Maths-related References

This background information covers multiple maths-related references in the play.

A-level examination

The A-level (Advanced Level) is a main school leaving qualification of the General Certificate of Education in England, Wales, Northern Ireland, the Channel Islands and the Isle of Man. It is available as an alternative qualification in other countries, where it is similarly known as an A-Level.

Students generally study for A-levels over a two-year period. For much of their history, A-levels have been examined by written exams taken at the end of these two years. A more modular approach to examination became common in many subjects starting in the late 1980s, and standard for September 2000 and later cohorts, with students taking their subjects to the half-credit "AS" level after one year and proceeding to full A-level the next year (sometimes in fewer subjects). In 2015, Ofqual decided to change back to a terminal approach where students sit all examinations at the end of the second year. AS is still offered, but as a separate qualification; AS grades no longer count towards a subsequent A-level.

Most students study three or four A-level subjects simultaneously during the two post-16 years (ages 16–18) in a secondary school, in a sixth form college, in a further and higher education college, or in a tertiary college, as part of their further education.

A-levels are recognized by many universities as the standard for assessing the suitability of applicants for admission in England, Wales, and Northern Ireland, and many such universities partly base their admissions offers on a student's predicted A-level grades, with the majority of these offers conditional on achieving a minimum set of final grades.

Structure
Prior to the 2015 government reforms of the A-level system, A-levels had (since the Curriculum 2000 reforms) consisted of two equally weighted parts: AS (Advanced Subsidiary) Level, usually assessed in the first year of study, and "A2 Level", usually assessed in the second year of study. It was also possible to take both AS Levels and A2 Levels for a subject in the same examination session - this was most common with Mathematics and Further Mathematics, where a student may have completed the entire Mathematics A-Level in their first year of study, followed by the entire Further Mathematics A-Level in their second. It was typical for an AS course to comprise two or three modules, with the A2 half of the course comprising two or three modules, for a total of four or six modules. The modules within each part may have been equally weighted or be of varying weights. Modules were either assessed by externally marked papers, or by school-assessed, externally moderated coursework.

Following the reforms, A-Levels and AS-Levels have been decoupled, with AS-Level results no longer counting towards the A-Level qualification. The AS-Level consists of the first half of the A-Level course and can be taught alongside the first year of the full A-Level course. Grades are determined by adding up the mark for each component (which is sometimes weighted) and applying a grade boundary.

A-Level Mathematics

Advanced Level (A-Level) Mathematics is a post-16 qualification taken in the United Kingdom (and occasionally other countries as well). In the UK, A-Level exams are traditionally taken by 17-18 year olds after a two-year course at a sixth form or college. Advanced Level Further Mathematics is often taken by students who wish to study a mathematics-based degree at university, or related degree courses such as physics or computer science.

Like other A-Level subjects, mathematics is assessed by examination at the end of the course. The syllabus seeks to develop skills in mathematical modelling, problem-solving, mathematical argument including mathematical language, and data analysis. It came to be regarded as one of the most beneficial A-Level subjects after the Russell Group of research-intensive universities in the United Kingdom published guidance for applicants in 2011 on their preferred A-Levels to prepare candidates for degree-level study at their institutions. Mathematics featured prominently in their list of A-Levels, which they described as "facilitating subjects". The Russell Group replaced its guidance with a new website in 2019, after a backlash led by the creative industries and criticism that it was unhelpful to disadvantaged applicants. But the impression that mathematics A-Level is respected and favored by selective universities and employers has persisted. Mathematics became the most popular A-Level subject by number of entries in 2014, overtaking English literature, and remained so for the next 12 years.

The linear structure and content were introduced for first teaching in September 2017, as part of far-reaching reforms to A-Levels and GCSEs introduced by Michael Gove during his tenure as Secretary of State for Education from 2010-14. These reforms replaced the modular system of assessment introduced in Curriculum 2000, whereby these qualifications were taught in modules, typically a total six of which three were taught in each year, with examinations after each whose results contributed to an overall final grade. The Gove changes also decoupled the AS-level qualification from A-levels, making AS levels a qualification in their own right.

