1.2.1 Algebraic vocabulary and notation

What you'll learn

How to distinguish expressions, equations, identities, formulae and inequalities.
How to identify terms, factors, coefficients, constants and variables.
How standard algebraic notation communicates operations, functions and sets of values.
How precise vocabulary helps you explain reasoning and avoid ambiguous working.

The building blocks of algebra

Algebra uses symbols to represent numbers and relationships. A letter may represent an unknown number, a quantity that can vary, or an arbitrary number chosen from a particular set.

Variables and constants

Definition

Variables and constants

A variable is a symbol, usually a letter, whose value can change or is not yet known. A constant is a fixed value.

In the expression $3 x + 7$ :

$x$ is a variable.
7 is a constant.
3 is the numerical multiplier of $x$ .

The same letter must represent the same value throughout a single argument unless it is explicitly redefined. Different letters can have equal values, but they do not have to.

Terms and coefficients

Definition

Terms and coefficients

A term is a part of an algebraic expression separated from other terms by addition or subtraction. A coefficient is a factor multiplying a variable or a product of variables.

For example, the expression

5x^2-3xy+y-8

has four terms: $5x^2$ , $- 3 x y$ , $y$ and $- 8$ .

Their coefficients are:

5 for the $x^2$ term;
$- 3$ for the $x y$ term;
1 for the $y$ term.

The final term, $- 8$ , is a constant term because it contains no variable.

Common Mistake

Keep the sign with the term

In $4x^2-7x+1$ , the second term is $- 7 x$ , not $7 x$ . Its coefficient is therefore $- 7$ .

Factors and powers

Definition

Factors

A factor is a number or algebraic expression multiplied by another quantity. To factorise an expression is to write it as a product of factors.

In $6x^2y$ :

6, $x^2$ and $y$ are factors;
$x$ is also a factor because $x2=x⋅xx^2=x\cdot x$ ;
the exponent, or index, 2 tells you how many factors of $x$ are multiplied together.

In $(x - 3) (x + 4)$ , the expressions $x - 3$ and $x + 4$ are factors.

Example

Describing an algebraic expression

Consider $7a^3b-2ab^2+5$ .

Separate at addition and subtraction signs, keeping each sign attached to its term. The terms are $7a^3b$ , $2ab^2$ and 5.
Identify the numerical factors multiplying the variables. The coefficients of the variable terms are 7 and $- 2$ .
Identify the term containing no variables. The constant term is 5.
Read the indices: $a^3$ means $a⋅a⋅aa\cdot a\cdot a$ , while $b^2$ means $b⋅bb\cdot b$ .

Expressions and mathematical statements

Expressions

Definition

Algebraic expression

An expression is a combination of numbers, variables and operations. It does not contain an equality or inequality sign and therefore does not make a claim that can be true or false.

Examples include:

$2x^2-5x+1$
$x+3x−1\frac{x+3}{x-1}$
$y+4\sqrt{y+4}$

You can simplify, expand, factorise or evaluate an expression. You do not “solve” an expression because it does not state a condition on the variable.

To evaluate an expression means to calculate its value for specified values of its variables.

Example

Evaluating an expression

Evaluate $2p^2-3pq$ when $p = - 2$ and $q = 5$ .

Substitute the given values using brackets: $2(-2)^2-3(-2)(5)$ .
Evaluate the power before multiplying: $2)^2=4$ , so the expression becomes $2 (4) - 3 (- 2) (5)$ .
Complete the multiplications and addition: $8 - (- 30) = 38$ .

Common Mistake

Use brackets when substituting negatives

If $x = - 3$ , then $x^2=(-3)^2=9$ . Writing $3^2$ without brackets means $3^2)=-9$ because powers are evaluated before the leading negative sign.

Equations

Definition

Equation

An equation states that two expressions are equal. A solution is a value, or set of values, that makes the equation true.

For example, $2 x + 3 = 11$ is true only when $x = 4$ . The set of all permitted input values being considered is called the domain, and the values satisfying the equation form its solution set.

The left-hand side and right-hand side of an equation are often abbreviated to LHS and RHS.

Identities

Definition

Identity

An identity is an equality that is true for every permitted value of its variable or variables. The symbol $≡\equiv$ may be used to show that two expressions are identically equal.

For example,

(x+2)2≡x2+4x+4.(x+2)^2\equiv x^2+4x+4.

This is true for every real value of $x$ . By contrast, $x+2)^2=9$ is an equation that is true only for particular values of $x$ .

