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Year 7 - Wednesday 10th Dec

Year 8 - Tuesday 2nd Dec

Year 9 - Tuesday 9th Dec GR

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Corbettmaths Practice Papers for 9-1 GCSE Maths

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Angles: compass directions Video 27b Practice Questions

Area: quarter-circle Video 47a Practice Questions

Averages: combined mean Video 53a Practice Questions Textbook Exercise

Averages: mode (frequency table) Video 56a

Averages: range (frequency tables) Video 57a

Averages: Quartiles Video 57b Practice Questions

Circles: perimeter - quarter-circle Video 62a Practice Questions

Circle theorems proof: right angle in a semi-circle Video 65a Practice Questions

Circle theorems proof: angles at circumference\centre Video 65b Practice Questions

Circle theorems proof: angles in the same segment Video 65c Practice Questions

Circle theorems proof: cyclic quadrilaterals Video 65d Practice Questions

Circle theorems proof: alternate segment theorem Video 65e Practice Questions

Circle theorems proof: radius and tangent proof Video 65f Practice Questions

Circle theorems proof: Textbook Exercise

Constructions: loci part 3 Video 77 Practice Questions Textbook Exercise

Coordinates: 3D Video 86 Practice Questions

Data handling cycle Video 89 Practice Questions

Dimensional analysis Video 97 Practice Questions

Division: long division Video 98a

Enlargements: ray method Video 109 Practice Questions

Equations: involving fractions Video 111 Practice Questions Textbook Exercise

Equations: x squared Video 116a Practice Questions

Equations: Think of a number Video 116b Practice Questions

Factorisation: splitting the middle Video 119a Textbook Exercise

Factorisation: difference of 2 squares Video 120 Practice Questions Textbook Exercise

Fractions: finding the midpoint Video 141a Practice Questions

Graphs composite bar charts Video 148a

Graphs: dual bar charts Video 148b

Graphs: misleading graphs Video 160a

Graphs: real life linear graphs Video 171a

Linear graphs: real life Video 198a

Money - See Video 400 or click here for collection

Multiplication: grid method Video 199 Practice Questions Textbook 1 Textbook 2

Multiplication: column method Video 200 Practice Questions Textbook 1 Textbook 2

Multiplication: times tables Video 204a Textbook Exercise

Number: odd and even numbers Video 221a Practice Questions

Number: product of primes (squares/cubes) Video 223a Practice Questions Textbook

Percentages: simple interest Video 236a Practice Questions Textbook Exercise

Proportion: unitary method Video 255a Practice Questions Textbook Exercise

Proportion: Graphs Video 255b

Proportion: Time Video 255c

Pythagoras: area of a regular hexagon Video 262 Practice Questions

Quadratic graphs: sketching using key points Video 265 Textbook Exercise

Quadratic graphs: finding turning point Video 265a Practice Questions

Quadratic graphs: line of symmetry Video 265b

Quadratics: solving (factorising) Video 266 Practice Questions Textbook 1 Textbook 2

Quadratics: solving graphically Video 267c Practice Questions Textbook Exercise

Quadratics: solving graphically advanced Video 267d Practice Questions

Questionnaires Video 268 Practice Questions

Ratio: express as fractions or % Video 269a Practice Questions Textbook Exercise

Ratio: given two ratios Video 271a Textbook Exercise

Ratio: difference between Video 271b Textbook Exercise

Ratio: expressing as 1:n Video 271c Textbook Exercise

Ratio: equations/ratios Video 271d

Ratio: solving problems 1 Video 271e Textbook Exercise

Sampling: using samples Video 281a

Sampling: random Video 282 Practice Questions

Scales and maps Video 283 Practice Questions

Sequences: nth term for fractional sequences Video 289 Textbook Exercise

Similar shapes: definition Video 291 Practice Questions

Similar shapes: further Video 294 Practice Questions

Simultaneous equations (substitution, both linear) Video 296 Practice Questions

Simultaneous equations (graphical) Video 297 Textbook Exercise

Surface area: mixture Video 309 Practice Questions

Surface area: other prisms Video 312

Transformations: mixture Practice Questions

Transformations of graphs Video 323 Practice Questions

Transformations of trigonometric graphs Video 324 Practice Questions

Trigonometry: sine rule (ambiguous case) Video 334a Textbook Exercise

Trigonometry: area of a triangle Video 337 Proof Practice Questions Textbook Exercise

Types of data: primary & secondary Video 343a Practice Questions Textbook Exercise

