Zero and One as Identities

The big idea

Zero and one play unique structural roles in arithmetic: zero is the only number that leaves a quantity unchanged under addition, and one is the only number that leaves a quantity unchanged under multiplication. These identity properties, together with zero’s role as an annihilator under multiplication and the impossibility of division by zero, are not curiosities, they are foundational to equivalence, simplification, and solving equations throughout algebra.

Why this matters for secondary maths

The identity properties of zero and one sit quietly beneath a remarkable range of mathematical techniques. Equivalent fractions work because multiplying numerator and denominator by the same number is multiplying by 1 (in the form n/n). Adding zero pairs to subtract a negative, as developed in Concept 5 ( Directed Number), works because adding zero changes nothing. Completing the square involves adding and subtracting the same value: adding zero in a strategically useful form. Rationalising a surd denominator involves multiplying by a ‘clever form of 1’.

The zero-product property, if $ab = 0$ , then $a = 0$ or $b = 0$ is the logical engine behind solving quadratic equations by factorising. Pupils who do not understand why this property is special (it is unique to zero: if $ab = 6$ , we cannot conclude that $a = 6$ or $b = 6$ ) will treat factorising as a procedural trick rather than a structural argument.

Division by zero is undefined, not merely ‘not allowed’. Understanding why it is undefined, rather than memorising a prohibition, connects to the multiplicative identity: if $a \times 0 = 0$ for all a, then there is no number that, when multiplied by 0, gives any non-zero result, so division by zero has no consistent answer. This reasoning is a powerful example of how mathematical structure constrains what is possible, and it prepares pupils for later work with asymptotes, limits, and algebraic fractions where division by zero must be excluded.

Without explicit attention to these ideas, pupils are left with a collection of disconnected rules: ‘anything times zero is zero’, ‘you can’t divide by zero’, ‘multiply top and bottom by the same thing’. Making the identity properties explicit transforms these rules into a connected set of structural facts that pupils can reason with rather than merely recite.

Key representations

The number line: zero as origin and identity

On the number line, zero is the reference point. The origin from which all other numbers derive their position. Adding zero to any number leaves it where it is: starting at 5 and moving 0 units means staying at 5. This is the additive identity made visible. The number line also supports understanding of additive inverses: +3 and −3 are equidistant from zero in opposite directions, and their sum is zero. Demonstrate in the classroom with additive inverse tool.

Strengths: Makes the additive identity concrete and visual. Supports the connection between zero and directed number.

Limitation: The multiplicative identity (multiplying by 1) is harder to show on a standard number line; a scaling model is more effective for this.

Scaling diagrams: one as the identity for multiplication

A bar or line segment scaled by a factor of 1 stays the same length. Scaling by a factor of 2 doubles it; scaling by $\frac{1}{2}$ halves it; scaling by 1 leaves it unchanged. This representation makes the multiplicative identity visible as a scaling operation. It also supports the idea of ‘multiplying by a clever form of 1’: scaling by $\frac{3}{3}$ does not change the value, because $\frac{3}{3} = 1$ .

Strengths: Connects the multiplicative identity to the scaling interpretation of multiplication developed in Concept Section 6 (Additive and Multiplicative Reasoning) and Concept Section 7 (The Area Model).

Limitation: Requires pupils to think of multiplication as scaling, which some may not yet have internalised.

Two-coloured counters: zero pairs

As established in Concept Section 5 (Directed Number), a positive counter and a negative counter form a zero pair: +1 + (−1) = 0. This representation makes the additive identity and additive inverses tangible. Adding any number of zero pairs to a collection does not change its value, because each pair sums to zero. This is the structural basis for key algebraic techniques: adding zero pairs to subtract a negative, and later, adding and subtracting the same value to complete the square.

Strengths: Directly connects the identity property to the manipulation pupils will do in directed number work and algebra.

