129 Fun Facts About Math

What is the sum of the interior angles of any triangle?

The sum of the interior angles of any triangle, regardless of its shape or size, is always 180 degrees. This is a fundamental concept in Euclidean geometry.

What defines a prime number?

A prime number is an integer greater than 1 with exactly two positive divisors, 1 and itself: 7 is prime, while 9 has the divisors 1, 3 and 9.

In algebra, what kind of equation is y = 3x + 2 an example of?

y = 3x + 2 is a linear equation: its variable appears only to the first power, so its graph is a straight line with slope 3 and y-intercept 2.

In calculus, what concept measures the instantaneous rate of change of a function?

The derivative measures a function's instantaneous rate of change; geometrically it is the slope of the tangent line at that point, so the derivative of x² is 2x.

What is the approximate value of the mathematical constant pi (π) to two decimal places?

Pi (π) is a mathematical constant representing the ratio of a circle's circumference to its diameter, approximately 3.14159. To two decimal places, it is 3.14.

Which mathematical theorem describes the relationship between the three sides of a right-angled triangle?

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In which number system is information represented using only two distinct symbols, typically 0 and 1?

Binary, the base-2 system, writes every number with 0 and 1 alone, which is why it underpins digital electronics and computers.

What formula is used to calculate the area of a circle?

The area of a circle is calculated by multiplying pi (π) by the square of its radius (r). The formula is A = πr².

In statistics, what term refers to the average of a set of numbers?

The mean, often called the average, is a measure of central tendency calculated by summing all the values in a dataset and dividing by the number of values.

Which sequence starts 0, 1, with each later number the sum of the two before it?

The Fibonacci sequence runs 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, each term the sum of the two before it.

What is a number that can be expressed as a fraction p/q, where p and q are integers and q is not zero?

A rational number is any number that can be written as a fraction, p/q, where both p and q are integers and q is not equal to zero.

Which branch of mathematics studies properties preserved under stretching and bending, but not tearing?

Topology, nicknamed 'rubber-sheet geometry', studies properties that survive stretching and bending: a coffee mug and a doughnut are topologically the same, each having exactly one hole.

What mathematical operation is the inverse of exponentiation?

A logarithm is the inverse operation of exponentiation. It answers the question: 'To what power must a given base be raised to produce a certain number?'

What is a geometric shape whose parts resemble the whole at various scales called?

A fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually exhibiting self-similarity, where parts of the shape resemble the whole.

Which of these numbers is an integer?

An integer is any positive or negative whole number, including zero. It does not include fractions or decimals.

What is a step-by-step procedure for solving a mathematical problem or performing a computation?

An algorithm in mathematics is a procedure, a description of a set of steps that can be used to solve a mathematical computation or problem.

In the octal number system, what is the base?

The octal number system is a base-8 system, meaning it uses eight distinct digits (0 through 7) to represent numbers.

What do we call a real number that cannot be written as a fraction p/q of integers?

An irrational number is a real number that cannot be expressed as a simple fraction of two integers. Its decimal representation is non-terminating and non-repeating.

What term names an expression of variables and coefficients using only +, −, × and non-negative integer exponents?

A polynomial is a mathematical expression consisting of variables, coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents.

In mathematics, what is a function that 'undoes' the operation of another function?

An inverse function essentially 'undoes' the effects of the original function. If f(x) produces y, then putting y into the inverse of f produces the output x.

How many sides does a hexagon have?

A hexagon is a polygon with six sides and six angles.

Which mathematician coined the term 'fractal' in 1975?

Benoît Mandelbrot coined 'fractal' in 1975 from the Latin fractus, 'broken', and popularised such shapes with computer images of the Mandelbrot set.

In algebra, what is a symbol for a value that is unknown or can change?

A variable is a symbol, typically a letter, that represents an unknown value or a value that can change in a mathematical expression or equation.

What number does the Roman numeral 'C' represent?