Types of graph: cubics Video 344 Practice Questions

Types of graph: exponential Video 345 Practice Questions

Types of graph: reciprocal Video 346 Practice Questions

Types of graph: mixture

Units: types of metric and imperial unit Video 347 Practice Questions

Units: Converting imperial units for length Video 348a Practice Questions

Units: Converting imperial units for mass Video 348b Practice Questions

Units: Converting imperial units for capacity Video 348c Practice Questions

Units: metric to imperial (length) Video 349d Practice Questions

Units: metric to imperial (mass) Video 349e Practice Questions

Units: metric to imperial (capacity) Video 349f Practice Questions

Vectors Video 353 Practice Questions

Views: front/side elevation and plan view Video 354 Practice Questions

Volume: Frustum Video 360a Textbook Exercise

Additional Topics

Flow Charts Video 364 Practice Questions

Geometric Proof Video 366

Identities: Equating coefficients Video 367 Practice Questions

Identities, Formula, Equations etc Video 367a Practice Questions

Intersecting Chord Theorem Video 368a Practice Questions

Intersecting Secant Theorem Video 368b Practice Questions

Quadratic graph (completing the square) Video 371

Equation of a Tangent to a Circle Video 372 Practice Questions

Iteration Video 373 Practice Questions

Iterative Processes Video 373a

Change of Sign Video 373a Practice Questions

Arithmetic progressions Video 374

Geometric progressions Video 375

Set notation Video 379

Venn Diagrams Video 380 Practice Questions

Shortest distance from point to a line Video 381 Textbook Exercise

Time series Video 382

Population Density Video 384a Practice Questions

Invariant Points Video 392 Textbook Exercise

Simultaneous Equations (3 Unknowns) Video 393 Textbook Exercise

Measuring Lines Video 397 Practice Questions

Drawing Lines Video 397a Practice Questions

Money: Coins Video 400a Practice Questions

Money: Change Video 400b

Money: Buying Video 400c Practice Questions

Money: Regular Payments Video 400d Practice Questions

Money: Bills Video 400e

Money: Discounts Video 400f Practice Questions

Money: VAT Video 400g Practice Questions

Money: Wages Video 400h Practice Questions

Money: Bank Statements Video 400i Practice Questions

Money: Tax Video 400j Practice Questions

Money: Cost per kg Video 400k Practice Questions

Money: Cost per litre Video 400l Practice Questions

Money: Cost per metre Video 400m Practice Questions

Money: Reading meters Video 400n

Money: Profit Video 400p Practice Questions

Money: Greatest/Maximum Video 400q Practice Questions

Money: Estimation Video 400r Practice Questions

Money: Midpoint Video 400s Practice Questions

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Videos, Practice Questions and Textbook Exercises on every Secondary Maths topic

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Maths Genie * Edexcel GCSE Maths Past Papers, Mark Schemes, Model Answers and Video Solutions

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Maths GCSE past papers (Foundation and Higher) for the Edexcel exam board with mark schemes, grade boundaries, model answers and video solutions.

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Stratford School Academy is a mixed, all ability, and non-faith secondary school for students aged 11 - 16. It is based in Forest Gate, London.

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A growing bank of randomly generated GCSE exam style questions with full worked solutions. Perfect for projecting in the classroom. Choose between single or split screen mode for ‘my turn, your turn’ worked examples.

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All the past grade boundaries for the 9 - 1 GCSE mathematics exam. All exam boards and tiers included.

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Almanac of Interesting Numbers – Mathigon

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Scroll along the number line, and explore hundreds of interesting curious properties and facts. Then zoom in and out to discover even more!

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Can you leave the supermarket, while keeping a 3m distance from everyone else?

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Tutorial

Factris is a classic risk reward high score game. Just like in Tetris, you have to arrange falling blocks using the arrow keys, and when you fill an entire row, it disappears to make space for new blocks. However, there are two key differences:

The blocks are always rectangles of varying sizes. Rather than rotating them, you can "refactor" them. For example, a block might start out as 12×1 tiles, but by pressing the up key, you can successively change its size to 6×2, 4×3, 3×4, 2×6 and 1×12.

You can exponentially more points if you complete multiple rows at once. Every tile in the first row is worth 10 points, every tile in the second row is worth 20 points, then 40 points, 80 points, 160 points, and so on.

There is no undo, so you need to pay close attention to the size of the rectangles coming next. Improve your numeracy, packing and knowledge of factors with Factris!
Factris was invented by Richard Lissaman at MEI. This version was designed and developed by Philipp Legner.

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Factris is a fun game that teaches about simple arithmetic, divisibility and factorisation.

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Learn about the countless hidden uses and applications which mathematics has in everyday life: From weather prediction to medicine, video games and music…

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Maryam Mirzakhani (مریم میرزاخانی‎, 1977 - 2017) was an Iranian mathematician and professor at Stanford University. She was the first woman to receive the Fields Medal, the highest award in mathematics.

Mirzakhani worked at the intersection of dynamical systems and geometry. She studied objects like hyperbolic surfaces and complex manifolds, but also contributed to many other areas of mathematics.

When solving problems, Mirzakhani would draw doodles and diagrams on large sheets of paper, to see the underlying patterns and beauty. Her daughter even described Maryam's work as "painting". At the age of 40, Mirzakhani died of breast cancer.

Born in Adelaide, Australia, Terence Tao (born 17 July) is sometimes called the "Mozart of mathematics". When he was 13, he became the youngest ever winner of the International Mathematical Olympiad, and when he was 24, he became the youngest tenured professor at the University of California, Los Angeles.

Tao has received the MacArthur Fellowship, the Breakthrough Prize in mathematics, as well as the Fields Medal, the highest award in mathematics, for "his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory".

Together with Ben Green, Tao proved the Green-Tao theorem, which states that there are arbitrarily long arithmetic sequences of prime numbers.

In 2003, the Russian mathematician Grigori Perelman (Григо́рий Перельма́нborn, born 1966) proved the Poincaré Conjecture, which, until then, was one of the most famous unsolved problems in mathematics.

The complex proof was verified by 2006, but Perelman declined two big awards that came with it: the $1 million Clay Millennium Prize, and the Fields Medal which is the highest recognition in mathematics. In fact, he said: "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo."

Perelman also made contributions to Riemannian geometry and geometric topology, and the Poincaré Conjecture is still the only one of the seven Millennium Prize problems to have been solved.

Yitang Zhang (张益唐, born 1955) was born in China and is now a professor of mathematics at the University of California.

Zhang discovered that there is a number k less than 70 million, so that there are infinitely many pairs of prime numbers that are exactly k apart. This was a groundbreaking discovery in number theory, for which he received the MacArthur award in 2014.

