Math

Sequences & Special Numbers

Fibonacci sequence

Each term is the sum of the two before it: F_n = F_n−1 + F_n−2, with F₁ = F₂ = 1 (often F₀ = 0 is included). learn more · MathWorld

• Origin: Leonardo of Pisa (“Fibonacci,” short for filius Bonacci, “son of Bonacci”) posed the rabbit-breeding problem in Liber Abaci (“Book of Calculation”), 1202. The same book popularized Hindu–Arabic numerals in Europe. The sequence was known centuries earlier in India (Pingala, Virahanka, Hemachandra) in the study of poetic meters. learn more
• Name: French mathematician Édouard Lucas attached Fibonacci's name to the sequence in the 1870s. Lucas also invented the Tower of Hanoi puzzle (1883), which takes 2ⁿ − 1 moves for n disks. learn more · Tower of Hanoi
• Handy facts: every 3rd Fibonacci number is even; F₁ + … + F_n = F_n+2 − 1; Cassini's identity F_n−1F_n+1 − F_n² = (−1)ⁿ; 144 is the only Fibonacci square other than 0 and 1. learn more

Golden ratio φ

The ratio of consecutive Fibonacci numbers F_n+1/F_n approaches the golden ratio φ = (1 + √5)/2 ≈ 1.6180339887. It satisfies φ² = φ + 1 and 1/φ = φ − 1 ≈ 0.618. Euclid called it dividing a line in “extreme and mean ratio.” The letter φ honors the sculptor Phidias. learn more · MathWorld

Binet's formula (closed form): F_n = (φⁿ − ψⁿ)/√5, where ψ = (1 − √5)/2 ≈ −0.618. Because |ψ| < 1, F_n is just φⁿ/√5 rounded to the nearest integer. learn more · MathWorld

Lucas numbers

Same rule, different start (L₁ = 1, L₂ = 3, or L₀ = 2): 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521… Ratios also approach φ, and L_n = φⁿ + ψⁿ = F_n−1 + F_n+1. learn more

Fibonacci in nature

• Spiral counts in sunflower heads (commonly 34 & 55 or 55 & 89), pinecones (8 & 13), pineapples, and daisies; leaf arrangement (phyllotaxis). learn more
• Family tree of a male honeybee (drone): ancestors in each generation are Fibonacci numbers. learn more

If you hear “rabbits,” “Liber Abaci,” “sunflower spirals,” or “each term is the sum of the previous two,” buzz Fibonacci. If you hear “approximately 1.618” or “φ squared equals φ plus one,” buzz the golden ratio.

The nautilus shell is a logarithmic spiral, but careful measurements show it is not a golden spiral — a popular myth. Also, Lucas numbers start 2, 1, 3… not 1, 1, 2. learn more

Figurate numbers

Catalan numbers

C_n = (2n)! / ((n+1)! n!) = C(2n, n)/(n+1): 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796 (C₀ through C₁₀). They count balanced strings of n pairs of parentheses, ways to triangulate a convex (n+2)-gon, binary trees with n nodes, and lattice paths that stay below the diagonal. Named for Belgian mathematician Eugène Catalan. learn more · MathWorld

Prime numbers

A prime has exactly two positive divisors, 1 and itself. There are 25 primes below 100: learn more

• Infinitely many primes: proved by Euclid (Elements, Book IX, Prop. 20). learn more
• Sieve of Eratosthenes: cross out multiples of each prime in turn to find all primes up to N. learn more
• Fundamental Theorem of Arithmetic: every integer > 1 factors uniquely into primes. learn more
• Twin primes differ by 2 (3 & 5, 5 & 7, 11 & 13, 17 & 19, 29 & 31…). Sophie Germain primes: p where 2p+1 is also prime (2, 3, 5, 11, 23, 29…). learn more · Sophie Germain primes
• Prime Number Theorem: the number of primes ≤ x is about x / ln x (proved 1896 by Hadamard and de la Vallée Poussin, independently). learn more
• Mersenne primes have the form 2^p − 1 (3, 7, 31, 127, 8191…), named for French friar Marin Mersenne. The largest known primes are Mersenne primes found by the GIMPS distributed-computing project; as of late 2024 the record was 2^136,279,841 − 1 (over 41 million digits). learn more · largest known prime

1 is not prime (and not composite). 2 is the only even prime. 51 = 3 × 17, 57 = 3 × 19, 91 = 7 × 13 — these “look prime” but aren't.