In the Roman numeral system, 'C' stands for one hundred.

Which ancient Greek mathematician is often called the 'Father of Geometry'?

Euclid, an ancient Greek mathematician, is widely referred to as the 'Father of Geometry' due to his influential work 'Elements'.

Which calculus operation finds the area under a curve between two limits?

Integration finds the area under a curve: the definite integral of f(x) from a to b sums infinitely many thin rectangles, so the integral of x from 0 to 1 is 1/2.

What is the probability of rolling a 3 on a standard six-sided die?

A standard six-sided die has six equally likely outcomes (1, 2, 3, 4, 5, 6), so the probability of rolling any specific number, like a 3, is 1 out of 6.

An angle that measures exactly 90 degrees is known as what type of angle?

A right angle is an angle that forms a square corner and measures exactly 90 degrees.

Which numeral system uses base 16, with the letters A to F standing for the values ten to fifteen?

Hexadecimal is base 16, using digits 0–9 and letters A–F; programmers favour it because each hex digit maps to exactly four binary bits, so a byte is always two hex digits.

In a set of numbers sorted from smallest to largest, what is the middle value called?

The median is the middle value once the numbers are arranged from smallest to greatest.

What is a sequence called when each term is found by adding a constant to the previous one?

An arithmetic progression is a sequence of numbers such that the difference between the consecutive terms is constant.

What is the value of 0 factorial, written 0!?

0! equals 1 by convention, so that n! = n × (n−1)! holds for n = 1 and there is exactly one way to arrange zero objects.

Who is credited with inventing the Cartesian coordinate system?

René Descartes introduced the coordinate system in La Géométrie (1637), letting problems of geometry be written as algebra and, later, calculus.

What is a mathematical structure consisting of rows and columns of numbers?

A matrix is a rectangular array of numbers, symbols or expressions arranged in rows and columns, used to represent mathematical objects or perform operations.

In statistics, what is the name for the value that appears most often in a data set?

The mode is the value that appears most often; a data set can have more than one mode, or none at all if every value appears equally often.

Who successfully proved Fermat's Last Theorem in the 1990s?

Wiles got there in 1994 by proving the Taniyama–Shimura conjecture, which Ribet had shown implies Fermat.

Which mathematical constant, denoted 'e', is the base of the natural logarithm, about 2.71828?

Euler's number is irrational and transcendental, and it is the base of both the natural logarithm and the exponential function.

Which irrational number, written phi, equals (1 + √5)/2, or about 1.618?

The Golden Ratio, phi, equals (1 + √5)/2 ≈ 1.618 and solves φ² = φ + 1; ratios of successive Fibonacci numbers such as 89/55 approach it.

What is a pattern of shapes covering a surface with no gaps or overlaps called?

A tessellation, or tiling, is the covering of a surface, often a plane, using one or more geometric shapes (tiles) with no overlaps and no gaps.

What is the absolute value of -7?

The absolute value of a number is its distance from zero on the number line, always resulting in a non-negative value. The absolute value of -7 is 7.

In mathematics, what is a quantity having both magnitude and direction?

A vector is a mathematical object that has both magnitude (or length) and direction, commonly represented by an arrow.

For the numbers 4 and 6, which of these equals 12?

The least common multiple of 4 and 6 is 12, the smallest positive number both divide into; their greatest common divisor, by contrast, is 2.

What mathematical concept refers to the arrangement of objects in a specific order?

A permutation is an arrangement of all or part of a set of objects, where the order of the arrangement matters.

What type of number includes a real part and an imaginary part?

A complex number is a number that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, satisfying i² = -1.

Which branch of mathematics studies curved spaces where the parallel postulate fails?

Non-Euclidean geometry replaces Euclid's parallel postulate: in hyperbolic geometry a line has infinitely many parallels through an outside point, and in elliptic geometry, such as a sphere's surface, none at all.

Which mathematician is famous for his work on game theory and the Nash equilibrium?