This is similar to the Twin Prime conjecture, which states that there are infinitely many pairs exactly 2 apart (for example 11 and 13) - but no one knows if this is true.

The British mathematician Sir Andrew Wiles (born 1953) is best known for proving Fermat's Last Theorem, which, until then, was one of the most famous unsolved problems in mathematics.

In 1637, Pierre de Fermat, wrote in the margin of a textbook that he had a wonderful proof that the equation a n n>2. Unfortunately, no one was able to find a proof - until Wiles, some 400 years later.+b n=c n has no integer solutions for

Wiles had been fascinated by the problem since the age of 10, and spent seven years working on it in solitude. He announced his solution in 1993, although a small gap in his argument took two more years to fix.

He was too old to receive the Fields medal, the highest award in mathematics, which has an age limit of 40. Instead, Wiles was awarded a special silver plaque for his work.

The problem asks for an algorithm to decide whether a given Diophantine equation (a polynomial equations with integer coefficients) has any integer-valued solutions.

William Paul Thurston (1946 - 2012) was an American mathematician and a pioneer in the fields of topology, manifolds and geometric group theory.

Thurston's Geometrization Conjecture is about describing the structure and geometry of different three-dimensional spaces. In 1982, he was awarded the Fields Medal for his study of 3D manifolds.

Paul Joseph Cohen (1934 - 2007) was an American mathematician who proved the continuum hypothesis, and that the axiom of choice is independent from the other Zermelo-Fraenkel axioms of set theory. He received the Fields medal for his work.

Alan Turing (1912 - 1954) was an English mathematician and is often called the "father of computer science".

During the Second World War, Turing played a critical role in breaking the Enigma code used by the German military, as part of the "Government Code and Cypher School" at Bletchley Park. This helped the Allies win the war, and may have saved millions of lives.

He also invented the Turing machine, a mathematical model of a general purpose computer, and the Turing test, which can be used to judge the ability of artificial intelligence.

Turing was gay, which was still a crime during his life, and meant that his groundbreaking accomplishments were never fully recognised. He committed suicide at the age of 41.

André Weil (1906 - 1998) was one of the most influential French mathematicians in the 20th century.

He was one of the founders of the Bourbaki group, a group of mathematicians working under the collective pseudonym Nicolas Bourbaki. The goal of the Bourbaki group was to unify all of mathematics with a formal, axiomatic foundation.

Weil believed that many problems in algebra and number theory had analogous versions in algebraic geometry and topology. These are known as Weil conjectures, and became the basis for both disciplines. They also have applications in fields like cryptography and computer science.

During the second World War, Weil fled to the United States and later joined the Institute for Advanced Study at Princeton University.

Kurt Gödel (1906 - 1978) was an Austrian mathematician who later immigrated to America, and is considered one of the greatest logicians in history.

At the age of 25, just after finishing his doctorate in Vienna, he published his two incompleteness theorems. These state that any (consistent and sufficiently powerful) mathematical system contains certain statements that are true but cannot be proven. In other words, mathematics contains certain problems that are impossible to solve.

This result had a profound impact on the development and philosophy of mathematics. Gödel also found an example of these "impossible theorems": the continuum hypothesis.

Mary Lucy Cartwright (1900 - 1998) was a British mathematician and one of the pioneers of Chaos theory. Together with Littlewood, she discovered curious solutions to a problem: an example of what we now call the Butterfly effect.

Elbert Cox (1895 - 1969) was the first African-American mathematician to receive a PhD. Universities in England and Germany refused to accept his thesis at the time, but Japan's Tohoku Imperial University did.

Cox taught at Howard University in the United States, he studied polynomial solutions to differential equations, generalised the Boole summation formula, and compared different grading systems.

Albert Einstein (1879 - 1955) was a German physicist, and one of the most influential scientists in history. He received the Nobel Prize for physics and TIME magazine called him the person of the 20th century.

Einstein triggered the most significant transformation in our view of the universe since Newton. He realised that classical, Newtonian physics was no longer enough to explain certain physical phenomenons.

At the age of 26, during his "miracle year", he published four groundbreaking scientific papers that explained the photoelectric effect and Brownian motion, introduced special relativity, and derived the formula E=m c 2, which states that energy ( E) and mass ( m) are equivalent.

G.H. Hardy (1877 - 1947) was a leading English pure mathematician. Together with John Littlewood, he made important discoveries in analysis and number theory, including the distribution of prime numbers.

In 1913, Hardy received a letter from Srinivasa Ramanujan, an unknown, self-taught clerk from India. Hardy immediately recognised his genius, and arranged for Ramanujan to travel to Cambridge where he was working. Together, they made important discoveries and authord numerous paper.

Hardy always disliked applied mathematics and expressed this in his personal account of mathematical thinking, the 1940 book A Mathematician's Apology.

Peano also developed a new, international language, Latino sine flexione, which was a simplified version of Latin.

The French mathematician Henri Poincaré (1854 - 1912) is often described as the last universalist, meaning that he worked in every field of mathematics known during his lifetime.

Poincaré is one of the founders of the field of Topology, and he came up with the Poincaré conjecture. This was one of the famous unsolved problems in mathematics, until it was proven in 2003 by Grigori Perelman

He also found a partial solution for the "three body problem", and discovered that the motion of three stars or planets in space can be completely unpredictable. This laid the foundation for modern Chaos theory.

Poincaré was the first to propose gravitational waves, and his work on Lorentz transformations was the basis upon which Albert Einstein built his theory of special relativity.