Example

Distinguishing an equation from an identity

Decide whether $3 (x - 2) + 4 = 3 x - 2$ is an identity.

Expand the left-hand side: $3 (x - 2) + 4 = 3 x - 6 + 4$ .
Collect the constant terms to obtain $3 x - 2$ .
The simplified left-hand side equals the right-hand side for every value of $x$ , so the statement is an identity: $3(x−2)+4≡3x−23(x-2)+4\equiv3x-2$ .

Key Idea

Equality and identity

An equation imposes a condition that may hold only for certain values. An identity records two equivalent forms of the same expression and holds throughout the permitted domain.

Formulae

Definition

Formula

A formula is an equation that expresses a general relationship between quantities. One variable may be described as the subject when it appears alone on one side.

For example, $A=πr2A=\pi r^2$ gives the area $A$ of a circle in terms of its radius $r$ . Here, $A$ is the subject.

Changing the subject rearranges the relationship without changing its meaning. From $A=πr2A=\pi r^2$ , with $r≥0r\geq0$ ,

r=Aπ.r=\sqrt{\frac{A}{\pi}}.

Inequalities

Definition

Inequality

An inequality compares quantities that are not necessarily equal. The symbols $<$ and $>$ mean “less than” and “greater than”; $≤\leq$ and $≥\geq$ include equality.

A strict inequality, such as $x < 4$ , excludes its boundary. A non-strict inequality, such as $x≤4x\leq4$ , includes it.

A chained inequality such as $2<x≤72<x\leq7$ means both $x > 2$ and $x≤7x\leq7$ .

Common Mistake

Reversing an inequality

When both sides of an inequality are multiplied or divided by a negative number, reverse the inequality sign. For example, $- 2 x < 6$ gives $x > - 3$ .

Writing operations clearly

In algebra, multiplication signs are usually omitted:

$a×ba\times b$ is written $ab$ ;
$3×x3\times x$ is written $3 x$ ;
$x×xx\times x$ is written $x^2$ .

The numerical coefficient normally comes first, followed by variables in alphabetical order, so write $6 ab$ rather than $a 6 b$ or $6 ba$ .

Division is usually written as a fraction. The expression

x+1x−2\frac{x+1}{x-2}

means that the whole of $x + 1$ is divided by the whole of $x - 2$ .

Common Mistake

Respect the fraction bar

The fraction $x+1x−2\frac{x+1}{x-2}$ is not the same as $x+1x−2x+\frac{1}{x}-2$ . A fraction bar groups its entire numerator and denominator, like brackets do.

Brackets control the order of operations. For example, $3 (x + 2)$ means that the whole expression $x + 2$ is multiplied by 3, whereas $3 x + 2$ multiplies only $x$ by 3.

Like terms and equivalent expressions

Definition

Like terms

Like terms contain exactly the same variables raised to exactly the same powers. Their coefficients may be added or subtracted.

Thus $4x^2$ and $7x^2$ are like terms, but $x^2$ and $x$ , or $x y$ and $x^2y$ , are not.

Two expressions are equivalent if they have the same value for every permitted value of their variables.

Example

Collecting like terms

Simplify $5x^2+3x-2x^2+4-7x$ .

Group terms with matching variable parts: $5x^2-2x^2)+(3x-7x)+4$ .
Combine the coefficients within each group: $5-2)x^2+(3-7)x+4$ .
Simplify to obtain $3x^2-4x+4$ .

Functions and mapping notation

Definition

Function

A function is a rule that assigns exactly one output to each permitted input. The notation $f (x)$ means the output of the function $f$ when the input is $x$ .

If $f(x)=2x^2-1$ , then $f(3)=2(3)^2-1=17$ . The notation $f (x)$ does not mean $f$ multiplied by $x$ .

A function can also be described by mapping notation:

f:x↦2x2−1.f:x\mapsto 2x^2-1.

This reads “the function $f$ maps $x$ to $2x^2-1$ ”.

Example

Using function notation

Given $f(x)=x^2+3x$ , find $f (a + 1)$ .

Treat the complete expression $a + 1$ as the input and replace every occurrence of $x$ : $f(a+1)=(a+1)^2+3(a+1)$ .
Expand both parts: $a+1)^2=a^2+2a+1$ and $3 (a + 1) = 3 a + 3$ .
Collect like terms to obtain $f(a+1)=a^2+5a+4$ .