John Nash was an American mathematician whose work in game theory, particularly the concept of the Nash equilibrium, earned him a Nobel Memorial Prize in Economic Sciences.

What type of number is equal to the sum of its proper positive divisors (excluding itself)?

A perfect number is a positive integer that is equal to the sum of its positive proper divisors (divisors excluding the number itself). For example, 6 is a perfect number because its proper divisors (1, 2, 3) sum to 6.

In the fraction 3/4, what is the 3 called?

The numerator, 3, counts how many equal parts you have, while the denominator, 4, names what kind of parts they are; its Latin root numerare means 'to count'.

What name is given to 7/4, where the top number is bigger than the bottom one?

An improper fraction, sometimes called top-heavy, is worth 1 or more, so it can always be rewritten as a whole number plus a smaller fraction.

In the American memory trick PEMDAS, what does the letter P stand for?

P stands for Parentheses: whatever is inside them is worked out first, from the innermost set outwards; British schools learn the same order as BODMAS, whose B is for Brackets.

In 4 + 5 × 2, which operation do you carry out first?

Multiplication comes before addition in the order of operations, so 4 + 5 × 2 = 4 + 10 = 14, not 18.

In the Roman numeral system's seven letters, what number does D stand for?

D means 500; the seven letters are I, V, X, L, C, D and M, and a smaller letter before a bigger one subtracts instead of adds.

Written in Roman numerals, what is 9?

IX is 9: a smaller symbol in front of a larger one is subtracted, so IX reads as ten minus one; the same trick gives XL (40), XC (90), CD (400) and CM (900).

A triangle whose three sides are all exactly the same length is called what?

An equilateral triangle has three sides of equal length, so its three angles are equal too, each measuring 60 degrees.

What is the name for a triangle whose three sides are all different lengths?

Scalene comes from the Greek skalenos, 'uneven': a scalene triangle has three sides of different lengths and therefore three different angles.

If a circle's radius is 6 cm, how long is its diameter?

The diameter always runs through the centre, so it is worth two radii end to end. Multiply it by pi and you get the distance all the way around.

What word describes the distance once around the outside of a circle?

The circumference is the distance around a circle, found with 2πr or by multiplying the diameter by pi; one full turn of a bicycle wheel covers exactly its circumference.

Which is the only even prime number?

Every other even number can be split into 2 times something smaller, which disqualifies it. And 1 is not prime at all — the definition rules it out on purpose.

How many sides does an octagon have?

The name comes from the Greek for 'eight angles'. Its interior angles add up to 1,080 degrees, which is why an American stop sign looks so close to a circle.

Which of these numbers is even?

A number is even if it can be split into two equal whole groups, so 18 is 9 plus 9. Zero counts as even too.

How many sides does a triangle have?

A triangle has three sides and three corners; 'tri-' is the Latin and Greek prefix for three, as in tricycle and tripod.

How many sides does a square have?

A square has four sides, all the same length, meeting at four right angles; the 'quad-' in quadrilateral means four.

What is the square root of 81?

The square root of 81 is 9, because 9 × 9 = 81; 8 × 8 is only 64 and 10 × 10 is 100.

How many square faces does a cube have?

A cube has six square faces, along with eight corners and twelve edges; a standard die is a cube with one to six spots per face.

What is 7 times 8?

7 × 8 = 56, one of the times-table facts people most often stumble on; 6 × 8 is 48 and 8 × 8 is 64.

How many things are in a dozen?

Eggs are often sold this way. Twelve is handy because it splits evenly into 2, 3, 4 and 6.

How many weeks are in a year?

A year contains 52 full weeks plus a day or so, which is why the same date shifts by one weekday from one year to the next.

How many hours are in a whole day?

A day has 24 hours: the twelve hours on a clock face go round twice, once for the morning and once for the afternoon and night.

How many days are in a leap year?

A leap year has 366 days: the extra day, February 29, keeps the calendar lined up with Earth's 365.24-day trip around the Sun.

