97 Fun Facts About Multiplication

Which property of multiplication says 3 × 4 and 4 × 3 give the same product, no matter the order?

Commutative Property, from the Latin commutare, 'to exchange': swapping the order of the factors never changes the product, so 3 × 4 and 4 × 3 are both 12.

What do you call the result of multiplying two or more numbers?

Product is the result of multiplying: 3 × 4 gives the product 12, just as 3 + 4 gives the sum 7 and 12 ÷ 4 gives the quotient 3.

What are the numbers being multiplied together called?

Factors, from the Latin for 'maker': in 3 × 4 = 12, the factors 3 and 4 make the product 12.

Which property says (2 × 3) × 4 and 2 × (3 × 4) give the same result, whatever the grouping?

Associative, because it's about how factors associate in groups; commutative is about their order.

Which property of multiplication lets a(b + c) be expanded to ab + ac?

The Distributive Property spreads a factor across every term in the brackets; it's why 7 × 12 can be done as 7 × 10 plus 7 × 2.

Which mathematician is widely credited with introducing the '×' symbol for multiplication in 1631?

William Oughtred used × in his 1631 algebra text Clavis Mathematicae, a compact Latin treatise that also introduced the :: sign for proportion.

Which multiplication method, also called the Italian or gelosia method, writes the digits round a box of cells and sums its diagonals?

Lattice multiplication breaks a big product into one-digit products inside the cells, then sums each diagonal with carries.

Which civilization multiplied by 'doubling and halving' using powers of two?

Egyptian scribes split one factor into powers of two and doubled the other, essentially binary arithmetic thousands of years before computers.

Which number is the multiplicative identity, leaving any number multiplied by it unchanged?

Multiplying any number by 1 leaves it unchanged, which is why 1 is called the multiplicative identity; 0 plays the same role for addition.

Which result states every integer over 1 is prime or a unique product of primes, up to order?

The fundamental theorem of arithmetic: every integer above 1 is a product of primes in exactly one way up to order, so 12 is 2 × 2 × 3 and nothing else.

Which easier operation do logarithm tables let you substitute for multi-digit multiplication?

Addition: since log(ab) = log a + log b, you add the two logarithms from the table and look up the antilog to get the product; division becomes subtraction.

Which Scottish mathematician built numbered calculating rods and named the technique rabdology?

John Napier described the rods in his book Rabdologiae; each rod carries one column of a multiplication table, reducing multiplying to a series of additions and dividing to subtractions.

Which hand-operated calculator uses logarithmic scales to multiply and divide?

The slide rule: moving one logarithmic scale along another adds the logarithms of two numbers, and the sum of the logarithms is the logarithm of the product.

In which year did Anatoly Karatsuba discover the first multiplication method faster than the grade-school one?

1960: Karatsuba, then a student, found his divide-and-conquer method that year; it was published in 1962 and was the first method to beat the schoolbook n² bound.

The oldest known multiplication tables, from about 4,000 years ago, were used by which civilization?

The Babylonians: Old Babylonian clay tablets from about 1800 BC carry multiplication tables in their base-60 system, where a complete table would need some 3,600 entries.

The world's earliest known decimal multiplication table, dated to 305 BC, was written on what?

The 21 slips are part of the Tsinghua Bamboo Slips collection from China's Warring States period.

In French, Italian and Russian, the multiplication table is named after which Greek?

Pythagoras: French 'table de Pythagore' and Italian 'tavola pitagorica' keep his name, though the attribution is traditional rather than proven; the oldest surviving Greek table dates from the 1st century AD.

The oldest surviving Greek multiplication table, on a 1st-century AD wax tablet, is housed where?

The British Museum houses the 1st-century AD tablet; wax tablets were the reusable notebooks of the ancient world, which is why so few survive at all.

Which property says the product of any two integers is itself an integer?

Closure: the integers are closed under multiplication, so 7 × (−3) = −21 stays an integer, whereas they are not closed under division, since 7 ÷ 3 is not.

In 1820, John Leslie recommended pupils memorize the times table up to what?

50 × 50: John Leslie's Philosophy of Arithmetic (1820) urged pupils to learn that far, and printed a full table up to 99 × 99 so numbers could be multiplied two digits at a time.

