Amultiplication algorithm is analgorithm (or method) tomultiply two numbers. Depending on the size of the numbers, different algorithms are more efficient than others. Numerous algorithms are known and there has been much research into the topic.

The oldest and simplest method, known sinceantiquity aslong multiplication orgrade-school multiplication, consists of multiplying every digit in the first number by every digit in the second and adding the results. This has atime complexity of${\displaystyle O(n^2)}$, wheren is the number of digits. When done by hand, this may also be reframed asgrid method multiplication orlattice multiplication. In software, this may be called "shift and add" due tobitshifts and addition being the only two operations needed.

In 1960,Anatoly Karatsuba discoveredKaratsuba multiplication, unleashing a flood of research into fast multiplication algorithms. This method uses three multiplications rather than four to multiply two two-digit numbers. (A variant of this can also be used to multiplycomplex numbers quickly.) Donerecursively, this has a time complexity of${\displaystyle O(n^{{\log }_{2}3})}$. Splitting numbers into more than two parts results inToom–Cook multiplication; for example, using three parts results in theToom-3 algorithm. Using many parts can set the exponent arbitrarily close to 1, but the constant factor also grows, making it impractical.

In 1968, theSchönhage–Strassen algorithm, which makes use of aFourier transform over amodulus, was discovered. It has a time complexity of${\displaystyle O(n\log n\log \log n)}$. In 2007,Martin Fürer proposed an algorithm with complexity${\displaystyle O(n\log n{2}^{\mathrm{\Theta }({\log }^{\ast }n)})}$. In 2014, Harvey,Joris van der Hoeven, and Lecerf proposed one with complexity${\displaystyle O(n\log n{2}^{3{\log }^{\ast }n})}$, thus making theimplicit constant explicit; this was improved to${\displaystyle O(n\log n{2}^{2{\log }^{\ast }n})}$ in 2018. Lastly, in 2019, Harvey and van der Hoeven came up with agalactic algorithm with complexity${\displaystyle O(n\log n)}$. This matches a guess by Schönhage and Strassen that this would be the optimal bound, although this remains aconjecture today.

Integer multiplication algorithms can also be used to multiply polynomials by means of the method ofKronecker substitution.

If apositional numeral system is used, a natural way of multiplying numbers is taught in schoolsaslong multiplication, sometimes calledgrade-school multiplication, sometimes called theStandard Algorithm:multiply themultiplicand by each digit of themultiplier and then add up all the properly shifted results. It requires memorization of themultiplication table for single digits.

This is the usual algorithm for multiplying larger numbers by hand in base 10. A person doing long multiplication on paper will write down all the products and then add them together; anabacus-user will sum the products as soon as each one is computed.

This example useslong multiplication to multiply 23,958,233 (multiplicand) by 5,830 (multiplier) and arrives at 139,676,498,390 for the result (product).

      23958233×         5830———————————————      00000000 ( =  23,958,233 ×     0)     71874699  ( =  23,958,233 ×    30)   191665864   ( =  23,958,233 ×   800)+ 119791165    ( =  23,958,233 × 5,000)———————————————  139676498390 ( = 139,676,498,390)

In some countries such asGermany, the above multiplication is depicted similarly but with the original product kept horizontal and computation starting with the first digit of the multiplier:^[1]

23958233 · 5830———————————————   119791165    191665864      71874699       00000000———————————————   139676498390

Below pseudocode describes the process of above multiplication. It keeps only one row to maintain the sum which finally becomes the result. Note that the '+=' operator is used to denote sum to existing value and store operation (akin to languages such as Java and C) for compactness.

multiply(a[1..p],b[1..q],base)// Operands containing rightmost digits at index 1product=[1..p+q]// Allocate space for resultforb_i=1toq// for all digits in bcarry=0fora_i=1top// for all digits in aproduct[a_i+b_i-1]+=carry+a[a_i]*b[b_i]carry=product[a_i+b_i-1]/baseproduct[a_i+b_i-1]=product[a_i+b_i-1]modbaseproduct[b_i+p]=carry// last digit comes from final carryreturnproduct

Somechips implement long multiplication, inhardware or inmicrocode, for various integer and floating-point word sizes. Inarbitrary-precision arithmetic, it is common to use long multiplication with the base set to 2^w, wherew is the number of bits in a word, for multiplying relatively small numbers. To multiply two numbers withn digits using this method, one needs aboutn² operations. More formally, multiplying twon-digit numbers using long multiplication requiresΘ(n²) single-digit operations (additions and multiplications).