In Roman numerals, what number does the letter X stand for?

X stands for 10 in Roman numerals, so XX is 20 and XXX is 30; the letter V, for 5, is half of it.

How many cents is a US quarter worth?

A quarter is 25 cents, a quarter of a 100-cent dollar; a dime is 10 cents and a nickel is 5.

How many different digit symbols does the everyday Hindu–Arabic number system use?

The everyday number system uses ten digits, 0 through 9, which is why it is called base 10 or decimal; computers use only two, 0 and 1.

How many years are in a millennium?

A millennium is 1,000 years, from the Latin mille, 'thousand'; ten centuries make one millennium.

Pythagoras founded his secretive school around 530 BC in which southern Italian city?

Pythagoras, born on Samos, settled in Croton (modern Crotone in Calabria) around 530 BC and founded his secretive brotherhood there.

Archimedes came from which Sicilian city?

Archimedes was born and lived in Syracuse, where he was killed when the Romans captured the city in 212 BC; he anticipated calculus with the method of exhaustion.

Newton shares credit for inventing calculus with which German polymath?

Leibniz also developed binary arithmetic; Newton's Principia appeared in 1687.

Leonhard Euler, who introduced the f(x) notation for functions, was from which country?

Euler was born in Basel, Switzerland, in 1707, and did most of his work abroad in St Petersburg and Berlin.

Fibonacci popularised the Hindu–Arabic numerals in Europe through which 1202 book?

Liber Abaci (1202) taught the Hindu–Arabic numerals to Europe; its rabbit-breeding problem is what produces the Fibonacci sequence.

The word 'algebra' derives from the title of a treatise by which Persian mathematician?

Al-Khwarizmi's treatise Al-Jabr (c. 820) gave systematic solutions of linear and quadratic equations, and its title, al-jabr ('restoring'), became the word 'algebra'.

Hypatia, the mathematician and philosopher killed in AD 415, lived in which city?

Hypatia taught philosophy and astronomy in Alexandria, Egypt, where a Christian mob murdered her in AD 415.

Ada Lovelace is often called the first programmer for her work on whose Analytical Engine?

Charles Babbage's proposed mechanical general-purpose computer was the Analytical Engine; Lovelace's notes on it earned her the 'first programmer' tag.

During World War II, Alan Turing worked at which British codebreaking centre?

Turing worked at Bletchley Park, the Government Code and Cypher School's wartime base, where his bombe machines helped break the German Enigma cipher.

Noether's theorem, named for Emmy Noether, links conservation laws to what?

Noether's theorem (1918) states that every continuous symmetry of a physical system yields a conservation law: symmetry under shifts in time gives conservation of energy, for example.

Which Cambridge mathematician recognised Ramanujan's genius and brought him to England?

Hardy arranged for the self-taught Indian prodigy to travel to Cambridge.

How many problems did David Hilbert publish in his famous 1900 list?

He presented ten of them at the Paris International Congress of Mathematicians.

How many Millennium Prize Problems did the Clay Mathematics Institute select in 2000?

The Clay Mathematics Institute named seven Millennium Prize Problems in 2000, each carrying a one-million-dollar prize; the Poincaré conjecture was the first to fall, in 2003.

Which Russian mathematician proved the Poincaré conjecture in papers posted in 2002–03?

Grigori Perelman completed Richard Hamilton's Ricci-flow programme in three arXiv papers (2002–03) and later declined both the Fields Medal and the Clay million-dollar prize.

Fields Medal winners must be under what age?

A Fields medallist must be under 40 on 1 January of the award year; two to four medals are given every four years at the International Congress of Mathematicians.

The Abel Prize, modelled on the Nobels, is awarded annually by the king of which country?

Norway's king presents the Abel Prize, established in 2002 in honour of Niels Henrik Abel, the Norwegian mathematician who died of tuberculosis at 26.