The Chinese eighty-one-sentence rhyme for learning products was historically known by what name?

The nine-nine table: in ancient times the rhyme began with 'nine nines are eighty-one' and counted down, which is where the name comes from.

Which algebraic property says that if ab = 0 then a or b is 0?

The zero-product property: a product is zero only when a factor is zero, which is why solving (x − 2)(x + 5) = 0 gives x = 2 or x = −5.

In what year did John Napier publish his description of the calculating rods now called Napier's bones?

1617: Rabdologiae came out three years after his 1614 logarithms, and Napier dedicated it to his patron Alexander Seton, Chancellor of Scotland.

In Napier's original design, his calculating rods had what cross-section?

Square: each of the four faces of a rod carried a different table, so a set of ten rods held forty tables.

Napier's Rabdology, describing his calculating rods, was printed in which city?

Edinburgh: Andrew Hart printed Rabdologiae there in 1617, and Napier was Laird of Merchiston, then just outside the city.

Beyond multiplying and dividing, advanced use of Napier's rods lets a user do what?

Extract square roots: Rabdologiae supplies an extra rod marked with squares so a user can pull square roots digit by digit; the rods are not logarithms.

Who, in 1698, proposed a middle dot for multiplication because × looked too much like x?

Gottfried Wilhelm Leibniz, in a 1698 letter to Johann Bernoulli, wrote that × was easily confounded with x and that he preferred an interposed dot, as in ZC·LM.

In computer programming, which symbol is the usual notation for multiplication?

The asterisk was on every keyboard when ASCII and EBCDIC had no × or dot, so 5*2 stuck.

In Germany, which symbol is the primary way to write multiplication?

The dot operator ⋅, the German Malpunkt: schoolbooks write 3 ⋅ 4 = 12, keeping × for vector products and dimensions; the same dot appears in SI units such as N⋅m.

Which ancient Egyptian text is the chief source for the doubling-and-adding method of multiplication?

The Rhind Papyrus (c. 1550 BC) shows the method: to compute 13 × 21 a scribe doubled 21 repeatedly and added the rows for 1, 4 and 8.

Which mathematician first described the modern method of multiplication with Hindu–Arabic numerals?

Brahmagupta, in the Brahmasphutasiddhanta of 628 AD, set out multiplication with Hindu-Arabic numerals along with rules for addition, subtraction, division and working with zero.

Who popularized place-value decimal arithmetic in the West in the 13th century?

Fibonacci: his Liber Abaci of 1202 showed Italian merchants how to calculate with Hindu-Arabic numerals and place value instead of Roman numerals and the abacus.

The 'grid method' of multiplication, splitting 34 × 13 into a two-by-two box of partial products, is a primary-school staple where?

England and Wales teach it, as do some US schools; 34 × 13 becomes 300 + 90 + 40 + 12 = 442.

The 2019 Harvey–van der Hoeven O(n log n) integer multiplication algorithm is based on which tool?

The fast Fourier transform: Harvey and van der Hoeven's 2019 algorithm rests on multidimensional FFTs, but only beats older methods for astronomically large numbers, so it is a theoretical rather than practical win.

Karatsuba's fast multiplication disproved which mathematician's conjecture that the schoolbook method was optimal?

Andrey Kolmogorov posed the problem at a Moscow seminar in 1960; Karatsuba, then a student, disproved the conjecture within a week.

Which multiplication algorithm, made practical in 1971, uses a Fourier transform over a modulus?

Schönhage–Strassen (1971) multiplies via a number-theoretic Fourier transform modulo 2^n + 1 and stayed the asymptotically fastest known method for 36 years, until Fürer's 2007 algorithm.

Martin Fürer improved the complexity of integer multiplication in 2007 while at which institution?

Penn State University: Fürer's 2007 algorithm ran in n log n · 2^O(log* n) time, closer to the O(n log n) bound that Schönhage and Strassen had conjectured optimal decades earlier.

'Peasant multiplication', which needs no memorized times tables, is another name for which approach?

The binary method: one number is repeatedly halved and the other doubled, and the rows beside odd numbers are added up, which is binary multiplication in disguise.

Who published a table of quarter squares from 1 to 1000 in 1817 as an aid to multiplication?