Perfect numbers

A perfect number equals the sum of its proper divisors: 6 = 1+2+3; 28 = 1+2+4+7+14. The first four — 6, 28, 496, 8128 — were known to the ancient Greeks; the fifth is 33,550,336. learn more

Euclid–Euler theorem: every even perfect number has the form 2^p−1(2^p − 1) where 2^p − 1 is a Mersenne prime (Euclid proved these are perfect; Euler proved all even perfect numbers look like this). So there is one even perfect number per Mersenne prime. Whether any odd perfect number exists is unsolved. learn more

Related: abundant numbers (divisors sum to more, smallest 12), deficient numbers (less), and amicable pairs like 220 & 284. abundant · amicable

Arithmetic & geometric sequences and series

• 1 + 2 + … + 100 = 5050 — the sum young Gauss supposedly found instantly by pairing 1+100, 2+99, … (50 pairs of 101). learn more
• Sums of powers: ∑k = n(n+1)/2, ∑k² = n(n+1)(2n+1)/6, ∑k³ = [n(n+1)/2]² (Nicomachus's theorem), sum of first n odd numbers = n². learn more · Nicomachus
• Classic: ½ + ¼ + ⅛ + … = 1 (a₁ = ½, r = ½). Repeating decimal 0.999… = 9/10 ÷ (1 − 1/10) = 1. learn more

Factorials & powers of 2

0! = 1 by definition (one way to arrange nothing). The “!” notation was introduced by Christian Kramp in 1808. 52! ≈ 8.07 × 10⁶⁷ (orderings of a deck of cards). learn more

learn more · 2¹⁰ = 1024 ≈ 10³ is why a “kilobyte” is sometimes 1024 bytes.

Harmonic series

1 + ½ + ⅓ + ¼ + … diverges (grows without bound), even though its terms go to 0. Classic proof by Nicole Oresme (c. 1350): group terms ½, (⅓+¼) > ½, (⅕+…+⅛) > ½, … so the sum exceeds any number. It grows like ln n; the difference approaches the Euler–Mascheroni constant γ ≈ 0.5772. By contrast, 1 + 1/4 + 1/9 + … = π²/6 (the Basel problem, solved by Euler, 1734). learn more · Basel problem

“Terms go to zero” does not mean a series converges — the harmonic series is the standard counterexample.

Collatz conjecture (3n + 1 problem)

Start with any positive integer. If even, halve it; if odd, multiply by 3 and add 1. Repeat. The conjecture (Lothar Collatz, 1937) says you always eventually reach 1. It is unsolved, though checked by computer for enormous numbers. Example: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. Starting at 27 takes 111 steps and climbs as high as 9232. learn more · MathWorld

Pascal's Triangle & the Binomial Theorem

Each entry is the sum of the two entries above it. Row n (starting with row 0) lists the binomial coefficients C(n, 0), C(n, 1), …, C(n, n). learn more · MathWorld

Row  0:                     1
Row  1:                   1   1
Row  2:                 1   2   1
Row  3:               1   3   3   1
Row  4:             1   4   6   4   1
Row  5:           1   5  10  10   5   1
Row  6:         1   6  15  20  15   6   1
Row  7:       1   7  21  35  35  21   7   1
Row  8:     1   8  28  56  70  56  28   8   1
Row  9:   1   9  36  84 126 126  84  36   9   1
Row 10: 1  10  45 120 210 252 210 120  45  10   1

Properties

Binomial theorem

Example: (x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴. The expansion of (x+y)ⁿ has n + 1 terms. The coefficient of x²y³ in (x+y)⁵ is C(5,3) = 10. With constants, include their powers: the coefficient of x² in (2x + 1)⁴ is C(4,2)·2² = 24. Newton generalized the theorem to fractional and negative exponents (binomial series). learn more · binomial series

Names around the world

“Yang Hui,” “Tartaglia,” “Khayyam,” or “each number is the sum of the two above it” → Pascal's triangle.

