This is the “#” band of the HBD2 Encyclopedia — every entry whose headword is a number or a symbol rather than a letter. It runs from 0 and the constants pi and the golden ratio up to the digits of the largest known prime and the speed of light, then on to the signs of mathematics (and their inventors), the planets and the zodiac, and the world’s currencies. Each entry is fact-checked.
Numbers and mathematical constants
Listed in value order, with the irrational constants φ, e, and π placed where they fall on the number line.
| 0 |
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|---|---|
| 1 |
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| 1.618… | The golden ratio (φ): the proportion of a line divided so that the whole is to the larger part as the larger part is to the smaller. |
| 2 |
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| 2.718… | e, Euler’s number — the base of the natural logarithm. |
| 3 |
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| 3.14159… | Pi (π): the ratio of a circle’s circumference to its diameter, an irrational number that never ends or repeats. |
| 4 |
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| 5 |
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| 6 |
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| 7 |
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| 8 |
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| 9 |
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| 10 |
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| 11 |
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| 12 |
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| 13 |
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| 14 |
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| 15 | The number of players on each side in rugby union. |
| 16 |
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| 17 |
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| 18 |
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| 19 |
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| 20 |
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| 21 |
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| 22 |
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| 23 | The number of pairs of chromosomes in a human body cell (46 in total); a sex cell carries a single set of 23. |
| 24 |
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| 25 |
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| 26 |
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| 27 |
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| 27.3 | The number of days the Moon takes to orbit Earth once against the stars (a sidereal month). |
| 28 |
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| 29 |
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| 29.5 | The number of days in a full cycle of lunar phases, from one new Moon to the next (a synodic month). |
| 30 |
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| 31 |
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| 32 |
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| 33 |
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| 34 |
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| 35 | The minimum age to serve as president or vice president of the United States. |
| 36 | The number of inches in a yard (91.44 cm). |
| 37 |
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| 39 |
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| 40 | The number of years marking a ruby jubilee. |
| 42 | In The Hitchhiker’s Guide to the Galaxy, the answer to the ultimate question of life, the universe and everything. |
| 46 |
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| 50 |
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| 52 | The number of cards in a standard deck, excluding jokers. |
| 54 |
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| 56 | The number of signers of the United States Declaration of Independence. |
| 60 |
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| 64 | The number of squares on a chessboard. |
| 65 | The number of years marking a sapphire jubilee. |
| 66 | The number of books in the Protestant Bible (39 in the Old Testament, 27 in the New). |
| 70 | The number of years marking a platinum jubilee. |
| 74 | The number of letters in the Khmer alphabet, the longest alphabet in the world. |
| 78 | The number of cards in a tarot deck: 22 Major Arcana and 56 Minor Arcana. |
| 88 |
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| 90 | The number of degrees in a right angle. |
