an identity?

An identity is an algebraic statement with an identity sign, ≡, between a left-hand side and a right-hand side that is true for all values of x

E.g. x + x ≡ 2x
This means x + x is identical to 2x, or that x + x can also be written as 2x

An identity cannot be solved
All numbers can be substituted into an identity and it will remain true

E.g. x + x ≡ 2x is true for x = 1, x = 2, x = 3 … (even x = -0.01, x = π etc)
Unlike with equations, where only the solutions work

E.g. 2x = 10 is not true for x = 1, x = 2, x = 3 … only for x = 5

Identities can be used to write algebraic expressions in different forms

E.g. find p and q if 3(x + y) + 2y ≡ px + qy

3(x + y) expands to 3x + 3y
The coefficient of x on the left is 3 and on the right is p, so p = 3
The coefficient of y on the left is 3 + 2 and on the right is q, so q = 5
Therefore 3(x + y) + 2y is identical to 3x + 5y
This method is called equating coefficients

What is an inequality?

An inequality compares a left-hand side to a right-hand side and states which one is bigger

x > y means x is greater than y
x ≥ y means x is greater than, or equal to, y
x < y means x is less than y
x ≤ y means x is less than, or equal to, y

E.g. x ≥ 8 means x can take any value that is greater than, or equal to, 8

This is the same as saying “8 or more”, or "at least 8"

The solutions of inequalities are usually, themselves, inequalities

x + 10 < 15 solves to give x < 5, so x is any number less than 5

Exam Tip

To fully understand the wording of an exam question you need to know the difference between an expression, equation, formula, identity and inequality.

Worked example

(a)

Write down the expression from the list below:

2x + 5 = 4 3x + 2x ≡ 5x 7x – 9 x = vt – w 4x – 1 ≥ 0

An expression does not have an equals, identity or inequality sign

7x – 9 is the expression

(b)

For how many values of x is the statement x² – 1 ≡ (x + 1)(x – 1) true?

no values of x two values of x (x = 1 and x = -1) all values of x

This is an identity (due to the ≡ symbol)
An identity is true for all values of x

all values of x

(c)

Find the whole numbers a and b such that 5(x – 2y) + ax + 3y ≡ 9x + by.

Expand the brackets (by multiplying x and -2y by 5)

5x – 10y + ax + 3y ≡ 9x + by

Both sides must be identical
Equate the coefficients of x (by setting the number of x’s on the left-hand side equal to the number of x’s on the right-hand side)

5 + a = 9

Solve this equation (by subtracting 5 from both sides)

a = 9 – 5

a = 4

Equate the coefficients of y (by setting the number of y’s on the left-hand side equal to the number of y’s on the right-hand side)

-10 + 3 = b

Solve this equation (by adding 3 to -10)

b = -7