Combinatorics & Probability

Permutations & combinations

Common C(n, k) values

Counting principles

• Handshake problem: if n people each shake hands once with everyone else, there are C(n,2) = n(n−1)/2 handshakes. 10 people → 45. (Same as games in a round-robin.) learn more ·MathWorld
• Diagonals of an n-gon: n(n−3)/2. Pentagon 5, hexagon 9, octagon 20, decagon 35. learn more
• Pigeonhole principle (Dirichlet's box principle): put n+1 pigeons in n holes and some hole holds two. Among 13 people, two share a birth month; among 367, two share a birthday. learn more
• Inclusion–exclusion: |A ∪ B| = |A| + |B| − |A ∩ B|; for three sets add back the triple intersection: |A∪B∪C| = |A|+|B|+|C| − |A∩B| − |A∩C| − |B∩C| + |A∩B∩C|. Numbers 1–100 divisible by 2 or 3: 50 + 33 − 16 = 67. learn more

“Permutation” = order matters (a lock “combination” is really a permutation!). If swapping two chosen items gives the same result, use combinations.

Probability basics

• P(event) = favorable outcomes / total equally likely outcomes, always between 0 and 1. learn more
• Complement: P(not A) = 1 − P(A). “At least one” problems are easiest this way: P(at least one 6 in 4 rolls) = 1 − (5/6)⁴ ≈ 0.518 (the Chevalier de Méré's problem that started the Pascal–Fermat letters of 1654). learn more
• Independent events: P(A and B) = P(A)·P(B). Mutually exclusive: P(A or B) = P(A) + P(B); in general P(A or B) = P(A) + P(B) − P(A and B). learn more
• Odds in favor = favorable : unfavorable (P = 1/4 means odds 1 : 3). learn more
• Expected value: E = ∑ (value × probability). One fair die: (1+2+…+6)/6 = 3.5; sum of two dice: 7. learn more
• Birthday problem: with just 23 people, the chance two share a birthday exceeds 50% (≈ 50.7%); with 70 it's about 99.9%. learn more
• Monty Hall problem: pick one of 3 doors; the host (who knows) opens a goat door; switching wins 2/3 of the time. Made famous by Marilyn vos Savant's 1990 Parade column. learn more
• Probability theory's founders: Pascal & Fermat (1654 letters on the “problem of points”); Cardano's earlier gambling book Liber de ludo aleae; Jacob Bernoulli's Ars Conjectandi (1713, law of large numbers); Kolmogorov's axioms (1933). learn more

Dice & cards facts

Two dice — 36 equally likely outcomes: learn more

Opposite faces of a standard die sum to 7. Doubles: 6/36 = 1/6.

Standard deck: 52 cards = 4 suits (♠ spades, ♥ hearts, ♦ diamonds, ♣ clubs) × 13 ranks (A, 2–10, J, Q, K). 26 red, 26 black; 12 face cards (J, Q, K); 4 aces. P(drawing a heart) = 1/4; P(an ace) = 1/13; P(a face card) = 3/13. 5-card hands: C(52,5) = 2,598,960. learn more

Geometry

Areas & perimeters

Inradius r = Area/s; circumradius R = abc/(4·Area). An inscribed angle is half its intercepted central angle; an angle inscribed in a semicircle is a right angle (Thales' theorem). learn more · Thales' theorem

Polygon names & angles

Sum of interior angles (n − 2)·180°; each interior angle of a regular n-gon (n − 2)·180°/n; each exterior angle 360°/n; exterior angles of any convex polygon sum to 360°. learn more · polygons

Also: 1000 sides = chiliagon (Descartes used it as an example of something you can conceive but not picture); 10,000 = myriagon. learn more

Only the equilateral triangle, square, and regular hexagon tile the plane by themselves (their interior angles 60°, 90°, 120° divide 360°). learn more

Volumes & surface areas

“Volume and surface area are both ⅔ of the circumscribing cylinder” → Archimedes, who considered it his greatest result and asked for a sphere inside a cylinder on his tomb (On the Sphere and Cylinder). learn more

Cavalieri's principle: solids with equal cross-sectional areas at every height have equal volumes (Bonaventura Cavalieri, 1635). learn more

Platonic solids & Euler's formula

The only five convex regular polyhedra (Euclid proves this at the end of Elements Book XIII). Plato's Timaeus linked them to the classical elements. learn more · MathWorld

Stated by Euler in 1752 (Descartes had noticed a related result). The number 2 is the Euler characteristic of a sphere; for a torus it's 0. A soccer ball is a truncated icosahedron (12 pentagons + 20 hexagons). learn more · truncated icosahedron

Duals swap faces and vertices: the cube (6 F, 8 V) and octahedron (8 F, 6 V) are duals; the dodecahedron and icosahedron are duals. Don't mix up the 12 and 20: the dodecahedron has 12 faces but 20 vertices.