| 95 | The number of theses Martin Luther posted in 1517, which helped to spark the Reformation. |
| 99 | In Islamic tradition, the ninety-nine names of God (al-Asma al-Husna). |
| 100 |
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| 108 |
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| 112 | The European Union’s common emergency telephone number, which GSM mobile phones worldwide also route to the local emergency service. |
| 114 | The number of surahs (chapters) in the Quran. |
| 117 | The Population Reference Bureau’s estimate, in billions, of all the people ever born on Earth. |
| 118 | The highest atomic number on the official IUPAC periodic table: oganesson. |
| 144 | The number of items in a gross (a dozen dozen). |
| 150 | Dunbar’s number — the rough upper limit on the stable social relationships a person can maintain. |
| 153 |
|
| 168 | The number of hours in a week (24 × 7). |
| 172 | Aaron Finch’s score for Australia against Zimbabwe in 2018, the highest individual innings in a men’s Twenty20 International. |
| 180 | The maximum score from three darts on a standard board (three treble-20s). |
| 204 | The total number of squares of every size on a chessboard (1² + 2² + … + 8²). |
| 206 | The number of bones in the adult human body. |
| 212 | The boiling point of water at standard pressure in degrees Fahrenheit (100 °C). |
| 264 | Rohit Sharma’s score for India against Sri Lanka in 2014, the highest individual innings in a men’s One Day International. |
| 300 | A perfect game in ten-pin bowling, scored by rolling twelve consecutive strikes. |
| 360 | The number of degrees in a full circle. |
| 365 |
|
| 400 | Brian Lara’s unbeaten score for the West Indies against England in 2004, the highest individual innings in Test cricket. |
| 451 | In Ray Bradbury’s Fahrenheit 451, the temperature at which book paper is said to catch fire and burn. |
| 500 | The lifespan, in years, that some classical accounts give the phoenix before it is reborn from its ashes. |
| 555 | The telephone exchange reserved for use in North American film and television (numbers such as 555-0100 to 555-0199), so no real line is dialed. |
| 640 | The number of acres in a square mile (2.59 km²). |
| 666 |
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| 777 |
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| 786 | In Islamic tradition, an auspicious number — the abjad numerical value of the phrase “Bismillah al-Rahman al-Rahim.” |
| 911 | The emergency telephone number across most of North America; the first 911 call was placed in Haleyville, Alabama, in 1968. |
| 969 | In Genesis, the age at which Methuselah died — the longest life span recorded in the Bible. |
| 999 | The emergency telephone number of the United Kingdom — the world’s first, introduced in London in 1937. |
| 1,001 | The number of nights over which Scheherazade tells her tales in the One Thousand and One Nights. |
| 1,679 | The number of binary digits in the Arecibo message, beamed into space in 1974 in the search for extraterrestrial intelligence. |
| 1,729 | The Hardy–Ramanujan, or “taxicab,” number — the smallest expressible as a sum of two cubes in two ways (1³ + 12³ = 9³ + 10³). |
| 2,520 | The smallest positive integer divisible by every whole number from 1 to 10. |
| 5,280 | The number of feet in a mile (1.61 km). |
| 6,174 | Kaprekar’s constant: repeatedly subtracting the ascending from the descending arrangement of a four-digit number’s digits leads here. |
| 10,000 | A myriad — the classical word for ten thousand. |
| 20,000 | In Jules Verne’s Twenty Thousand Leagues Under the Sea, the distance the submarine Nautilus travels — the length of the voyage, not its depth. |
| 24,601 | Jean Valjean’s prisoner number in Victor Hugo’s Les Misérables. |
| 86,400 | The number of seconds in a day (24 × 60 × 60). |
| 41,024,320 | The number of digits in the largest known prime number, 2136,279,841 − 1 — a Mersenne prime found by the GIMPS project in October 2024. |
| 299,792,458 | The speed of light in a vacuum, in meters per second — an exact value, because the meter itself has been defined from it since 1983. |
Mathematical symbols
The working signs of arithmetic, algebra, geometry, analysis, and logic — from the everyday to the specialist’s shorthand, each credited to its inventor where history records a first use.
| +︎ |
|
|---|---|
| −︎ | The minus sign. In Widmann’s 1489 book the pair first marked surplus and deficit in trade, and only later settled into addition and subtraction. |
| ±︎ | Plus-or-minus: a tolerance, or both square roots at once. William Oughtred used it in his Clavis Mathematicae (1631). |