Number sets and restrictions

Common number sets include:

$N\mathbb{N}$ : natural numbers, usually the positive integers; check whether zero is included in the context.
$Z\mathbb{Z}$ : integers.
$Q\mathbb{Q}$ : rational numbers, which can be written as a fraction of two integers with non-zero denominator.
$R\mathbb{R}$ : real numbers.

The notation $x∈Rx\in\mathbb{R}$ means that $x$ is an element of the real numbers. The notation $x∉Zx\notin\mathbb{Z}$ means that $x$ is not an integer.

An expression may have restrictions. For example, $1x−4\frac{1}{x-4}$ is undefined when $x = 4$ , so its real domain excludes 4. Similarly, $x−2\sqrt{x-2}$ is real only when $x≥2x\geq2$ .

Tip

State excluded values

When simplifying an algebraic fraction, record values excluded by the original denominator. Cancelling a factor does not make an originally undefined value permissible.

Exam technique

In the exam

Use precise verbs: simplify an expression, solve an equation, prove an identity and evaluate a function.
Keep signs attached to terms, and use brackets whenever you substitute a negative number or a multi-term expression.
Check domains and excluded values before dividing, cancelling factors or taking square roots.
Use equality signs only between genuinely equal expressions; do not join unrelated stages of working with $=$ .

Self review

Check yourself

In $4x^3-5xy+9$ , what are the terms, coefficients, variables and constant term?
What is the difference between an equation, an identity and a formula?
If $g (x) = 3 x - 2$ , what do $g (- 4)$ and $g (t + 1)$ mean?

What you'll learn

How to distinguish expressions, equations, identities, formulae and inequalities.
How to identify terms, factors, coefficients, constants and variables.
How standard algebraic notation communicates operations, functions and sets of values.
How precise vocabulary helps you explain reasoning and avoid ambiguous working.

The building blocks of algebra

Algebra uses symbols to represent numbers and relationships. A letter may represent an unknown number, a quantity that can vary, or an arbitrary number chosen from a particular set.

Variables and constants

Definition

Variables and constants

A variable is a symbol, usually a letter, whose value can change or is not yet known. A constant is a fixed value.

In the expression $3 x + 7$ :

$x$ is a variable.
7 is a constant.
3 is the numerical multiplier of $x$ .

The same letter must represent the same value throughout a single argument unless it is explicitly redefined. Different letters can have equal values, but they do not have to.

Terms and coefficients

Definition

Terms and coefficients

A term is a part of an algebraic expression separated from other terms by addition or subtraction. A coefficient is a factor multiplying a variable or a product of variables.

For example, the expression

5x^2-3xy+y-8

has four terms: $5x^2$ , $- 3 x y$ , $y$ and $- 8$ .

Their coefficients are:

5 for the $x^2$ term;
$- 3$ for the $x y$ term;
1 for the $y$ term.

The final term, $- 8$ , is a constant term because it contains no variable.

Common Mistake

Keep the sign with the term

In $4x^2-7x+1$ , the second term is $- 7 x$ , not $7 x$ . Its coefficient is therefore $- 7$ .

Factors and powers

Definition

Factors

A factor is a number or algebraic expression multiplied by another quantity. To factorise an expression is to write it as a product of factors.

In $6x^2y$ :

6, $x^2$ and $y$ are factors;
$x$ is also a factor because $x2=x⋅xx^2=x\cdot x$ ;
the exponent, or index, 2 tells you how many factors of $x$ are multiplied together.

In $(x - 3) (x + 4)$ , the expressions $x - 3$ and $x + 4$ are factors.

Example

Describing an algebraic expression

Consider $7a^3b-2ab^2+5$ .

Separate at addition and subtraction signs, keeping each sign attached to its term. The terms are $7a^3b$ , $2ab^2$ and 5.
Identify the numerical factors multiplying the variables. The coefficients of the variable terms are 7 and $- 2$ .
Identify the term containing no variables. The constant term is 5.
Read the indices: $a^3$ means $a⋅a⋅aa\cdot a\cdot a$ , while $b^2$ means $b⋅bb\cdot b$ .

Expressions and mathematical statements

Expressions

Definition

Algebraic expression

An expression is a combination of numbers, variables and operations. It does not contain an equality or inequality sign and therefore does not make a claim that can be true or false.

Examples include:

$2x^2-5x+1$
$x+3x−1\frac{x+3}{x-1}$
$y+4\sqrt{y+4}$

You can simplify, expand, factorise or evaluate an expression. You do not “solve” an expression because it does not state a condition on the variable.