Pythagorean theorem & triples

Euclid's Elements Book I, Proposition 47 (the “windmill” proof). Known much earlier to the Babylonians (tablet Plimpton 322, c. 1800 BC, lists triples) and in China as the gougu theorem. Hundreds of proofs exist — see Garfield and Loomis under Unexpected Mathematicians. learn more · Plimpton 322

Euclid's formula generates triples: (m² − n², 2mn, m² + n²) for m > n > 0. m=2, n=1 gives 3-4-5. learn more

Special right triangles

Triangle centers

The Euler line passes through the orthocenter, centroid, circumcenter (and nine-point center); the centroid sits between, twice as far from the orthocenter as from the circumcenter. The incenter is generally not on it (only for isosceles triangles). In an equilateral triangle all centers coincide. The nine-point circle passes through the 3 side midpoints, 3 altitude feet, and 3 midpoints from the orthocenter to the vertices. learn more · nine-point circle

Match the clue word to the center: “medians” → centroid; “perpendicular bisectors” or “equidistant from the vertices” → circumcenter; “angle bisectors” or “inscribed circle” → incenter; “altitudes” → orthocenter.

Coordinate formulas

• Distance: d = √((x₂−x₁)² + (y₂−y₁)²) (in 3D add (z₂−z₁)²). learn more
• Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2). learn more
• Circle: (x−h)² + (y−k)² = r², center (h, k). learn more
• Distance from point to line Ax + By + C = 0: |Ax₀ + By₀ + C| / √(A²+B²). learn more
• Shoelace formula (polygon area from vertices): A = ½|∑(x_iy_i+1 − x_i+1y_i)|. learn more
• Conic sections (Apollonius named them): circle, ellipse x²/a² + y²/b² = 1, parabola y = ax², hyperbola x²/a² − y²/b² = 1. Eccentricity: circle 0, ellipse 0–1, parabola 1, hyperbola > 1. learn more

Trigonometry

SOH-CAH-TOA & reciprocal functions

Secant is the reciprocal of cosine and cosecant is the reciprocal of sine — the “co-” pairs with the one without “co-”.

Unit-circle values

On the unit circle, the point at angle θ is (cos θ, sin θ). learn more · exact values

Pattern for sin at 0°, 30°, 45°, 60°, 90°: √0/2, √1/2, √2/2, √3/2, √4/2. Cosine runs the same list backward.

Degrees ↔ radians

180° = π radians. Degrees → radians: multiply by π/180. Radians → degrees: multiply by 180/π. One radian ≈ 57.2958° (the angle whose arc equals the radius). A full circle is 2π rad = 360°. learn more

ASTC quadrant rule

“All Students Take Calculus,” going counterclockwise from Quadrant I, tells which functions are positive: learn more

Identities

Law of sines & law of cosines

Use the law of sines for AAS/ASA (and the ambiguous SSA case); use the law of cosines for SAS and SSS. law of sines · law of cosines

Periods & amplitudes

For y = A sin(Bx − C) + D: amplitude = |A|, period = 2π/|B|, phase shift = C/B, vertical shift = D. learn more

Example: y = 3 sin(2x) has amplitude 3 and period π. Trig history: Hipparchus (table of chords, “father of trigonometry”); the word “sine” comes via a Latin mistranslation (sinus, “bay/fold”) of an Arabic transliteration of Sanskrit jya. learn more

Algebra & Number Sense

Quadratics

Discriminant Δ = b² − 4ac: positive → two real roots; zero → one repeated real root; negative → two complex-conjugate roots. A perfect-square Δ (with rational coefficients) gives rational roots. Vertex at x = −b/2a. learn more · discriminant

Vieta's formulas (François Viète): for ax² + bx + c, sum of roots = −b/a, product = c/a. For a cubic ax³ + bx² + cx + d: sum = −b/a, sum of pairwise products = c/a, product = −d/a. Example: x² − 7x + 10 has roots summing to 7 and multiplying to 10 (roots 2 and 5). learn more

Watch the sign: sum of roots is −b/a, not b/a. For x² + 5x + 6, the roots (−2, −3) sum to −5.

History: cubic solved by del Ferro/Tartaglia and published by Cardano (Ars Magna, 1545); quartic by Ferrari; the general quintic has no formula in radicals (Abel–Ruffini theorem; Galois explained why). learn more

Factoring patterns

Cube factoring mnemonic SOAP: signs go Same, Opposite, Always Positive.