| ×︎ | The multiplication sign, introduced by William Oughtred in 1631. |
| ⋅︎ | The raised multiplication dot. Leibniz pushed for it from 1698, objecting that Oughtred’s cross was too easily confused with the letter x — both signs survive today. |
| ÷︎ | The obelus. Originally an ancient editor’s mark for doubtful passages, it was first used for division by Johann Rahn in 1659. |
| ⁄︎ | The fraction slash, or solidus, for setting fractions on a single line of type — a printer’s economy that mathematics adopted in the nineteenth century. |
| =︎ |
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| ≠︎ | Not equal: an equals sign canceled by a stroke. |
| ≈︎ | Approximately equal — the double tilde. |
| ≡︎ | The triple bar: identically equal. In number theory it is Gauss’s sign for congruence, from the Disquisitiones Arithmeticae (1801). |
| ≅︎ | Congruent — a similarity tilde stacked on an equals sign; the combination goes back to Leibniz. |
| ∝︎ | Proportional to; in use since the eighteenth century. |
| <︎ | Less than. With its mirror image, it was introduced by Thomas Harriot and published posthumously in 1631. |
| >︎ | Greater than — Harriot’s companion sign. The open side always faces the larger quantity. |
| ≤ ≥︎ | Less-than-or-equal and greater-than-or-equal, added by the French scientist Pierre Bouguer in 1734. |
| ≪ ≫︎ | Much less than, and much greater than. In analytic number theory the pair is called Vinogradov’s notation, after Ivan Vinogradov. |
| %︎ |
|
| ‰︎ | The per-mille sign: parts per thousand, the percent sign’s finer sibling — and, like it, an evolution of commercial shorthand rather than any one person’s invention. |
| √︎ | The radical sign for roots, introduced by Christoph Rudolff in 1525 — possibly a stylized letter r. |
| ∞︎ | The infinity symbol, or lemniscate, introduced by John Wallis in 1655. |
| π︎ |
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| e︎ |
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| i︎ | The imaginary unit — the square root of −1. Euler introduced the symbol in 1777. |
| φ︎ | The golden ratio, 1.618… (listed above). The letter is credited to Mark Barr, honoring the sculptor Phidias, whose work was said to embody the proportion. |
| ∠︎ | The angle sign, coined by Pierre Hérigone in his Cursus mathematicus (1634) — a book that introduced dozens of symbols at a stroke. |
| ⊥︎ | Perpendicular — another of Hérigone’s 1634 coinages. In logic it doubles as “bottom,” the always-false proposition. |
| ∥︎ | Parallel — among the oldest signs still in service, with roots in Greek mathematics. |
| ′ ″︎ | The prime and double prime: Lagrange’s notation for first and second derivatives, and the everyday signs for feet and inches, or arc minutes and seconds. |
| ẋ︎ | Newton’s dot — the fluxion, his sign for a rate of change in time. Mechanics still writes velocities this way. |
| dx︎ | Leibniz’s differential: an infinitesimally small change in x, from the same 1675 manuscripts as the integral sign. |
| ∂︎ | The partial-derivative sign, a curled d — used by Adrien-Marie Legendre in 1786, and made standard by Carl Jacobi’s advocacy half a century later. |
| Δ︎ | The capital delta: a finite change or difference in a quantity; in analysis, the Laplacian. |
| δ︎ | The small delta: Lagrange’s sign for a variation (1755), and the arbitrarily small quantity paired with epsilon in the definition of a limit. |
| ε︎ | Epsilon: an arbitrarily small positive quantity — a convention fixed by Augustin-Louis Cauchy’s analysis lectures of the 1820s. |
| lim︎ | The limit. Simon l’Huilier abbreviated it to lim. in 1786; the arrow form, x → a, spread from G. H. Hardy’s usage in 1908. |
| Σ︎ | The summation sign — Greek capital sigma, for “sum” — introduced by Euler in 1755. |
| ∏︎ | The product sign, sigma’s multiplicative sibling — Greek capital pi. Carl Friedrich Gauss used it in 1812. |
| ∫︎ | The integral sign — an elongated S, for the Latin summa — devised by Gottfried Wilhelm Leibniz in 1675. |
| ∮︎ | The contour integral, for integrating around a closed loop; credited to the physicist Arnold Sommerfeld. |
| ∇︎ | Nabla, or del — the gradient operator, introduced by William Rowan Hamilton. The name, after an ancient harp of similar shape, was suggested to the physicist P. G. Tait by the scholar William Robertson Smith. |
| | |︎ | Absolute-value bars — distance from zero — introduced by Karl Weierstrass in 1841. |
| ⌊⌋ ⌈⌉︎ | Floor and ceiling — round down, round up — coined by Kenneth Iverson in 1962, in the same book that gave the programming language APL its name. |
| ∘︎ | Function composition, read “after”: (f ∘ g)(x) means f(g(x)). |