Charles Lutwidge Dodgson (1832 - 1898) is best know under his pen name Lewis Carroll, as the author of Alice's Adventures in Wonderland and its sequel Through the Looking-Glass.

However, Carroll was also a brilliant mathematician. He always tried to incorporate puzzles and logic into his children's stories, making them more enjoyable and memorable.

Richard Dedekind (1831 - 1916) was a German mathematician and one of the students of Gauss. He developed many concepts in set theory, and invented Dedekind cuts as the formal definition of real numbers. He also gave the first definitions of number fields and rings, two important constructs in abstract algebra.

While still in his teens, Galois proved that there is no general solution for polynomial equations of degree five or higher - simultaneously with Niels Abel.

Unfortunately, other mathematicians who he shared these discoveries with repeatedly misplaced or simply returned his work, and he failed his school and university exams while concentrating on much more complex work.

At the age of 20, Galois was shot in a duel (some say a feud over a woman), and later died of his wounds. During the night before his death, he summarised his mathematical discoveries in a letter to a friend. It would take other mathematicians many years to fully understand these letters, and realise the impact of his work.

Augustus De Morgan (1806 - 1871) was a British mathematician and logician. He studied the geometric properies of complex numbers, formalised mathematical induction, suggested quaternions, and came up with new mathematical notation.

The De Morgan laws explain how to transform logical relationships in set theory, for example A∩B‾A∪B‾=A‾∩B‾. =A‾∪B‾ and

William Rowan Hamilton (1805 - 1865) was an Irish mathematician and child prodigy. He invented quaternions, the first example of a "non-commutative algebra", which has important applications in mathematics, physics and computer science.

He first came up with the idea while walking along the Royal Canal in Dublin, and carved the fundamental formula into a stone bridge he passed: i 2=j 2=k 2=ijk=−1.

Hamilton also made significant contributions to physics, including optics and Newtonian mechanics.

Niels Henrik Abel (1802 - 1829) was an important Norwegian mathematician. Even though he died at the age of 26, he made groundbreaking contributions to a wide range of topics.

At the age of 16, Abel proved the binomial theorem. Three years later, he proved that it is impossible to solve quintic equations - by independently inventing group theory. This had been an open problem for over 350 years! He also worked on elliptic functions and discovered Abelian functions.

Abel spent his life in poverty: he had six siblings, his father died when he was 18, he was unable to find a job at a university, and many mathematicians initially dismissed his work. Today, one of the highest awards in mathematics, the Abel Prize is named after him.

In theory, these machines could automatically perform certain calculations stored on cards or tape. However, due to the high production costs, they were never fully completed during Babbage's lifetime. In 1991, a functional replica was constructed at the Science Museum in London.

August Ferdinand Möbius (1790 - 1868) was a German mathematician and astronomer. He studied under Carl Friedrich Gauss in Göttingen and is best known for his discovery of the Möbius strip: a non-orientable two-dimensional surface with only one side. (However, it was independently discovered by Johann Benedict Listing just a few months earlier.)

Many other concepts in mathematics are named after him, including the Möbius plane, Möbius transformations, the Möbius function μ n in number theory, and the Möbius configuration of two mutually inscribed tetrahedra.

Joseph Fourier (1768 - 1830) was a French mathematician, and a friend and advisor of Napoleon. In addition to his mathematical research, he is also credited with the discovery of the greenhouse effect.

While travelling to Egypt, Fourier became particularly fascinated with heat. He studied heat transfer and vibrations, and discovered that any periodic function can be written as an infinite sum of trigonometric functions: a Fourier series.

Adrien-Marie Legendre (1752 - 1833) was an important French mathematician. He studied elliptic integrals and their usage in physics. He also found a simple proof that π is irrational, and the first proof that π 2 is irrational.

Gaspard Monge (1746 - 1818) was a French mathematician. He is considered the father of differential geometry, having introduced the concept of lines of curvature on surfaces in three-dimensional space (e.g. on a sphere). Monge also invented orthographic projection and descriptive geometry, which allows the representation of three-dimensional objects using two-dimensional drawings.

During the French Revolution, Monge served as Minister of the Marine. He helped reform the French education system and found the École Polytechnique.

Maria Gaetana Agnesi (1718 - 1799) was an Italian mathematician, philosopher, theologian, and humanitarian. Agnesi was the first western woman to write a mathematics textbook. She was also the first woman to be appointed professor at a university.

Her textbook, the Analytical Institutions for the use of Italian youth combined differential and integral calculus, and was an international success.

Agnesi also studied a bell-shaped curve described by the equation y=a 3 x 2+a 2. This function is now called the Witch of Agnesi. The strange name might come from a pun in the Italian language, were the word "versiera" for "witch" sounds similar to the ropes used when sailing.

Émilie du Châtelet (1706 - 1749) was a French scientist and mathematician. As a women, she was often excluded from the scientific community, but she built friendships with renown scholars, and had a long affair with the philosopher Voltaire.

She applied her mathematical ability while gambling, and used her winnings to buy books and laboratory equipment, and made important advanced regarding the concepts like energy and energy conservation.

Around the age of 42, Du Châtelet became pregnant again. At the time, without adequate healthcare, this was very dangerous for women of her age. She was also working on a French translation of Newton's book Principia, which containes the basic laws of physics.

Du Châtelet was determined to finish the translation, as well as a detailed commentary with additions and clarifications, and often worked 18 hours per day. She died just a few days after giving birth to a daughter, but her completed work was published posthumously, and is still used today.

Christian Goldbach (1690 - 1764) was a Prussian mathematician and contemporary of Euler, Leibniz and Bernoulli. He was tutor of Russian Tsar Peter II, and is remembered for his "Goldbach Conjecture".