Limitation: Only addresses the additive identity and additive inverse. Does not represent the multiplicative identity.

Worked examples

These may seem obvious and not necessary as examples, and that may be true if pupils have a secure knowledge. However, it is easy to assume that all pupils, at least by secondary school age, understand these things implicitly when that may not be the case. Spending the time to establish these can be worth while.

Example 1: The additive identity

Calculate 47 + 0 and 0 + 47.

Both give 47. Adding zero to any number, in either order, leaves the number unchanged. On the number line: start at 47, move 0 units, you stay at 47. This is the additive identity: a + 0 = 0 + a = a for every number a.

Example 2: The multiplicative identity

Calculate 38 × 1 and 1 × 38.

Both give 38. Multiplying any number by 1, in either order, leaves the number unchanged. As a scaling operation: scaling a length of 38 by a factor of 1 gives a length of 38. This is the multiplicative identity: $a \times 1 = 1 \times a = a$ for every number a.

Example 3: The zero-product property (zero as annihilator)

Calculate 247 × 0.

The answer is 0, regardless of the size of the other number. Using the area model: a rectangle with one side of length 0 has no area. Using repeated addition: zero groups of 247, or 247 groups of zero, both give 0. This is the annihilator property: $a \times 0 = 0 \times a = 0$ for every number a.

The crucial follow-up question: if I tell you that two numbers multiply to give zero, what can you conclude? At least one of them must be zero. This is the zero-product property, and it is the reason factorising is useful for solving equations.

Example 4: Multiplying by a ‘clever form of 1’

Write $\frac{3}{5}$ as an equivalent fraction with denominator 20.

Multiply by $\frac{4}{4}$ (which equals 1): $\frac{3}{5} \times \frac{4}{4} = \frac{12}{20}$ . The value has not changed because multiplying by 1 never changes the value. We chose the form of 1 (namely $\frac{4}{4}$ ) that produces the denominator we want. This is not a trick, it is a direct application of the multiplicative identity.

Example 5: Why division by zero is undefined

What is 6 ÷ 0?

If 6 ÷ 0 had an answer, say k, then that would mean 0 × k = 6. But we have just established that 0 × k = 0 for every number k. There is no number that, when multiplied by 0, gives 6. So 6 ÷ 0 has no answer; it is undefined.

What about 0 ÷ 0? If 0 ÷ 0 = k, then 0 × k = 0, which is true for every k. So 0 ÷ 0 does not give a unique answer; it is also undefined (indeterminate).

The bridge to algebra

The identity properties of zero and one are not just arithmetic facts; they are structural principles that pupils will rely on throughout algebra, often without realising it. Making the connection explicit helps pupils see algebraic techniques as logical applications of ideas they already understand.

Equivalent fractions and algebraic fractions:

Arithmetic: $\frac{3}{5} = \frac{12}{20}$ because $\frac{3}{5} \times \frac{4}{4} = \frac{12}{20}$ , and $\frac{4}{4} = 1$ .

Algebra: $\frac{x}{3} = \frac{2 x}{6}$ because $\frac{x}{3} \times \frac{2}{2} = \frac{2 x}{6}$ , and $\frac{2}{2} = 1$ .

The technique is the same: multiply by a form of 1 to produce an equivalent expression with a different denominator. This underpins adding algebraic fractions with different denominators throughout KS4.

The zero-product property and solving by factorising:

Arithmetic: if a × b = 0, then a = 0 or b = 0.

Algebra: if $(x - 3) (x + 5) = 0$ , then $x - 3 = 0$ or $x + 5 = 0$ , giving $x = 3$ or $x = - 5$ .

This is the logical argument behind solving quadratics by factorising. It works because zero is the unique annihilator. The only number whose presence as a factor guarantees a product of zero.

Adding zero strategically:

Arithmetic: $402 - 198 = 402 - 200 + 2 = 204$ . We subtracted 2 extra and then added 2 back, a net change of zero.