Goldbach's conjecture states that every even number greater than 2 is the sum of two what?

Goldbach's conjecture says every even number above 2 is the sum of two prime numbers, as 20 = 7 + 13; Goldbach proposed it in a 1742 letter to Euler.

The four colour theorem was the first major theorem proved using what?

Appel and Haken's 1976 proof checked nearly 2,000 map configurations by computer, and many mathematicians initially rejected it because no human could verify it by hand.

In what year did Kurt Gödel publish his incompleteness theorems?

Gödel published his incompleteness theorems in 1931, aged 25, showing that any consistent formal system rich enough for arithmetic contains true statements it cannot prove.

The Erdős number measures 'collaborative distance' from Paul Erdős via what?

Erdős travelled with two suitcases, turning up unannounced to collaborate.

Which Scottish mathematician is best known as the discoverer of logarithms?

John Napier published his logarithms in 1614; he also invented the calculating rods called 'Napier's bones' and helped popularise the decimal point.

Euclid's Elements is a collection of how many books?

Euclid's Elements, written around 300 BC, runs to 13 books covering plane and solid geometry, number theory and incommensurable magnitudes.

Pi Day was founded in 1988 by Larry Shaw, an employee of which San Francisco institution?

Larry Shaw, a physicist at San Francisco's Exploratorium science museum, started Pi Day there in 1988; the US House of Representatives recognised March 14, 2009 as National Pi Day.

The imaginary unit i is defined as a solution to which equation?

x² = −1 has no real solution, since squaring any real number never gives a negative result; i is defined as a solution to it.

Which 17th-century mathematician first used the ∞ symbol for infinity?

John Wallis introduced ∞ for infinity in his 1655 work De sectionibus conicis; the symbol is also called a lemniscate, or 'lazy eight' in livestock branding.

Which Indian mathematician is credited as the first to formalise the number zero?

Brahmagupta's Brāhmasphuṭasiddhānta (AD 628) was the first text to treat zero as a number in its own right and to set out rules for arithmetic with it.

Diagrams showing logical relations between sets were popularised in the 1880s by whom?

John Venn introduced his overlapping-circle diagrams in an 1880 paper; Venn diagrams are used widely in set theory, logic and probability.

In the Monty Hall problem, what should the contestant do to maximise their odds?

Switching doors wins 2/3 of the time, because the first pick is right only 1/3 of the time; Marilyn vos Savant's 1990 Parade column made the puzzle famous.

In a group of 23 people, what is the approximate chance that two share a birthday?

With 23 people the chance of a shared birthday is about 50.7%, the counterintuitive 'birthday paradox'; with 57 people it passes 99%.

Euler's 1736 proof about the Seven Bridges of Königsberg founded which branch of maths?

Euler showed no walk could cross each of Königsberg's seven bridges exactly once, a result regarded as the first theorem of graph theory and a forerunner of topology.

The Rubik's Cube was invented in 1974 by a professor of what?

Ernő Rubik was a professor of architecture at Budapest's Academy of Applied Arts and Crafts, teaching interior design, when he built the cube in 1974.

What was Sudoku originally called?

Howard Garns's puzzle appeared in Dell magazines in 1979 as 'Number Place'; Nikoli introduced it to Japan in 1984 as Sūdoku, roughly 'single digit'.

Eratosthenes is best remembered as the first known person to calculate what?

Eratosthenes estimated Earth's circumference around 240 BC from the different noon shadow angles at Syene and Alexandria; he also devised the prime-number sieve that bears his name.

Mersenne primes are primes of what form?

They are named after a 17th-century French Minim friar.

The Mersenne prime that GIMPS announced in October 2024 has roughly how many decimal digits?

Luke Durant's find, M136279841, runs to 41,024,320 decimal digits, roughly 41 million; it was the largest known prime when GIMPS announced it in October 2024.

The Collatz conjecture's sequences are nicknamed after what weather phenomenon?

Lothar Collatz introduced the idea in 1937.