Abraham de Moivre (1667 - 1754) was a French mathematician who worked in probability and analytic geometry. He is most remembered for de Moivre's formula, which links trigonometry and complex numbers.

De Moivre discovered the formula for the normal distribution in probability, and first conjectured the central limit theorem. He also found a non-recursive formula for Fibonacci numbers, linking them to the golden ratio φ.

Pierre de Fermat (1607 - 1665) was a French mathematician and lawyer. He was an early pioneer of calculus, as well as working in number theory, probability, geometry and optics.

In 1637, he wrote a short note in the margin of one of his textbooks, claiming that the equation a n n>2, and that he had a "marvellous proof, which this margin is too narrow to contain". This became known as +b n=c n has no integer solutions for Fermat's Last Theorem, and one of the most famous unsolved problems in mathematics - until it was finally proven in 1994.

René Descartes (1596 - 1650) was a French mathematician and philosopher, and one of the key figures in the Scientific Revolution. He refused to accept the authority of previous philosophers, and one of his best-known quotes is "I think, therefore I am".

Descartes is the father of analytical geometry, which allows us to describe geometric shapes using algebra. This was one of the prerequisites, which allowed Newton and Leibnitz to invent calculus a few decades later.

He is credited with the first use of superscripts for powers or exponents, and the cartesian coordinate system is named after him.

Marin Mersenne (1588 - 1648) was a French mathematician and priest. Because of the frequent exchanges with his contacts in the scientific world during the 17th century, he has been called the "the post-box of Europe".

Today we mostly remember him for the Mersenne primes, prime numbers that can be written as 2 n−1. Most of the largest known primes are of this type. He also studied acoustics and the harmonics of a vibrating string, and wrote about theology and philosophy.

Galileo Galilei (1564 - 1642) was an Italian astronomer, physicist and engineer. He used one of the first telescopes to make observations of the night sky, where he discovered the four largest moons of Jupiter, the phases of Venus, sunspots, and much more.

Galileo, sometimes called the "father of modern science", also studied the motion of objects in free fall, kinematics, material science, and invented the thermoscope (an early thermometer).

He was a vocal proponent of Heliocentrism, the idea that the Sun was at the centre of our solar system. This eventually led to him being tried by the Catholic Inquisition: Galileo was forced to recant and spent the rest of his life under house arrest.

François Viète (1540 - 1603) was a French mathematician, lawyer, and advisor to Kings Henry III and IV of France. He made significant advances in Algebra, and first introduced the use of letters to represent variables.

Viète discovered the connection between the roots and coefficients of a polynomial, called Viète's formula. He also wrote books about geometry and trigonometry, including calculating π to 10 decimal places using a polygon with 393216 sides.

Pedro Nunes (1502 - 1578) was a Portuguese mathematician and astronomer. As Royal Cosmographer of Portugal he taught navigational skills to many sailors and explorers.

Nunes first noticed that if a ship always follows the same compass bearing, it won't travel on a straight line or great circle. Instead, it will follow a path called a rhumb line or loxodrome, which spirals towards the North or South pole.

Nunes also tried to calculate which day in the year has the fewest hours of sunlight, he disproved previous attempts to solve classical geometry problems like trisecting an angle, and he invented a system for measuring fractional parts of angles.

The Italian Gerolamo Cardano (1501 - 1576) was one of the most influential mathematicians and scientists of the Renaissance. He investigated hypercycloids, published Tartaglia's and Ferrari's solution for cubic and quartic equations, was the first European to systematically use negative numbers, and even acknowledged the existence of imaginary numbers (based on −1).

Cardano also made some early progress in probability theory and introduced binomial coefficients and binomial theorem to Europe. He invented many mechanical devices, including combination locks, gyroscopes with three degrees of freedom, and drive shafts (or Cardan shafts) that are still used in vehicles today.

Nicolaus Copernicus (1473 - 1543) was a Polish mathematician, astronomer and lawyer. During his life, most people believed in the Geocentric model of the universe, with Earth at the centre and everything else rotating around it.

Copernicus created a new model, where the sun is at the centre, and Earth moves around it on a circle. He also predicted that Earth rotates around its axis once every day. Afraid that it would upset the Catholic church, he only published the model just before his death - triggering what is now called the Copernican Revolution.

Copernicus also worked as a diplomat and physician, and made important contributions to economics.

Zhu Shijie (朱世杰, 1249 - 1314) was one of the greatest Chinese mathematicians. In his book Jade Mirror of the Four Unknowns, he showed how to solve 288 different problem using systems of polynomial equations and four variables (called Heaven, Earth, Man and Matter).

Zhu made extensive use of Pascal's triangle. He also invented rules for solving systems of linear equations - predating our modern matrix methods by many centuries.

Qin first developed a method for numerically solving polynomial equations, which is now known as Horner's method. He found a formula for the area of a triangle based on the length of its three sides, calculated the sum of arithmetic series, and introduced a symbol for "zero" into Chinese mathematics.

Qin also invented Tianchi basins, which were used to measure rainfall and gather meteorological data important for farming.

Leonardo Pisano, commonly known as Fibonacci (1175 - 1250) was an Italian mathematician. He is best known for the number sequence named after him: 1, 1, 2, 3, 5, 8, 13, ...

Fibonacci is also responsible for popularising the Arabic numerals (0, 1, 2, 3, 4, ...) in Europe, which was still using Roman numerals (I, V, X, D, ...) in the 12th century CE. He explained the decimal system in a book called "Liber Abaci", a practical textbook for merchants.