Algebra: completing the square uses the same idea. To rewrite $x^{2} + 6 x$ as a perfect square, we add and subtract 9: $x^{2} + 6 x + 9 - 9 = (x + 3)^{2} - 9$ . The +9 and −9 together add zero, preserving the value while revealing structure.

This technique depends on the additive identity: adding zero changes nothing, but adding zero in a useful form can reveal structure that was previously hidden.

Multiplying by 1 to rationalise:

Later in KS4, pupils rationalise surd denominators by multiplying by a ‘clever 1’. For example, $\frac{1}{2} \times \frac{2}{2} = \frac{2}{2}$ . The principle is the same as creating equivalent fractions: multiplying by 1 (in the form $\frac{\sqrt2}{\sqrt2}$ ) does not change the value.

Key vocabulary

Term	Definition
Additive identity	The number 0. Adding 0 to any number leaves it unchanged: $a + 0 = 0 + a = a$ .
Multiplicative identity	The number 1. Multiplying any number by 1 leaves it unchanged: $a \times 1 = 1 \times a = a$ .
Additive inverse	The number that, when added to a given number, gives zero. The additive inverse of $a$ is $- a$ , because $a + (- a) = 0$ .
Multiplicative inverse (reciprocal)	The number that, when multiplied by a given number, gives 1. The multiplicative inverse of $a$ (where $a \neq = 0$ ) is $\frac{1}{a}$ , because $a \times \frac{1}{a} = 1$ .
Zero pair	A pair of numbers that sum to zero, such as +3 and −3. Used in directed number work.
Annihilator	A number that, when used in an operation, always produces a fixed result regardless of the other input. Under multiplication, 0 is the annihilator: $a \times 0 = 0$ .
Zero-product property	The principle that if $a \times b = 0$ , then $a = 0$ or $b = 0$ . This property is unique to zero.
Undefined	A mathematical expression that has no meaningful value. Division by zero is undefined because no number satisfies the required relationship.

What we don’t say


Avoid	Why	Say instead
‘You can’t divide by zero’ (without explanation)	This presents a prohibition without reasoning. Pupils deserve to understand why: it contradicts the annihilator property. Understanding the reason builds mathematical thinking.	‘Division by zero is undefined because there is no number that, when multiplied by zero, gives a non-zero result. The structure of multiplication makes it impossible.’
‘Anything times zero is zero’ (said casually and left unexplored)	While true, this statement is often recited without appreciation of its structural significance. The annihilator property is what makes the zero-product property work, which in turn is what makes solving by factorising possible.	‘Multiplying any number by zero gives zero. This is called the annihilator property, and it has a powerful consequence: if a product is zero, at least one of the factors must be zero.’
‘Just multiply top and bottom by the same thing’ (for equivalent fractions)	This hides the underlying reason: we are multiplying by 1 in a particular form. If pupils do not understand that $\frac{n}{n} = 1$ , they cannot explain why the procedure works or extend it to algebraic fractions.	‘We multiply by 1, written as $\frac{n}{n}$ , which doesn’t change the value but gives us the denominator we need. This works because $\frac{n}{n} = 1$ , and multiplying by 1 leaves the value unchanged.’
‘Zero is nothing’	Zero is not ‘nothing’. It is a number with specific properties. It is the additive identity, the origin of the number line, and the boundary between positive and negative numbers. Calling it ‘nothing’ undermines its mathematical significance.	‘Zero is the number that, when added to any other number, leaves it unchanged. It is the starting point on the number line and the boundary between positive and negative numbers.’

Common misconceptions and how to surface them

Misconception 1: Division by zero gives zero.

Some pupils reason that ‘since anything times zero is zero, anything divided by zero must be zero’. This is a logical error. The annihilator property says $a \times 0 = 0$ for all $a$ ; it does not say $0 \times a = a$ for all $a$ , which is what would be needed for division by zero to give zero. Ask: ‘If 6 ÷ 0 = 0, then 0 × 0 should equal 6. Does it?’ The contradiction reveals the error.