Antoine Voisin's 1817 table ran from 1 to 1000; later quarter-square tables reached 100,000 (Samuel Laundy, 1856) and 200,000 (Joseph Blater, 1888).

The symbol ∏ used for the product of a sequence derives from which Greek letter?

Pi: ∏ is the capital Greek letter pi, standing for 'product', just as the summation sign ∑ is capital sigma for 'sum'.

In a multiplication, what is the technical name for the number that is being multiplied?

The multiplicand is the number being multiplied and the multiplier is the number it is multiplied by; since order does not matter the distinction is mostly academic.

In algebra, what is the 3 in the expression 3xy² called?

The coefficient is the multiplier of the variable part; the 2 on the y is the exponent.

Which mathematician introduced the n! notation for the factorial in 1808?

Christian Kramp introduced n! in his 1808 Élémens d'arithmétique universelle; Legendre published his formula for the prime factorization of factorials that same year.

The Ishango bone, sometimes cited as a hint of prehistoric multiplication, was found where?

Central Africa: the bone was found at Ishango in the Congo near Lake Edward; dated to roughly 18,000–20,000 BC, its notched columns are debated by archaeologists.

Andrew Booth invented his signed-binary multiplication algorithm in 1950 while researching what?

Booth was working at Birkbeck College in London; his algorithm handles two's complement numbers and is still taught in computer architecture.

Jakow Trachtenberg developed his rapid mental-calculation system while held where?

A Nazi concentration camp: the Odessa-born engineer devised the system there to keep his mind occupied, and after the war taught it at his Mathematical Institute in Zurich.

Henri Genaille, who invented carry-free multiplication rulers in 1891, worked as what?

A railway engineer: Genaille's rulers are a variant of Napier's bones that show the carry graphically, so products can be read off with no mental arithmetic.

Volker Strassen's 1969 algorithm speeds up which kind of multiplication?

Matrix multiplication: Strassen's method beats the standard O(n³) algorithm for large matrices by recursively splitting each matrix into 2 × 2 blocks.

Which French mathematician posed the 1885 problem that led Genaille to devise his rulers?

Édouard Lucas posed the problem in 1885; Genaille answered with his rulers in 1891 and Lucas popularised them, which is why they are known as Genaille–Lucas rulers.

Writing * for multiplication in programming originated in which language?

FORTRAN, from 1957, adopted * because early character sets had no × or ⋅ while every keyboard had an asterisk; C, BASIC and nearly every later language kept it.

The earliest known × for multiplication appears in a 1618 appendix to which work?

Mirifici Logarithmorum Canonis Descriptio: the anonymous appendix to the 1618 English edition of Napier's logarithm book has been attributed to William Oughtred, who used the same cross in his own 1631 algebra text.

In botany, a × placed between two plant names marks the plant as what?

A hybrid: Crocosmia × crocosmiiflora, for instance, is a cross between two other Crocosmia species. When the symbol is unavailable, a plain letter x is often substituted.

When a historian dates an event '1225×1232', with a multiplication cross between two years, what does it mean?

It happened sometime between those years: no earlier than the first and no later than the second, the cross standing for uncertainty.

In the APL programming language, × applied to a single argument computes what?

The sign function: APL reuses its symbols, so the two-argument × multiplies while the one-argument (monadic) form returns −1, 0 or 1 depending on the sign of its argument.

How old was Anatoly Karatsuba when, as a student, he found his fast multiplication method?

23: Karatsuba, born in January 1937, found the method in the autumn of 1960, within a week of Kolmogorov stating his conjecture at a Moscow seminar.

Karatsuba's 1962 paper listed which second author, whose own result it also contained?

Yuri Ofman: Kolmogorov himself wrote the 1962 Doklady article 'Multiplication of multidigit numbers on automata', combining Karatsuba's and Ofman's results; Karatsuba only learned of it when reprints arrived.

Karatsuba's trick replaces four half-size multiplications with how many?

Three: the middle term comes from (a + b)(c + d) minus the two outer products, at the cost of a few extra additions; applied recursively it beats the n² schoolbook method.

The Toom-3 algorithm cuts nine sub-multiplications down to how many?

Splitting each number into three parts generalises Karatsuba, which is the Toom-2 case. Stephen Cook later tidied the method in his 1966 PhD thesis.

Strassen's 1969 method multiplies two 2×2 matrices using how many products, not eight?