Jia Xian (賈憲, c. 1010 - 1070) was a Chinese mathematician during the Song dynasty. He described Pascal's triangle, more than six centuries before Pascal, and used it to calculate square and cube roots.

Hasan Ibn al-Haytham (أبو علي، الحسن بن الحسن بن الهيثم, c. 965 - 1050) lived in Cairo during the Islamic Golden Age, and studied mathematics, physics, astronomy, philosophy, and medicine. He was a proponent of the scientific method: the belief that any scientific hypothesis must be verified using experiments or mathematical logic - centuries before European scientists during the Renaissance.

Al-Haytham was particularly interested in optics and visual perception. He also derived a formula for the sum of fourth powers (1 4+2 4+3 4+...+n 4), and he studied the link between algebra and geometry.

Zu Chongzhi (祖沖之, 429 - 500 CE) was a Chinese astronomer, mathematician, writer, politician and inventor.

He calculated Pi accurately to 7 decimal places - a record which was not surpassed until 800 years later. To do this, he approximated a circle with a 24,576-sided polygon.

Zu also discovered the formula 4 3 πr 3 for the volume of a sphere. His precise astronomical observations allowed him to create a new, more accurate calendar and to predict solar eclipses. He also calculated that Jupiter takes almost 12 years to orbit the sun.

While we know today this model is incorrect, Ptolemy's scientific impact is indisputable. He developed trigonometric tables with many practical applications, which remained the most accurate for many centuries. He also created detailed maps of the Earth, and wrote about music theory and optics.

Heron of Alexandria ( Ἥρων ὁ Ἀλεξανδρεύς, c. 10 - 70 CE) was a Greek mathematician and engineer. He lived in the city of Alexandria in Egypt, and is one of the greatest "experimenter" of antiquity.

His inventions include windmills, pantograph, as well as a radial steam turbine called aeolipile or Hero's engine. Hero's formula allows you to calculate the area of any triangle, using just the length of its three sides.

Hipparchus of Nicaea (Ἵππαρχος, c. 190 - 120 BCE) was a Greek astronomer and mathematicians, and one of the greatest astronomers of antiquity.

Hipparchus made detailed observations of the night sky and created the first comprehensive star catalog in the western world. He is considered the father of trigonometry: he constructed trigonometric tables and used these to reliably predict solar eclipses. He also invented the astrolabe and solved different problems in spherical trigonometry.

Archimedes (c. 287 - 212 BCE) was an ancient Greek scientist and engineer, and one of the greatest mathematicians of all time. He discovered many concepts of calculus and worked in geometry, analysis and mechanics.

While taking a bath, Archimedes discovered a way to determine the volume of irregular objects using the amount of water they displaced when submerged. He was so excited by this discovery that he ran out on the street, still undressed, yelling "Eureka!" (Greek for "I have found it!").

As an engineer, he built ingenious defence machines during the siege of his home city Syracuse in Sicily. After two years, the Romans finally managed to enter, and Archimedes was killed. His last words were "Do not disturb my circles" - which he was studying at the time.

Euclid taught mathematics in Alexandria, but not much else is known about his life.

Aristotle (Ἀριστοτέλης, c. 384 - 322 BCE) was a philosopher in Ancient Greece. Together with his teacher Plato, he is considered the "Father of Western Philosophy". He was also the private tutor of Alexander the Great.

Aristotle wrote about science, mathematics, philosophy, poetry, music, politics, rhetoric, linguistics, and many other subjects. His work was highly influential during the Middle Ages and into the Renaissance, and his views on ethics and other philosophical questions are still being discussed today.

Aristotle is also the first known person to formally study logic, including its applications in science and mathematics.

Zeno of Elea (c. 495 - 430 BCE) was a Greek philosopher who his known for his famous paradoxes, which have fascinated mathematicians for centuries.

One example is the paradox of motion: imagine that you want to run a 100 meter race. You first have to run half the distance (50 meters). But before doing that, you have to run a quarter of the distance (25 meters). Before running a quarter, you have to run 1 8th, 1 16th, and so on. This is an infinite number of tasks, which means that you'll never arrive!

Pythagoras of Samos (c. 570 - 495 BCE) was a Greek philosopher and mathematician. He is best known for proving Pythagoras' Theorem, but made many other mathematical and scientific discoveries.

Pythagoras tried to explain music in a mathematical way, and discovered that two tones sound "nice" together (consonant) if the ratio of their frequencies is a simple fraction.

He also founded a school in Italy where he and his students worshipped mathematics almost like a religion, while following a number of bizarre rules - but the school was eventually burned down by their adversaries.

The Ishango Bone is possibly the oldest mathematical artefact still in existence: it was discovered in 1950, in the Democratic Republic of Congo in central Africa, and is named after the region where it was found. It is dates back to the Upper Paleolithic period of human history, and is approximately 20,000 years old.

The bone is 10 cm long and contains a series of notches, which many scientists believe were used for counting. The grouping of the notches might even suggest some more advanced mathematical understanding, like decimal numbers or prime numbers.

In ancient Mesopotamia, almost 10,000 years ago, scribes and merchants started using small, three-dimensional clay objects as counters, to represent certain quantities, units or goods. Thousands of these were found on archaeological sites across the Middle East, like these from Tepe Gawra in Iraq (from around 4000 BCE):

The cone, sphere and flat disc were used to represent small, medium and large measures of grain. The tetrahedron probably measured the amount of work done in one day.