| ℘︎ | Weierstrass’s ℘, the letter he drew for his elliptic function — notation so personal that Unicode names the character after him. |
| ∴︎ | The “therefore” sign: three dots in a triangle. Johann Rahn used it in the same 1659 book that debuted the division sign. |
| ∵︎ | Because — the upside-down companion of the “therefore” sign. |
| ∧ ∨︎ | Logical and, and logical or. The vee, from the Latin vel, entered logic with Russell and Whitehead’s Principia Mathematica (1910); the matching wedge followed in the 1930s. |
| ¬︎ | Logical not; the hooked negation sign spread with the formal-logic school of the 1930s. |
| ⇒ ⇔︎ | Implies, and if and only if — the double arrows, spread in the mid-twentieth century with Bourbaki among the popularizers. |
| ∈︎ | Set membership, “is an element of” — introduced, as a Greek epsilon, by Giuseppe Peano in 1889. |
| ⊂ ⊆︎ | The subset signs — contained in, or contained in or equal. The curved cup goes back to Ernst Schröder’s logic of 1890. |
| ∪ ∩︎ | Union and intersection — everything in either set, and everything in both — introduced by Giuseppe Peano in 1888. |
| ∅︎ | The empty set: the Scandinavian letter Ø, borrowed by André Weil for the Bourbaki group in 1939. |
| ∀︎ | The universal quantifier, “for all” — an inverted A, introduced by Gerhard Gentzen in 1935. |
| ∃︎ | The existential quantifier, “there exists” — a reversed E, introduced by Giuseppe Peano in 1897. |
| ℕ ℤ ℚ ℝ ℂ︎ |
|
| ℵ₀︎ | Aleph-null, the smallest infinity — the count of the counting numbers. Georg Cantor chose the Hebrew aleph for his infinite cardinals in the 1890s. |
| ∎︎ | The tombstone, or halmos: end of proof. Paul Halmos borrowed the mark from magazine typography, where it closes an article — mathematicians named it after him, and it has largely retired the old Q.E.D. |
Astronomical and zodiac symbols
The classical signs for the Sun, the Moon, the planets, and the asteroids — many of which doubled as alchemical symbols for metals — followed by the twelve signs of the zodiac.
| ☉︎ |
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|---|---|
| ☽︎ |
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| ⊕︎ |
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| ☿︎ |
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| ♀︎ |
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| ♂︎ |
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| ♃︎ |
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| ♄︎ |
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| ♅︎ | An astronomical symbol for Uranus. |
| ♆︎ | The astronomical symbol for Neptune. |
| ♇︎ | A symbol for Pluto, now classified as a dwarf planet. |
| ⚳︎ | The historical astronomical symbol for the dwarf planet Ceres. |
| ⚴︎ | The historical astronomical symbol for the asteroid Pallas. |
| ⚵︎ | The historical astronomical symbol for the asteroid Juno. |
| ⚶︎ | The historical astronomical symbol for the asteroid Vesta. |
| ♈︎ | The zodiac sign Aries, the Ram. |
| ♉︎ | The zodiac sign Taurus, the Bull. |
| ♊︎ | The zodiac sign Gemini, the Twins. |
| ♋︎ | The zodiac sign Cancer, the Crab. |
| ♌︎ | The zodiac sign Leo, the Lion. |
| ♍︎ | The zodiac sign Virgo, the Maiden. |
| ♎︎ | The zodiac sign Libra, the Scales. |
| ♏︎ | The zodiac sign Scorpio, the Scorpion. |
| ♐︎ | The zodiac sign Sagittarius, the Archer. |
| ♑︎ | The zodiac sign Capricorn, the Sea-Goat. |
| ♒︎ | The zodiac sign Aquarius, the Water-Bearer. |
| ♓︎ | The zodiac sign Pisces, the Fishes. |
Common symbols and currency signs
Everyday punctuation and reference marks, then a selection of national currency signs.
| &︎ |
|
|---|---|
| *︎ | The asterisk, from the Greek for “little star.” |
| @︎ |
|
| ^︎ | The caret, used to mark an insertion point or to denote exponentiation. |
| °︎ | The degree sign, for angles and for temperature. |
| !︎ |
|
| ~︎ | The tilde, used to mean “approximately” and as a diacritic over letters such as ñ. |
| #︎ |
|
| $︎ |
|
| ¢︎ | The cent sign, one hundredth of a dollar. |
| €︎ |
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| £︎ |
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| ₹︎ |
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| ¥︎ |
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| ₪︎ | The Israeli new shekel. |
| ₽︎ | The Russian ruble. |
| R︎ | The South African rand. |
| ₩︎ | The South Korean won. |
| ৳︎ | The Bangladeshi taka. |
| ₨︎ | The rupee sign (Rs) used by Pakistan, Sri Lanka, Nepal, Mauritius, and others. |
| R$︎ | The Brazilian real. |
| ₱︎ | The Philippine peso. |
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