50 Fun Facts About High School Math

How many people in a room make it more likely than not that two share a birthday, the famous birthday paradox?

With 23 people the chance is 50.7%, because there are 253 different pairs who might match, not just 23 people compared with you.

In the Monty Hall problem, what is the chance of winning the car by switching doors after the host reveals a goat?

Switching wins 2/3 of the time, a result so counter-intuitive that nearly 1,000 PhDs wrote to Parade magazine insisting Marilyn vos Savant had got it wrong.

Which Greek, whose followers reportedly could not eat beans, gives his name to the right-triangle rule a² + b² = c²?

Pythagoras's rule has been proved more ways than possibly any other theorem, including a proof published by future US president James Garfield.

Which constant, celebrated on March 14 (also Einstein's birthday), is a circle's circumference divided by its diameter?

NASA's Jet Propulsion Laboratory uses π to just 15 decimal places even for interplanetary navigation, enough to size a circle 40 billion miles across with an error of little more than half an inch.

After twenty heads in a row, what is the probability that a fair coin lands heads on the next flip?

The chance stays 50% because a coin has no memory; believing a tail is 'due' is the gambler's fallacy, also called the Monte Carlo fallacy after a 1913 roulette run of 26 blacks.

A price is cut by 20% and then cut by 20% again; what percentage of the original price remains after both reductions?

Two 20% cuts leave 64% because the second cut comes off the already-reduced price: 0.8 × 0.8 = 0.64, not the 0.6 that adding the cuts suggests.

What is the repeating decimal 0.999…, the limit of the sequence 0.9, 0.99, 0.999 and so on, in ordinary arithmetic?

0.999… is exactly 1: tripling 1/3 = 0.333… gives 0.999… on one side and 1 on the other, so the two decimals are simply two names for one number.

What do the numbers 1 to 100 add up to, famously found fast by pairing 1 with 100, 2 with 99?

The total is 5,050 because the fifty pairs 1 + 100, 2 + 99 … 50 + 51 each make 101, and 50 × 101 = 5,050.

Which 9th-century Persian scholar's Latinised name gave us 'algorithm', and his book Al-Jabr the word 'algebra'?

Al-Khwarizmi's Al-Jabr, written around 820, set out the first systematic solution of linear and quadratic equations, with 'al-jabr' meaning restoring or completion.

Which Pisan merchant's son brought the sequence 1, 1, 2, 3, 5, 8 to Europe in 1202 via a puzzle about rabbits?

Fibonacci did not invent the sequence: Indian scholars had described it centuries earlier, and his 1202 Liber Abaci mattered more for spreading the ten Indian digits, zero included, across Europe.

What is b² − 4ac called, the expression under the quadratic formula's square root, whose sign counts the real roots?

A negative discriminant means the quadratic's graph never touches the x-axis, so its two roots are complex conjugates rather than real numbers.

In the line equation y = mx + b, what is m, a quantity that is undefined for a vertical line?

Nobody knows for certain why m was chosen for the slope; the letter first appears in an 1844 English textbook that wrote the line as y = mx + b.

What is 2 + 3 × 4 under the usual order of operations, the convention that ranks multiplication above addition?

Multiplying first gives 2 + 12 = 14; many simple calculators work strictly left to right and would show 20, which is why the convention matters.

A colony doubles every hour; by what factor has it grown after ten hours, a number every computer scientist knows?

Ten doublings give 2¹⁰ = 1,024, which is why a kilobyte traditionally meant 1,024 bytes rather than a round 1,000.

Which curve, the graph of every quadratic function, did Galileo show is also the path of a thrown projectile?

A parabola reflects every ray arriving parallel to its axis through one focus, which is why satellite dishes and car headlight reflectors use the shape.

Which French philosopher's La Géométrie, appended to his 1637 Discourse on the Method, united algebra and geometry?

Descartes also coined the word 'imaginary' for square roots of negative numbers, intending it as a put-down of a concept he thought useless.