Exponent & logarithm rules

e ≈ 2.718281828, base of the natural log (ln). Useful values: log₁₀2 ≈ 0.3010, log₁₀3 ≈ 0.4771, ln 2 ≈ 0.693, ln 10 ≈ 2.303. Logarithms were invented by John Napier (1614); Henry Briggs made common (base-10) logs. Rule of 72: money doubles in about 72/(interest rate %) years. e · rule of 72

Compound interest: A = P(1 + r/n)^nt; continuous: A = Pe^rt. learn more

Lines & slope

Parallel lines have equal slopes; perpendicular slopes are negative reciprocals (product −1). Horizontal line: slope 0; vertical line: slope undefined. learn more

Absolute value

|x| is the distance from 0, so it's never negative. |x − a| < b ⇔ a − b < x < a + b; |x − a| > b ⇔ x < a − b or x > a + b. Solve |2x − 3| = 7: 2x − 3 = ±7, so x = 5 or x = −2. The graph of y = |x| is a V. learn more

Squares, cubes & roots to memorize

√2 ≈ 1.41421   √3 ≈ 1.73205   √5 ≈ 2.23607   (√10 ≈ 3.162). √2 · √3 · √5

Squaring numbers ending in 5: take the tens digit n, compute n(n+1), append 25. 65²: 6·7 = 42 → 4225. 1729 = 1³ + 12³ = 9³ + 10³, the Hardy–Ramanujan “taxicab number.” learn more

Number bases

In base b, place values are powers of b. Binary (base 2) uses 0–1; hexadecimal (base 16) uses 0–9 then A=10, B=11, C=12, D=13, E=14, F=15. binary · hexadecimal · positional notation

Shortcut: each hex digit = 4 binary digits (1101 0110₂ = D6₁₆ = 214). Babylonians used base 60 (sexagesimal — hence 60 minutes, 360°); the Maya used base 20 (vigesimal). sexagesimal · Maya numerals

Divisibility rules

learn more · GCD × LCM = product of two numbers; Euclid's algorithm finds the GCD by repeated division. Euclidean algorithm

Complex Numbers

i = √−1, so i² = −1. A complex number is a + bi (a = real part, b = imaginary part). Descartes coined “imaginary” (dismissively); Euler introduced the symbol i (1777); Gauss popularized “complex number.” The complex plane is also called the Argand diagram (Jean-Robert Argand, 1806; Caspar Wessel, 1799). learn more · imaginary unit · complex plane

Powers of i (cycle of 4)

To find iⁿ, divide n by 4 and use the remainder: i²⁰²⁶: 2026 = 4·506 + 2 → i² = −1. The sum of any four consecutive powers of i is 0. learn more

Arithmetic, conjugate, modulus

• Add: (a+bi) + (c+di) = (a+c) + (b+d)i. Multiply: (a+bi)(c+di) = (ac − bd) + (ad + bc)i. Example (2+3i)(1−i) = 2 − 2i + 3i − 3i² = 5 + i. learn more
• Conjugate: conj(a + bi) = a − bi; (a+bi)(a−bi) = a² + b² (real). Divide by multiplying top and bottom by the conjugate. Non-real roots of real polynomials come in conjugate pairs. learn more
• Modulus (absolute value): |a + bi| = √(a² + b²); |3 + 4i| = 5. Argument θ = the angle from the positive real axis. learn more

Polar form, Euler's formula, De Moivre

Multiplying complex numbers multiplies moduli and adds angles. Euler's identity is often voted “the most beautiful equation in mathematics.” Surprising fact: iⁱ = e^−π/2 ≈ 0.2079, a real number. Abraham de Moivre was a French Huguenot who lived in London. Euler's formula · Euler's identity · De Moivre

Roots of unity

The n solutions of zⁿ = 1 are e^2πik/n, k = 0, 1, …, n−1 — equally spaced points on the unit circle forming a regular n-gon. They sum to 0 (for n ≥ 2). Cube roots of unity: 1, −½ ± (√3/2)i. Fourth roots: 1, i, −1, −i. learn more

Fundamental Theorem of Algebra: every degree-n polynomial has exactly n complex roots (counting multiplicity). Gauss's 1799 doctoral dissertation gave a (nearly complete) proof. learn more