The number 6174 is known as whose constant?

Kaprekar's constant: take any four-digit number with at least two different digits, subtract its digits sorted ascending from descending, repeat, and you reach 6174 within seven steps.

Which Platonic solid has twelve pentagonal faces?

The dodecahedron has twelve regular pentagons for faces; the icosahedron has twenty triangles, and there are only five Platonic solids in all.

Euler's identity links how many fundamental constants?

Euler's identity, e^(iπ) + 1 = 0, links five constants: 0, 1, π, e and i; a 1990 Mathematical Intelligencer poll named it the most beautiful theorem.

Which lowercase Greek letter most commonly represents standard deviation?

Lowercase sigma (σ) denotes a population's standard deviation, the square root of the variance σ²; mu (μ) is used for the mean.

How many radians equal 360 degrees?

A full 360-degree turn is 2π radians: one radian is the angle whose arc length equals the radius, and 2π radii fit around the circle.

What do you call a quadrilateral whose four sides are all the same length?

A rhombus has four sides of equal length; a square is the special case with right angles, and other names include diamond and lozenge.

What is the smallest pair of amicable numbers?

220 and 284 are amicable: 220's proper divisors sum to 284, and 284's (1, 2, 4, 71, 142) sum to 220; the pair was known to the Pythagoreans.

Under what male pseudonym did Sophie Germain correspond with Lagrange and Gauss?

Germain wrote as 'Monsieur Le Blanc', borrowing a former student's name for fear her work would be dismissed as a woman's; she later won the Paris Academy's 1816 prize on elasticity.

The Banach–Tarski paradox says a solid ball can be cut up and reassembled into what?

It relies on the axiom of choice and a finite number of disjoint pieces.

The 'butterfly effect' is closely associated with which mathematician-meteorologist?

Edward Lorenz, an MIT meteorologist, found in 1961 that tiny rounding changes transformed his weather model's output; his 1972 talk asked whether a butterfly's wings could set off a tornado.

Conway's Game of Life, devised in 1970, is an example of what?

The Game of Life is a cellular automaton: each cell on a grid lives or dies by counting its neighbours; Conway published it in Scientific American in October 1970.

In Zeno's most famous paradox, which swift runner can never overtake a tortoise?

In Zeno's paradox, Achilles can never overtake the tortoise because each time he reaches its last position it has moved on; it illustrates the puzzle of infinite divisibility.

Who coined the word 'googol' in 1920?

Milton Sirotta also suggested 'googolplex': one followed by zeroes until you get tired.

Which young French mathematician, founder of group theory, died in a duel in 1832?

He had only recently been released from prison.

Niels Henrik Abel proved it impossible to solve which type of equation in radicals?

Abel's 1824 proof showed the general quintic (degree-five) equation has no solution in radicals, unlike the quadratic, cubic and quartic; Ruffini had published an incomplete proof earlier.

Who became the first woman to win the Fields Medal, in 2014?

Maryam Mirzakhani, an Iranian-born Stanford professor, won the 2014 Fields Medal for work on the geometry of Riemann surfaces; she died of cancer in 2017 at 40.

Katherine Johnson's orbital calculations were critical to whose crewed spaceflights?

She worked 33 years at NASA and its predecessor NACA.

The Arithmetica, which inspired Fermat's famous marginal note, was written by whom?

Diophantus of Alexandria wrote the Arithmetica around AD 250; ten of its thirteen books survive, and Fermat's note was scribbled beside its problem on splitting a square.

Which is the best-known primitive Pythagorean triple?

(3, 4, 5) is the smallest Pythagorean triple, since 3² + 4² = 9 + 16 = 25 = 5²; (6, 8, 10) is a triple too but not primitive.

After 6, what is the next perfect number?

28 is perfect: its proper divisors 1, 2, 4, 7 and 14 add up to 28; the definition goes back to Euclid's Elements, and the next ones are 496 and 8128.