Which Scottish laird published the first logarithms in 1614, so astronomers could multiply big numbers by adding?

Napier's invention, said Laplace, 'doubled the life of the astronomer' by reducing months of calculation to a few days.

Which number, about 1.618 and written with the Greek letter phi, is the positive solution of x² = x + 1?

The golden ratio is the hardest number to approximate with fractions: its continued fraction is all 1s, so approximations such as 13/8 converge as slowly as possible.

Which constant, about 2.718, is the base of natural logarithms and the limit of (1 + 1/n)ⁿ as n grows without bound?

The constant e was found by Jacob Bernoulli in 1683 while studying compound interest: compounding a 100% yearly rate ever more often tops out at a factor of 2.718.

Which number, the length of a 1 × 1 square's diagonal, was probably the first ever proved to be irrational?

√2's irrationality so unsettled the ancient Greeks that legend has Hippasus drowned at sea for revealing that some lengths cannot be written as fractions.

How many degrees do a flat triangle's three angles add up to, a total that grows larger on a sphere?

On a sphere a triangle's angles add up to more than 180° and can reach as much as 540°; the surplus is called the spherical excess.

What is the sum of the interior angles of an octagon, the shape of the familiar red stop sign?

An octagon's angles total 1,080° because it splits into six triangles from one corner; each corner of a regular octagon measures 135°.

Besides the equilateral triangle and square, which regular polygon is the only one that tiles a floor by itself?

Regular hexagons tile with three meeting at every corner, and a honeycomb uses them because the shape makes the most efficient use of space and wax.

Which polygon, the shape of Britain's 50p coin, is the smallest regular one that compass and straightedge cannot draw?

The heptagon became a coin shape in 1969 with the 50p, the world's first seven-sided coin; its curved sides have constant width, so it rolls through vending machines.

Which side of a right triangle, always the longest, is the denominator of both SOH and CAH in the SOHCAHTOA mnemonic?

Hypotenuse comes from the Greek for 'stretching under' the right angle; the word reached English in the 1570s by way of Latin and French.

How far apart are the points (0, 0) and (6, 8) on a coordinate grid, by the distance formula?

The distance is 10 because √(6² + 8²) = √100; the two points and the origin's axes form a 6-8-10 triangle, a scaled-up 3-4-5.

An arc subtends 40° at a circle's centre; what angle does it subtend at a point on the remaining circumference?

The inscribed angle is always half the central angle on the same arc, so 40° at the centre becomes 20° at the edge, wherever on the arc the point sits.

How many Platonic solids exist, the shapes Kepler nested inside one another in 1596 to model the Solar System?

The five are the tetrahedron, cube, octahedron, dodecahedron and icosahedron; Plato matched four of them to earth, air, fire and water.

Whose Elements, compiled around 300 BC in Alexandria, contains the classic proof that the prime numbers never run out?

Euclid's Elements has appeared in over a thousand editions since its first printing in 1482, a record said to be second only to the Bible.

Which Greek, reputed to have predicted a 585 BC eclipse, is credited with proving every angle in a semicircle is 90°?

Thales is said to have measured the Great Pyramid's height from its shadow, at the moment when his own shadow equalled his height.

Which trig function, opposite over adjacent, is named from the Latin 'touching' because its line touches the circle?

Tangent's partner secant comes from the Latin secans, 'cutting', because the secant line cuts through the unit circle that the tangent line only touches.

Which rule for any triangle's third side collapses to a² + b² = c² when the included angle is a right angle?

The law of cosines appears in Book II of the Elements around 300 BC as a statement about rectangle areas, long before sines or cosines existed.

Roughly how many degrees make one radian, the angle whose arc is as long as the circle's radius?

One radian is about 57.3° because a circle's circumference is about 6.28 radii long, so a full 360° turn holds only 6.28 radians.

Which Greek astronomer, famed for discovering the precession of the equinoxes, is called the founder of trigonometry?