Specification
There are three papers which must all be taken in the same year. There are three overarching themes - “Argument, language and proof”, “Problem solving” and “Modelling” throughout the assessment.

Each board structures the three papers as follows:

AQA
Paper 1: Pure Mathematics
Paper 2: Content on Paper 1 plus Mechanics
Paper 3: Content on Paper 1 plus Statistics

Edexcel
Paper 1: Pure Mathematics 1
Paper 2: Pure Mathematics 2
Paper 3: Statistics and Mechanics

OCR A
Paper 1: Pure Mathematics
Paper 2: Pure Mathematics and Statistics
Paper 3: Pure Mathematics and Mechanics

OCR B (MEI)
Paper 1: Pure Mathematics and Mechanics
Paper 2: Pure Mathematics and Statistics
Paper 3: Pure Mathematics and Comprehension

Grading
It was suggested by the Department for Education that the high proportion of candidates who obtain grade A makes it difficult for universities to distinguish between the most able candidates. As a result, the 2010 exam session introduced the grade A*—which serves to distinguish between the better candidates.

Prior to the 2017 reforms, the A* grade in maths was awarded to candidates who achieved an A (480/600) in their overall A Level, as well as achieving a combined score of 180/200 in modules Core 3 and Core 4. For the reformed specification, the A* is given by a more traditional grade boundary based on the raw mark achieved by the candidate over their papers.

List of subjects

1. Core Mathematics:
Covers foundational topics like algebra, calculus, trigonometry, and coordinate geometry.

2. Further Mathematics:
Expands upon Core Mathematics with additional areas such as complex numbers, matrices, differential equations, and numerical methods.

3. Pure Mathematics:
Explores advanced topics in algebra, calculus, and mathematical proofs.

4. Applied Mathematics:
Focuses on practical applications of mathematical concepts to solve real-world problems in various fields.

5. Mechanics:
Focuses on the study of motion, forces, and vectors, particularly relevant for physics or engineering interests.

6. Statistics:
Involves collecting, analyzing, and interpreting data, including topics like probability, hypothesis testing, regression analysis, and sampling.

7. Discrete Mathematics:
Deals with separate and distinct mathematical structures, including topics such as combinatorics, graph theory, and algorithms.

8. Decision Mathematics:
Applies mathematical techniques to solve real-world problems related to optimization, networks, and decision-making.

9. Financial Mathematics:
Applies mathematical concepts to analyse financial markets, investments, and risk management.

Results
The proportion of candidates in England, Wales and Northern Ireland acquiring these grades in 2025 are below, presented as cumulative percentages.

Prime numbers

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

The property of being prime is called primality. A simple but slow method of checking the primality of a given number ⁠n, called trial division, tests whether ⁠⁠n⁠ is a multiple of any integer between 2 and√( n). Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of October 2024 the largest known prime number is a Mersenne prime with 41,024,320 decimal digits.

There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says roughly that the probability of a randomly chosen large number being prime is inversely proportional to its number of digits, that is, to its logarithm.

Several historical questions regarding prime numbers are still unsolved. These include Goldbach's conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many pairs of primes that differ by two. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals.

The first 25 prime numbers (all the prime numbers less than 100) are: 

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

No even number ⁠n⁠ greater than 2 is prime because any such number can be expressed as the product ⁠2×n/2⁠. Therefore, every prime number other than 2 is an odd number, and is called an odd prime. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are even, and decimal numbers that end in 0 or 5 are divisible by 5.

Cubes of cardinal numbers

In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874–1884. Cardinality can be used to compare an aspect of finite sets. For example, the sets {1,2,3} and {4,5,6} are not equal, but have the same cardinality, namely three. Cubes of cardinal numbers are found by multiplying a counting number by itself three times: n3 = n x n x n, representing the volume of a cube. Cubes of the first 10 cardinal numbers: 

1³ = 1
2³ = 8
3³ = 27
4³ = 64
5³ = 125
6³ = 216
7³ = 343
8³ = 512
9³ = 729
10³ = 1,000