To evaluate an expression means to calculate its value for specified values of its variables.

Example

Evaluating an expression

Evaluate $2p^2-3pq$ when $p = - 2$ and $q = 5$ .

Substitute the given values using brackets: $2(-2)^2-3(-2)(5)$ .
Evaluate the power before multiplying: $2)^2=4$ , so the expression becomes $2 (4) - 3 (- 2) (5)$ .
Complete the multiplications and addition: $8 - (- 30) = 38$ .

Common Mistake

Use brackets when substituting negatives

If $x = - 3$ , then $x^2=(-3)^2=9$ . Writing $3^2$ without brackets means $3^2)=-9$ because powers are evaluated before the leading negative sign.

Equations

Definition

Equation

An equation states that two expressions are equal. A solution is a value, or set of values, that makes the equation true.

The left-hand side and right-hand side of an equation are often abbreviated to LHS and RHS.

Identities

Definition

Identity

An identity is an equality that is true for every permitted value of its variable or variables. The symbol $≡\equiv$ may be used to show that two expressions are identically equal.

For example,

(x+2)2≡x2+4x+4.(x+2)^2\equiv x^2+4x+4.

This is true for every real value of $x$ . By contrast, $x+2)^2=9$ is an equation that is true only for particular values of $x$ .

Example

Distinguishing an equation from an identity

Decide whether $3 (x - 2) + 4 = 3 x - 2$ is an identity.

Expand the left-hand side: $3 (x - 2) + 4 = 3 x - 6 + 4$ .
Collect the constant terms to obtain $3 x - 2$ .
The simplified left-hand side equals the right-hand side for every value of $x$ , so the statement is an identity: $3(x−2)+4≡3x−23(x-2)+4\equiv3x-2$ .

Key Idea

Equality and identity

An equation imposes a condition that may hold only for certain values. An identity records two equivalent forms of the same expression and holds throughout the permitted domain.

Formulae

Definition

Formula

A formula is an equation that expresses a general relationship between quantities. One variable may be described as the subject when it appears alone on one side.

For example, $A=πr2A=\pi r^2$ gives the area $A$ of a circle in terms of its radius $r$ . Here, $A$ is the subject.

Changing the subject rearranges the relationship without changing its meaning. From $A=πr2A=\pi r^2$ , with $r≥0r\geq0$ ,

r=Aπ.r=\sqrt{\frac{A}{\pi}}.

Inequalities

Definition

Inequality

An inequality compares quantities that are not necessarily equal. The symbols $<$ and $>$ mean “less than” and “greater than”; $≤\leq$ and $≥\geq$ include equality.

A strict inequality, such as $x < 4$ , excludes its boundary. A non-strict inequality, such as $x≤4x\leq4$ , includes it.

A chained inequality such as $2<x≤72<x\leq7$ means both $x > 2$ and $x≤7x\leq7$ .

Common Mistake

Reversing an inequality

When both sides of an inequality are multiplied or divided by a negative number, reverse the inequality sign. For example, $- 2 x < 6$ gives $x > - 3$ .

Writing operations clearly

In algebra, multiplication signs are usually omitted:

$a×ba\times b$ is written $ab$ ;
$3×x3\times x$ is written $3 x$ ;
$x×xx\times x$ is written $x^2$ .

The numerical coefficient normally comes first, followed by variables in alphabetical order, so write $6 ab$ rather than $a 6 b$ or $6 ba$ .

Division is usually written as a fraction. The expression

x+1x−2\frac{x+1}{x-2}

means that the whole of $x + 1$ is divided by the whole of $x - 2$ .

Common Mistake

Respect the fraction bar

The fraction $x+1x−2\frac{x+1}{x-2}$ is not the same as $x+1x−2x+\frac{1}{x}-2$ . A fraction bar groups its entire numerator and denominator, like brackets do.

Brackets control the order of operations. For example, $3 (x + 2)$ means that the whole expression $x + 2$ is multiplied by 3, whereas $3 x + 2$ multiplies only $x$ by 3.

Like terms and equivalent expressions

Definition

Like terms

Like terms contain exactly the same variables raised to exactly the same powers. Their coefficients may be added or subtracted.

Thus $4x^2$ and $7x^2$ are like terms, but $x^2$ and $x$ , or $x y$ and $x^2y$ , are not.

Two expressions are equivalent if they have the same value for every permitted value of their variables.

Example

Collecting like terms

Simplify $5x^2+3x-2x^2+4-7x$ .