Hipparchus compiled the first known trigonometric table, listing chord lengths for angles, which he needed to work out the orbits of the Moon and Sun.

Which Greek of Syracuse asked for a sphere inside a cylinder to be carved on his tomb, marking his favourite discovery?

Archimedes was killed by a Roman soldier during the sack of Syracuse in 212 BC despite orders that he be spared; Cicero later found the tomb overgrown with bushes.

Which Alexandrian librarian, first to calculate Earth's circumference, gives his name to a prime-finding 'sieve'?

Eratosthenes' sieve crosses out the multiples of each prime in turn, starting from 2, and is still one of the most efficient ways to list the smaller primes.

Which Swiss mathematician, author of some 866 publications, was the first to write f(x) for a function applied to x?

Euler also introduced the capital sigma Σ for summation, and is regarded as the most prolific contributor in the history of mathematics and science.

Which German, called the 'Prince of Mathematicians', gives his name to the bell-shaped curve of the normal distribution?

Gauss settled as a teenager in 1796 a problem open since the ancient Greeks — which regular polygons can be drawn with compass and straightedge — adding the 17-gon to the list.

Which self-taught Indian mathematician famously told G. H. Hardy that the taxi number 1729 was far from dull?

Ramanujan compiled nearly 3,900 results with almost no formal training; 1729 is the smallest number that is a sum of two cubes two ways, 1³ + 12³ and 9³ + 10³.

Which average do reports of incomes prefer to the mean, since a handful of billionaires can drag the mean upward?

The median is the middle value, so in the list 1, 2, 3, 4, 1,000 it stays at 3 while the mean leaps to 202.

Which measure of spread, the square root of the variance, was named in print by Karl Pearson in 1894?

About 68% of values in a normal distribution lie within one standard deviation of the mean, 95% within two and 99.7% within three.

Opposite faces of an ordinary die add up to which number, forcing the 1, 2 and 3 faces to share a corner?

Seven is also the likeliest total when two dice are thrown: six of the 36 possible combinations make it, from 1 + 6 through to 6 + 1.

In how many different orders can five people line up for a photo, the number written as 5! in maths?

5! = 5 × 4 × 3 × 2 × 1 = 120; the exclamation-mark notation for factorials was introduced by Christian Kramp in 1808.

Which French prodigy, builder of a calculator before he was 19, founded probability theory in 1654 letters on gambling?

Pascal's triangle of binomial coefficients was known in China and Persia centuries before him; his treatise on it appeared only after his death, in 1665.

Which French magistrate wrote in a 1637 margin note that he had a proof 'too large to fit', unproved until 1994?

Fermat's Last Theorem resisted every attempt for over three and a half centuries until Andrew Wiles completed a proof in 1994, using mathematics Fermat could never have known.

Whose conjecture, that every even number above 2 is the sum of two primes, is verified to 4 × 10¹⁸ but unproved?

Goldbach's 'weak' version — every odd number above 5 is a sum of three primes — has had a proof by Harald Helfgott since 2013, but the even case remains open.

Which number is the smallest 'perfect' number, one equal to the sum of its proper divisors, with 28 being the next?

6 = 1 + 2 + 3; every perfect number ever found is even, and nobody knows whether an odd perfect number exists.

Which Alexandrian engineer, describer of the first recorded steam engine, found a triangle's area from its sides alone?

Heron's formula is area = √(s(s−a)(s−b)(s−c)), where s is half the perimeter, so no height or angle is needed; his aeolipile was a rocket-like steam engine.

Which Italian, left with a speech impediment by a sabre wound in the 1512 sack of Brescia, solved the cubic equation?

Tartaglia revealed his cubic solution to Gerolamo Cardano in 1539 on a promise of secrecy, which Cardano broke by publishing it, sparking a famous feud.

Which Indian mathematician's 628 treatise first gave rules for arithmetic with zero and negative numbers?

Brahmagupta got nearly everything right, but claimed that zero divided by zero is zero — a result modern mathematics leaves undefined.