These two tablets from Susa in Iran were created around 3200 BCE and used a more advanced technique: the counters were pressed into the clay while it was still soft, to create a record:

Again, the triangular and circular impressions represent smaller and larger measures of grain. The patterns across the rest of the tablet were the official seals of the scribes.

These simple markings actually laid the foundations for cuneiform, one of the first writing system in history.

This is the oldest known clay tablet with mathematican computations - it was created around 2700 BCE in Sumer, one of the earliest civilisations that flourished in the Middle East.

It shows a multiplication table in cuneiform, which may have been used by student scribes to learn mathematics.

This tablet shows a multiplication table that was created around 2600 BCE in the Sumerian city of Shuruppak. It is one of the oldest mathematical tablets we have ever discovered.

The table has three columns. The dots in the first two columns represent distances ranging from around 6 meters to 3 kilometres. The third column contains the product of the first two, which is the area of a rectangle with the given dimensions.

Sumer was a region of ancient Mesopotamia in the Middle East. They invented Cuneiform as one of the earliest writing systems, by pressing small, wedge-shaped markers into clay tablets like this one. They also developed the base-60 number system.

This Babylonian clay tablet, called Plimpton 322, was created around 1750 BCE in Sumeria, during the reign of Hammurabi the Great.

The exact purpose of the tablet has been debated by archeologists. Some think that it was a "teachers aid", designed to help generate right-angled triangles. Others think it may be a very early trigonometry table.

This circular tablet from the Yale Babylonian Collection, called YBC 7289, was created around 1800 - 1600 BCE in ancient Babylon. It shows the geometric diagram of a square with its diagonals.

The cuneiform numerals indicate that one side of the square is 30 units long, and show how to find the length of the diagonal: units.

The tablet shows that Babylonian scribes knew Pythagoras' theorem, more than 1000 years before Pythagoras was even born. They were also able to calculate square roots and had an estimate for accurate to 6 decimal places. It is the highest computational accuracy ever seen in the ancient world!

While this simple tablet may have just been a practice exercise by a novice scribe, its mathematical and historical importance is enormous.

These two clay tablets from the Yale Babylonian Collection were created between 1800 and 1600 BCE, and contain exercises by student scribes, to calculate the area of different geometric shapes.

Tablet YBC 7290 shows how to calculate the area of a trapezium, by multiplying the average of the bases and the average of the sides.

Tablet YBC 11120 shows how to calculate the area of a circle, using the approximation

The Rhind Papyrus is one of the most famous mathematical documents from ancient Egypt. It was written around 1550 BCE by a scribe called Ahmose, who is maybe the earliest contributor to maths in history, whose name we still know today.

The papyrus is around 2 meters long and contains 84 problems about multiplication, division, fractions, and geometry. It was probably used as a kind of "textbook" by other scribes.

One of the most notable sections is a table. This shows how you can write rational numbers of the form where n is an odd number, as sums of unit fractions.

The papyrus is named after Scottish antiquarian Alexander Henry Rhind, who purchased it in Luxor, Egypt. Today, most of its remains are located at the British Museum in London.

Menna was a chief scribe in ancient Egypt, and in charge of measuring the size of fields for farming, inspected crop yields, reporting to the Pharaoh's central field administration, and calculating taxes.

The wall paintings in his tomb show the different measuring and calculating techniques used more than 3,000 years ago. For example, in the first row, you can see how long distances were measured using ropes with knots at regular intervals.

The tomb was built around 1420 BCE in the Valley of the Kings.

Here you can see a set of 21 Bamboo Strip that were created around 2300 years ago in China. When arrenged correctly, they form a multiplication table in base 10, written in ancient Chinese calligraphy.

While earlier civilisations like the Babylonians created multiplications tables in base 60, this is by far the oldest known decimal multiplication table - and it looks very similar to what we still use today.

Around 300 BCE, Euclid of Alexandria wrote The Elements, collection of 13 books that contained mathematical definitions, postulates, theorems and proofs, and covering topics like geometry and number theory.

It is one of the most famous books ever written, and one of the most influential works in the history of mathematics. Copies were used as textbooks for thousands of years and studied all around the world, with thousands of new editions published

No original copies of the Elements still exist today. This small papyrus fragment dates back to around 100 AD, and may be a part of the oldest existing copy of Euclid's work.

A palimpsest is a scroll or parchment from which the text has been washed or scraped off so that it can be reused. This method was common in the Middle Ages - even for documents by brilliant scientists and mathematicians.

Archimedes of Syracuse lived in the 3rd Century BCE and was one of the greatest mathematicians in history. A Greek copy of some of his work, created around 1000 CE in Byzantium, was later overwritten by Christian monks in Palestine. More recently, forgers added pictures to increase the value of the documents.

In 1998, scientists started studying the Archimedes Palimpsest, and used X-rays, ultraviolet and infrared light to uncover the hidden original text.

The Suàn shù shū (筭數書), which means Book on Numbers and Computation, is one of the oldest mathematical manuscripts from China. It was written around 200 BCE and consists of 200 strips of bamboo.

There are 69 problems, each with a solution, covering topics like arithmetic, fractions, integer factorisation, geometric sequences, inverse proportions, unit conversion, and error handling. Geometry problems show how to find the area of circles and rectangles, as well as the volume of three-dimensional solids, while assuming that

The inscription on this stone includes the oldest known use of the number zero: it dates back to the Khmer civilisation in Cambodia, around the year 683 CE.

Part of the text contains the number 605. Can you spot the dot that represents the zero? The inscription was only recently rediscovered by Amir Aczel.