Seven: Strassen's 1969 algorithm uses 7 multiplications and 18 additions instead of 8 and 4, and applied recursively to block matrices it beats the n³ schoolbook approach, running in about O(n^2.81).

Matrix multiplication was first described in 1812 by which French mathematician?

Binet introduced it to represent the composition of linear maps, which is why the operation is not commutative.

Andrew Booth devised his signed binary multiplication algorithm at which London college?

Birkbeck College in Bloomsbury, where Booth was researching crystallography at the time; his 1950 algorithm is a staple of computer architecture courses.

The 1964 Wallace tree hardware multiplier came from a computer scientist of which country?

Australia: Chris Wallace, later of Monash University, designed it; layers of adders sum the partial products until only two numbers remain, making multiplication nearly as fast as addition.

Who first conceived a multiply–accumulate unit, in a 1909 design for an analytical engine?

Ludgate, an Irish accountant's clerk working alone in Dublin, also used the unit for division by multiplying with a reciprocal series.

Which 8-bit processor, introduced in 1978, was unusual in having a multiply instruction?

The Motorola 6809's MUL instruction multiplied two 8-bit registers into a 16-bit result; most early microprocessors made programmers write shift-and-add routines, and the instruction only became common with 16-bit chips.

The FOIL mnemonic for multiplying two binomials first appeared in a textbook in what year?

1929: William Betz's Algebra for Today introduced the mnemonic; in the US 'foil' has since become a verb meaning to expand a product of binomials.

In the FOIL method for multiplying binomials, what does the O stand for?

First, Outer, Inner, Last: the first term of the first binomial times the second term of the second is the outer product.

Which scribe copied out the Rhind papyrus, the main source for Egyptian doubling multiplication?

Ahmes copied the Rhind papyrus around 1550 BC from an older text; it and the Moscow papyrus preserve the method, which needs only doubling, halving and adding.

The grid method became standard in English primary schools through which 1990s initiative?

The National Numeracy Strategy of 1999 introduced a daily numeracy hour; pupils usually move on to long multiplication but keep the grid as a fallback.

Which Neopythagorean author included a times table in his Introduction to Arithmetic?

Nichomachus of Gerasa (c. 60–120 AD) included a multiplication table in his Introduction to Arithmetic; his Neopythagorean leanings helped attach Pythagoras's name to the table.

The Warring States decimal times table at Tsinghua University multiplies numbers up to?

99.5: its 21 strips handle any whole or half number from 0.5 up to 99.5; Nature reported it in 2014 as the world's oldest decimal multiplication table.

How many distinct numbers appear in the 1 to 10 times table?

42: of the 100 products in the 10 × 10 table, most repeat (3 × 4 and 2 × 6 are both 12), leaving 42 distinct values from 1 to 100.

Which US body's 1989 standards urged less rote memorisation of times tables?

The National Council of Teachers of Mathematics later clarified in 2006 that basic facts must still be learned, without settling how.

Which secondary school maths teacher created Times Tables Rock Stars in 2010?

Bruno Reddy played rock and wielded an inflatable guitar at the start of lessons while firing quick multiplication questions at pupils.

The 1965 book Vedic Mathematics presents its speed-calculation tricks as how many sutras?

Bharati Krishna Tirtha claimed to have found them in a lost appendix to the Atharvaveda; scholars find no such Vedic source and note the methods use decimals unknown in Vedic times.

In 1980 the 'human computer' Shakuntala Devi multiplied two random 13-digit numbers at Imperial College London in how long?

28 seconds for a 26-digit answer, earning a Guinness record whose certificate was only issued in 2020, seven years after her death.

The only surviving stepped reckoner, the first four-function calculator, is in which city?

Hanover: the 16-digit brass machine is kept at the Gottfried Wilhelm Leibniz Library, the state library of Lower Saxony; its carry mechanism was beyond what 1690s craftsmen could make reliable.

Who patented the arithmometer, the first commercially successful mechanical calculator?

Thomas de Colmar patented it in 1820; serious production only began in 1851, and for almost forty years it was the only mechanical calculator in commercial production anywhere.

Dorr Felt built the first comptometer prototype in 1884 using what as the outer casing?