Calculus

Limits

lim_x→a f(x) = L means f(x) gets arbitrarily close to L as x approaches a. Cauchy and Weierstrass made this rigorous with the ε–δ (epsilon–delta) definition. Famous limits: lim_x→0 (sin x)/x = 1; lim_n→∞ (1 + 1/n)ⁿ = e; lim_x→∞ 1/x = 0. A function is continuous at a if lim_x→a f(x) = f(a). learn more · ε–δ

Derivatives

The derivative is the instantaneous rate of change = slope of the tangent line. Velocity is the derivative of position; acceleration is the derivative of velocity. learn more · Khan Academy

learn more · Mean value theorem (and its special case, Rolle's theorem)

Integrals

The definite integral ∫_a^b f(x) dx is the signed area under the curve, defined as a limit of Riemann sums. The “+ C” (constant of integration) is required for indefinite integrals. learn more · lists of integrals · Riemann sum

Fundamental Theorem of Calculus

Differentiation and integration are inverse operations. Isaac Barrow (Newton's teacher) proved a geometric version; Newton and Leibniz developed it systematically. learn more

L'Hôpital's rule

If f(x)/g(x) gives 0/0 or ∞/∞ as x → a, then lim f/g = lim f′/g′ (if the latter exists). Example: lim_x→0 sin x / x = lim cos x / 1 = 1. Named for the Marquis de l'Hôpital, whose 1696 book Analyse des infiniment petits was the first calculus textbook — but the rule was really discovered by his paid tutor Johann Bernoulli. learn more

L'Hôpital's rule differentiates numerator and denominator separately — it is not the quotient rule.

Newton vs. Leibniz

Isaac Newton developed his “method of fluxions” around 1665–66 but published late (parts in Principia, 1687; full account posthumously). Gottfried Wilhelm Leibniz developed calculus independently in the 1670s and published first (1684, “Nova Methodus pro Maximis et Minimis”). A bitter priority dispute followed; a 1712 Royal Society committee — secretly guided by Newton, its president — ruled for Newton. Today both are credited with independent discovery. British loyalty to Newton's clumsier notation is said to have slowed British mathematics for a century. learn more

learn more · Taylor series: e^x = 1 + x + x²/2! + x³/3! + …; sin x = x − x³/3! + x⁵/5! − … (Maclaurin series are Taylor series centered at 0). Taylor series

“Fluxions” → Newton. “Monads,” “binary arithmetic,” “stepped reckoner,” or the dy/dx notation → Leibniz. “First calculus textbook” → L'Hôpital.

Statistics

The median is resistant to outliers; the mean is not. Adding one billionaire to a room barely moves the median income but drags the mean way up.

Standard deviation

Standard deviation (σ) measures spread: roughly the typical distance of values from the mean. Variance = σ² = the average squared deviation from the mean; σ = √variance. (Sample standard deviation divides by n − 1 instead of n.) Data 2, 4, 4, 4, 5, 5, 7, 9: mean 5, variance 4, σ = 2. Adding a constant to every value doesn't change σ; multiplying every value by k multiplies σ by |k|. A z-score z = (x − μ)/σ counts standard deviations from the mean. learn more · z-score

The 68-95-99.7 (empirical) rule

For a normal distribution (bell curve, Gaussian distribution): about 68% of data lies within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. Example: IQ scores (mean 100, σ = 15) → 68% between 85 and 115, 95% between 70 and 130. learn more · normal distribution

• The normal curve was studied by de Moivre (1733) and Gauss (hence “Gaussian”); the Central Limit Theorem explains why averages tend to be normal. learn more
• Correlation coefficient r ranges from −1 to 1 (Pearson). Correlation does not imply causation. learn more
• Simpson's paradox: a trend in several groups reverses when the groups are combined. learn more
• Benford's law: in many real data sets, the leading digit is 1 about 30% of the time. learn more

Mathematicians

Written the way clues appear: the hardest (most obscure) facts first, the giveaway last. Dates for ancient figures are approximate (c. = circa, fl. = flourished). learn more · MacTutor History of Mathematics

Ancient & medieval

Early modern (1500–1800)

19th century

20th & 21st centuries

“A mathematician is a machine for turning coffee into theorems” is usually attributed to Erdős, but he credited it to his friend Alfréd Rényi. learn more

Don't confuse the Bernoullis: Jacob = probability/Ars Conjectandi/law of large numbers; Johann = brachistochrone/taught Euler/L'Hôpital's rule; Daniel (Johann's son) = fluid flow (Bernoulli's principle).