Many ancient civilisations, like the Greeks and Romans, did not have a "zero" in their numeral system. From Cambodia, the concept was passed to India, where the Hindu-Arabic numeral system originated. From there, it spread to the Middle East and Europe, and we still use it today.

Some ancient American civilisations like the Maya also used zero in their calendars, but their numbers systems did not survive colonisation.

The title of the book Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa'l-muqābala (الْكِتَابْ الْمُخْتَصَرْ فِيْ حِسَابْ الْجَبْرْ وَالْمُقَابَلَة, short just __Al-Jabr__‎) translates to The Compendious Book of Calculations by Completion and Balancing.

It was written by the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī around 820 CE, and established Algebra as a new area of mathematics. In fact, the name algebra derived from the word al-ğabr in the title of the book.

Al-Khwārizmī is often called the father of algebra. In the book, he shows how to solve linear and quadratic equations, how to calculate the area and volume of certain geometric shapes, and he introduces the concept of "balancing" when solving equations.

Maqalah fi al-jabra wa-al muqabalah, which means Demonstration of Problems in Algebra, is a manuscript written by the Persian mathematician Omar Khayyam, around 1100 CE.

Khayyam managed to classify and solve all cases of cubic equations, using the intersection of conic sections. For example, on this page he shows how to solve equations of the form using the intersection of a circle and a hyperbola.

He also explored a triangle of binomial coefficients. In Iran, this triangle is called the Khayyam triangle, while in Europe and America it is more commonly known as Pascal's traingle.

The Lilāvatī is the first volume of a series of books written by Bhāskara II, one of the greatest mathematicians and astronomers in medieval India. It was published around 1150, when he was 36 years old.

Bhāskara wrote the book for his daughter, and the title actually means "playful". He writes about problem-solving, number sequences, Pythagoras' theorem, combinatorics, and many other topics.

These two pages show a problem about a pet peacock standing on a column, which can be solved using Pythagoras' theorem.

In the following volumes, Bhāskara also writes about algebra and astronomy. The combined work is called Siddhānta-Śiromani, which is Sanskrit for Crown of Treatises.

Very few Mayan documents have survived until today: one of them is the Dresden codex. It was created in the 13th century and describes Mayan mathematics and astronomy.

The Mayan number system had base 20 - using both fingers and toes for counting. Every digit from 1 to 19 consists of circles (representing 1) and horizontal lines (representing 5). Can you work out what all the numbers on this page are?

The Dresden Codex was used as a divination almanac, to record the date of astronomical events important for certain rituals. This fragment may contain the dates of eclipses of the planet Venus.

The Liber Abaci, Latin for Book of Calculation, was published in 1202 by Leonardo Fibonacci, the son of an Italian merchant. Together with his father, he spent his youth travelling around the Mediterranean.

He studied mathematics from Islamic scholars and learned about new ideas like algebra and the Hindu-Arabic numerals, both of which greatly simplified business transactions. When he returned to Italy, Fibonacci wrote a book about everything he learned.

He first introduced our current number system to Europe, which was still using Roman numerals at the time, and explained how to convert between both systems. In later chapters, he explains how to calculate profit and interest, how to approximate irrational numbers, how to determine whether a number is prime, and many other topics in mathematics. Most famously, he shows how rabbit populations might grow using the numbers 1, 1, 2, 3, 5, 8, ... These numbers are now known as Fibonacci numbers.

The Siyuan Yujian (四元玉鉴), which means Jade Mirror of the Four Unknowns, is a masterpiece of Chinese mathematics, published in 1303 by Zhu Shijie. It consists of four individual books and 288 different problems.

Zhu shows how to solve problems using systems of polynomial equations with up to four unknown variables, 天 (Heaven), 地 (Earth), 人 (Man) and 物 (Matter). He explains how to eliminate variables and how to find the side length of two and three-dimensional shapes given their volume or area.

To solve some of these problems, Zhu even used the numbers in Pascal's triangle, more than 300 years before Pascal was born!

Zhu also published a number of other mathematics texts, like the Suanxue Qimeng (New Arithmetic Enlightenment) in 1299. This textbook is written in verse, like many similar books at the time, which makes it wasy to memorise the arithemtic calculations.

Quipu are a recording system that was used by the Incan civilisation in South America around 1400 - 1560. They consist of many strings with small knots, all of which are attached to one larger rope. The type and position of the knots, as well as the colour of the strings, was used to record numbers, dates and maybe even text.

The Incans used a decimal number system like we do today. The position of a knot indicates the place value (ones, tens, hundreds, ...). Different types of knots (e.g. figure-8 knots and long-knots) represents the digit from 0 to 9.

When the Italian mathematician Luca Pacioli needed illustrations for his book De divina proportione (published in 1509), he asked Leonardo Da Vinci, a renown artist and former student.

Da Vinci created 60 different images of polyhedra. He often made a solid version, as well as a transparent version that only shows the edges, which was a completely new way to represent these 3-dimensional solids.

The Codex Mendoza is a description of the Aztec civilisation, which was commissioned in 1541 by Antonio de Mendoza. Its three sections explain the history and daily life of the Aztec people and list the different rulers and towns that were conquered.

The codex also contains examples of the Aztec calendar system, which you can see along the blue bar. Each of the symbols represents a date, and consists of a small image combined with several small circles.

The Aztec calendar used 20 day signs represented by a small image (crocodile, wind, house, lizard, snake, rabbit, water, etc.), together with up to 13 circles. This gives a cycle of 20 × 13 = 260 days.

Can you see which dates are be represented by the symbols on this page?

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Maths Genie is a free GCSE and A Level revision site. It has past papers, mark schemes and model answers to GCSE and A Level exam questions.