A macaroni box, with meat skewers, staples and rubber bands for the insides; the key-driven comptometer stayed in production from 1887 into the mid-1970s.

Curt Herzstark drew up his palm-sized Curta calculator while imprisoned in which camp?

Buchenwald: camp officials let Herzstark keep working so the device could be a gift to Hitler after the war. It was later manufactured in Liechtenstein and nicknamed the pepper grinder.

In the 6-to-10 finger trick, the touching fingers and those below count as what?

Tens: for 7 × 8, touch the 7-finger to the 8-finger; the 2 + 3 touching-and-below fingers give 50, and the 3 × 2 fingers above give 6, so 56.

A full set of Genaille–Lucas multiplication rulers consists of how many strips?

Eleven: an index strip plus one strip for each digit 0 to 9; each digit strip carries a column of triangles that shows the carry graphically, so products can be read off by sight.

The standard 19th-century slide rule layout came in 1859 from a lieutenant in which force?

Amédée Mannheim's rule added a cursor and single-decade C and D scales, and the French Artillery adopted it.

Which brand of aluminium slide rule did Buzz Aldrin carry to the Moon on Apollo 11?

Pickett: Aldrin's aluminium N600-ES model sold at auction in 2007; the rule flown on Apollo 13 belongs to the National Air and Space Museum.

The HP-35 pocket scientific calculator launched in 1972 at what US price?

$395: justifiable for engineers but not students; by 1976 the TI-30 sold for under $25 and the slide rule was finished.

Cayley tables, the multiplication tables of group theory, first appeared in which year?

1854: Arthur Cayley's paper 'On the theory of groups' that year simply called them tables; a group is abelian exactly when its table is symmetric about the main diagonal.

Gibbs and Heaviside introduced the dot and × notation for vector products in which year?

1881: Hamilton's quaternion product of 1843 already contained both operations; Gibbs and Heaviside independently separated them into today's dot and cross notation.

Peano's 1889 treatise axiomatising arithmetic was written in which language?

Latin: Peano wrote the 1889 booklet Arithmetices principia in Latin, partly in his new logical symbols, and defined multiplication recursively from the successor operation.

Montgomery modular multiplication, introduced in 1985, is fast because it avoids what?

Division: numbers are kept in Montgomery form so the only division needed is by a power of two, which a computer does with a bit shift; no trial division by the modulus is performed.

Plato used which factorial as an ideal community's population, partly for divisibility?

5,040 is 7! and has 60 divisors, handy for splitting citizens into groups. There is otherwise little evidence of Greek study of factorials.

The word 'exponent' for repeated multiplication was coined in 1544 by which mathematician?

Michael Stifel used the term in his 1544 Arithmetica integra; it comes from the Latin exponere, 'to put forth'. Descartes later introduced superscript notation in La Géométrie but wrote squares as xx.

Robert Recorde's 16th-century term 'zenzizenzizenzic' referred to which power of a number?

Zenzic meant squared, so squaring three times gives the eighth power. His fifth power was the sursolid and his seventh the second sursolid.

Archimedes proved the law 10^a × 10^b = 10^(a+b) in which of his works?

The Sand Reckoner: Archimedes needed the rule to handle enormous powers of ten while estimating how many grains of sand would fill the universe.

Laplace said logarithms, by shortening long multiplications, doubled the life of whom?

The astronomer: Laplace called logarithms an admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer and spares him the errors of long calculations.

In base 10, a perfect square can never end in which digit?

7: squares end only in 0, 1, 4, 5, 6 or 9, a quick check that rules out 2, 3, 7 and 8 before you bother with a square root.

What name describes an algorithm, like Harvey–van der Hoeven's, that only wins on impossibly large inputs?

Galactic algorithm: Lipton and Regan coined the term for methods whose advantage appears only on inputs larger than anything in the galaxy; Harvey and van der Hoeven's multiplication wins only above 2^(1729^12) bits.

The dot product of two vectors yields what kind of quantity?

A scalar: the dot product multiplies matching components and adds them, so (1, 2, 3) · (4, 5, 6) = 4 + 10 + 18 = 32, a single number with no direction.

In the German layout of long multiplication, work starts at which digit of the multiplier?

The leftmost: the problem is written on a single line and the partial products are stacked starting from the multiplier's first digit, the reverse of the usual English-language order.