Group terms with matching variable parts: $5x^2-2x^2)+(3x-7x)+4$ .
Combine the coefficients within each group: $5-2)x^2+(3-7)x+4$ .
Simplify to obtain $3x^2-4x+4$ .

Functions and mapping notation

Definition

Function

A function is a rule that assigns exactly one output to each permitted input. The notation $f (x)$ means the output of the function $f$ when the input is $x$ .

If $f(x)=2x^2-1$ , then $f(3)=2(3)^2-1=17$ . The notation $f (x)$ does not mean $f$ multiplied by $x$ .

A function can also be described by mapping notation:

f:x↦2x2−1.f:x\mapsto 2x^2-1.

This reads “the function $f$ maps $x$ to $2x^2-1$ ”.

Example

Using function notation

Given $f(x)=x^2+3x$ , find $f (a + 1)$ .

Treat the complete expression $a + 1$ as the input and replace every occurrence of $x$ : $f(a+1)=(a+1)^2+3(a+1)$ .
Expand both parts: $a+1)^2=a^2+2a+1$ and $3 (a + 1) = 3 a + 3$ .
Collect like terms to obtain $f(a+1)=a^2+5a+4$ .

Number sets and restrictions

Common number sets include:

$N\mathbb{N}$ : natural numbers, usually the positive integers; check whether zero is included in the context.
$Z\mathbb{Z}$ : integers.
$Q\mathbb{Q}$ : rational numbers, which can be written as a fraction of two integers with non-zero denominator.
$R\mathbb{R}$ : real numbers.

The notation $x∈Rx\in\mathbb{R}$ means that $x$ is an element of the real numbers. The notation $x∉Zx\notin\mathbb{Z}$ means that $x$ is not an integer.

An expression may have restrictions. For example, $1x−4\frac{1}{x-4}$ is undefined when $x = 4$ , so its real domain excludes 4. Similarly, $x−2\sqrt{x-2}$ is real only when $x≥2x\geq2$ .

Tip

State excluded values

When simplifying an algebraic fraction, record values excluded by the original denominator. Cancelling a factor does not make an originally undefined value permissible.

Exam technique

In the exam

Use precise verbs: simplify an expression, solve an equation, prove an identity and evaluate a function.
Keep signs attached to terms, and use brackets whenever you substitute a negative number or a multi-term expression.
Check domains and excluded values before dividing, cancelling factors or taking square roots.
Use equality signs only between genuinely equal expressions; do not join unrelated stages of working with $=$ .

Self review

Check yourself

In $4x^3-5xy+9$ , what are the terms, coefficients, variables and constant term?
What is the difference between an equation, an identity and a formula?
If $g (x) = 3 x - 2$ , what do $g (- 4)$ and $g (t + 1)$ mean?

What you'll learn

The building blocks of algebra

Variables and constants

Terms and coefficients

Factors and powers

Expressions and mathematical statements

Expressions

Equations

Identities

Formulae

Inequalities

Writing operations clearly

Like terms and equivalent expressions

Functions and mapping notation

Number sets and restrictions

Review 1.2.1 Algebraic vocabulary and notation by teaching Genie

1.2.1 Algebraic vocabulary and notation

What you'll learn

The building blocks of algebra

Variables and constants

Terms and coefficients

Factors and powers

Expressions and mathematical statements

Expressions

Equations

Identities

Formulae

Inequalities

Writing operations clearly

Like terms and equivalent expressions

Functions and mapping notation

Number sets and restrictions

Review 1.2.1 Algebraic vocabulary and notation by teaching Genie

1.2.1 Algebraic vocabulary and notation

What you'll learn

The building blocks of algebra

Variables and constants

Terms and coefficients

Factors and powers

Expressions and mathematical statements

Expressions

Equations

Identities

Formulae

Inequalities

Writing operations clearly

Like terms and equivalent expressions

Functions and mapping notation

Number sets and restrictions

Review 1.2.1 Algebraic vocabulary and notation by teaching Genie

1.2.1 Algebraic vocabulary and notation Revision Guide

1.2.1 Algebraic vocabulary and notation

What you'll learn

The building blocks of algebra

Variables and constants

Terms and coefficients

Factors and powers

Expressions and mathematical statements

Expressions

Equations

Identities

Formulae

Inequalities

Writing operations clearly

Like terms and equivalent expressions

Functions and mapping notation

Number sets and restrictions

Review 1.2.1 Algebraic vocabulary and notation by teaching Genie

1.2.1 Algebraic vocabulary and notation Revision Guide