Misconception 2: $0 \times a = a$ (confusing the identity with the annihilator).

Pupils sometimes confuse the additive identity ( $a + 0 = a$ ) with multiplication by zero. They may write 5 × 0 = 5, applying the ‘leaves it unchanged’ idea to the wrong operation. Clarify: ‘Adding zero leaves a number unchanged. Multiplying by zero gives zero. Adding zero and multiplying by zero do very different things.’ A quick sorting activity, ‘which of these equal 7: $7 + 0$ , $7 \times 0$ , $7 \times 1$ , $7 - 0$ ?’, surfaces this confusion.

Misconception 3: Multiplying by 1 is pointless.

Pupils may think that since multiplying by 1 ‘doesn’t do anything’, it is mathematically useless. In fact, it is one of the most frequently used techniques in secondary and higher mathematics. Present a challenge: ‘How can I write $\frac{3}{5}$ with a denominator of 20 without changing its value?’ The answer involves multiplying by $\frac{4}{4} = 1$ . Ask: ‘Why does multiplying top and bottom by 4 not change the value?’ The answer is that $\frac{4}{4} = 1$ , and multiplying by 1 preserves value.

Misconception 4: The zero-product property applies to numbers other than zero.

If $a \times b = 12$ , pupils may think they can conclude $a = 12$ or $b = 12$ . This is false (for example, $a = 3$ and $b = 4$ ). The zero-product property is unique to zero. Ask: ‘I know two numbers multiply to give 12. Can I say one of them must be 12?’ Then: ‘I know two numbers multiply to give 0. Can I say one of them must be 0?’ The contrast between the two cases makes the special status of zero clear.

Diagnostic questions

Question 1: Which of these are equal to 8? Explain each.

8 + 0 8 × 1 8 × 0 8 − 0 0 + 8 1 × 8 0 × 8

What this reveals: A pupil who correctly identifies 8 + 0, 8 × 1, 8 − 0, 0 + 8, and 1 × 8 as equal to 8, and recognises that 8 × 0 and 0 × 8 equal 0, understands the distinction between the additive identity, the multiplicative identity, and the annihilator. A pupil who marks 8 × 0 = 8 or 8 × 1 = 0 is confusing these roles.

Question 2: Tom says ‘6 ÷ 0 = 0 because anything divided by zero is zero.’ Explain why Tom is wrong.

What this reveals: A pupil who can explain that if 6 ÷ 0 = 0 then 0 × 0 should equal 6, which it does not, demonstrates structural reasoning. A pupil who simply says ‘you can’t divide by zero’ without being able to explain why is reciting a rule without understanding. The framework aims for the first response: reasoning from the annihilator property, not from a memorised prohibition.

Progression spine

Stage	Key ideas	Notes
Primary (Y1–6)	Adding zero to a number. Multiplying by 1. Multiplying by 0. Early fraction equivalence. Understanding zero as a placeholder in place value.	Pupils meet these ideas informally but rarely name them as identity properties. The framework’s Year 7 treatment makes implicit knowledge explicit.
Year 7	Explicit naming of the additive and multiplicative identities. The annihilator property. The zero-product property. Equivalent fractions as ‘multiplying by 1’. Division by zero as undefined, with structural explanation. Connection to zero pairs.	The goal is to equip pupils with precise language and structural understanding for properties they have used informally since primary school.
Years 8–11	Solving quadratics by factorising (zero-product property). Equivalent and algebraic fractions (‘multiplying by a clever 1’). Completing the square (adding zero in a useful form). Rationalising denominators. Asymptotic behaviour where denominators approach zero.	The identity properties recur throughout KS3 and KS4. Pupils who understand the structural basis in Year 7 recognise each new application as a familiar principle, not a new trick.