Non-Euclidean geometry trio: Lobachevsky (Russian), Bolyai (Hungarian), and Gauss (who claimed he'd found it first but didn't publish). Riemann later developed elliptic/spherical geometry.

Unexpected Mathematicians

Politicians, writers, nurses, and artists who did real math — a favorite quiz-bowl crossover with history and literature.

James A. Garfield's proof of the Pythagorean theorem

In 1876, while serving as a U.S. Representative from Ohio (he was in the House 1863–1880), James A. Garfield — later the 20th President — devised an original proof of the Pythagorean theorem. It was published in the New-England Journal of Education (April 1, 1876), which noted it was something members of both parties could agree on. Garfield had been a teacher and principal at the Western Reserve Eclectic Institute (now Hiram College) before the Civil War. learn more · Pythagorean theorem · Cut-the-Knot proofs

The proof (trapezoid method):

1. Take a right triangle with legs a, b and hypotenuse c. Place a second, congruent copy rotated 90° so the two triangles sit end to end; their legs line up to form one straight side of length a + b.
2. Connect the free ends. The figure is a trapezoid with parallel sides a and b and height a + b.
3. The trapezoid is made of three right triangles: the two congruent ones (each area ½ab) and an isosceles right triangle with both legs c (area ½c²). The angle between the two c's is 90° because the two acute angles of the original triangle sum to 90°.
4. Set the areas equal:

(It is exactly half of the classic “square inside a square” proof figure.)

“A future U.S. president,” “a trapezoid,” “1876,” “New-England Journal of Education” → James A. Garfield. He was later assassinated by Charles Guiteau (shot July 2, 1881; died September 19, 1881).

Quiz questions sometimes loosely call Garfield a “senator” when he wrote the proof. He was a Representative. The Ohio legislature elected him to the U.S. Senate in January 1880, but he won the presidency that November and never served in the Senate.

Other surprising mathematicians

A popular story says Queen Victoria enjoyed Alice and asked for Carroll's next book, receiving a math text. Dodgson himself publicly denied this ever happened.

Famous Problems & Prizes

Millennium Prize Problems

Announced by the Clay Mathematics Institute in Paris on May 24, 2000: seven problems, $1 million each. learn more · Clay Institute

1. Birch and Swinnerton-Dyer conjecture
2. Hodge conjecture
3. Navier–Stokes existence and smoothness
4. P versus NP
5. Poincaré conjecture — solved (Perelman; prize declined)
6. Riemann hypothesis
7. Yang–Mills existence and mass gap

Hilbert's 23 problems

Presented by David Hilbert at the International Congress of Mathematicians in Paris, August 1900 (he discussed 10 in the talk; the full list of 23 was published). Highlights: #1 continuum hypothesis (shown independent by Gödel 1940 and Cohen 1963); #2 consistency of arithmetic (affected by Gödel's incompleteness); #8 Riemann hypothesis (still open); #10 algorithm for Diophantine equations (proved impossible by Yuri Matiyasevich, 1970). learn more

Seven Bridges of Königsberg

Can you walk through Königsberg (Prussia; today Kaliningrad, Russia) crossing each of its seven bridges over the Pregel River exactly once? Euler proved in 1736 that it's impossible: such a walk requires 0 or 2 land masses with an odd number of bridges, and Königsberg had 4. This founded graph theory (and foreshadowed topology); the path is now called an Eulerian path. learn more · Eulerian path

Three classical construction problems

Using only a compass and straightedge, the Greeks tried — and all three were later proved impossible: learn more

Related: a regular n-gon is constructible iff n is a power of 2 times distinct Fermat primes (3, 5, 17, 257, 65537) — Gauss–Wantzel theorem. So the heptagon (7) and nonagon (9) are not constructible; the 17-gon is. learn more

“Delian problem” or “oracle at Delos” → doubling the cube. “Lindemann” or “π is transcendental” → squaring the circle.

Fields Medal & Abel Prize

There is no Nobel Prize in mathematics. The legend that Nobel skipped math because a mathematician (Mittag-Leffler) had an affair with his wife is false — Alfred Nobel never married. learn more

Constants, Symbols & the Greek Alphabet

Famous constants

π and e are both transcendental (not roots of any integer polynomial); √2 and φ are irrational but algebraic (roots of x² − 2 and x² − x − 1). learn more

Who introduced the symbols

learn more about the history of notation

Greek alphabet

24 letters, from alpha to omega (“the alpha and the omega” = beginning and end). learn more · Greek letters in math