When implemented in software, long multiplication algorithms must deal with overflow during additions, which can be expensive. A typical solution is to represent the number in a small base,b, such that, for example, 8b is a representable machine integer. Several additions can then be performed before an overflow occurs. When the number becomes too large, we add part of it to the result, or we carry and map the remaining part back to a number that is less thanb. This process is callednormalization. Richard Brent used this approach in his Fortran package, MP.^[2]

Computers initially used a very similar algorithm to long multiplication in base 2, but modern processors have optimized circuitry for fast multiplications using more efficient algorithms, at the price of a more complex hardware realization.^{[citation needed]} In base two, long multiplication is sometimes called"shift and add", because the algorithm simplifies and just consists of shifting left (multiplying by powers of two) and adding. Most currently available microprocessors implement this or other similar algorithms (such asBooth encoding) for various integer and floating-point sizes inhardware multipliers or inmicrocode.^{[citation needed]}

On currently available processors, a bit-wise shift instruction is usually (but not always) faster than a multiply instruction and can be used to multiply (shift left) and divide (shift right) by powers of two. Multiplication by a constant anddivision by a constant can be implemented using a sequence of shifts and adds or subtracts. For example, there are several ways to multiply by 10 using only bit-shift and addition.

((x<<2)+x)<<1# Here 10*x is computed as (x*2^2 + x)*2(x<<3)+(x<<1)# Here 10*x is computed as x*2^3 + x*2

In some cases such sequences of shifts and adds or subtracts will outperform hardware multipliers and especially dividers. A division by a number of the form${\displaystyle 2^n}$ or${\displaystyle 2^n\pm 1}$ often can be converted to such a short sequence.

In addition to the standard long multiplication, there are several other methods used to perform multiplication by hand. Such algorithms may be devised for speed, ease of calculation, or educational value, particularly when computers ormultiplication tables are unavailable.

Thegrid method (or box method) is an introductory method for multiple-digit multiplication that is often taught to pupils atprimary school orelementary school. It has been a standard part of the national primary school mathematics curriculum in England and Wales since the late 1990s.^[3]

Both factors are broken up ("partitioned") into their hundreds, tens and units parts, and the products of the parts are then calculated explicitly in a relatively simple multiplication-only stage, before these contributions are then totalled to give the final answer in a separate addition stage.

The calculation 34 × 13, for example, could be computed using the grid:

  300   40   90 + 12 ————  442

followed by addition to obtain 442, either in a single sum (see right), or through forming the row-by-row totals

(300 + 40) + (90 + 12) = 340 + 102 = 442.

This calculation approach (though not necessarily with the explicit grid arrangement) is also known as thepartial products algorithm. Its essence is the calculation of the simple multiplications separately, with all addition being left to the final gathering-up stage.

The grid method can in principle be applied to factors of any size, although the number of sub-products becomes cumbersome as the number of digits increases. Nevertheless, it is seen as a usefully explicit method to introduce the idea of multiple-digit multiplications; and, in an age when most multiplication calculations are done using acalculator or aspreadsheet, it may in practice be the only multiplication algorithm that some students will ever need.

Lattice, or sieve, multiplication is algorithmically equivalent to long multiplication. It requires the preparation of a lattice (a grid drawn on paper) which guides the calculation and separates all the multiplications from theadditions. It was introduced to Europe in 1202 inFibonacci'sLiber Abaci. Fibonacci described the operation as mental, using his right and left hands to carry the intermediate calculations.Matrakçı Nasuh presented 6 different variants of this method in this 16th-century book, Umdet-ul Hisab. It was widely used inEnderun schools across the Ottoman Empire.^[4]Napier's bones, orNapier's rods also used this method, as published by Napier in 1617, the year of his death.

As shown in the example, the multiplicand and multiplier are written above and to the right of a lattice, or a sieve. It is found inMuhammad ibn Musa al-Khwarizmi's "Arithmetic", one of Leonardo's sources mentioned by Sigler, author of "Fibonacci's Liber Abaci", 2002.^{[citation needed]}

• During the multiplication phase, the lattice is filled in with two-digit products of the corresponding digits labeling each row and column: the tens digit goes in the top-left corner.
• During the addition phase, the lattice is summed on the diagonals.
• Finally, if a carry phase is necessary, the answer as shown along the left and bottom sides of the lattice is converted to normal form by carrying ten's digits as in long addition or multiplication.

The pictures on the right show how to calculate 345 × 12 using lattice multiplication. As a more complicated example, consider the picture below displaying the computation of 23,958,233 multiplied by 5,830 (multiplier); the result is 139,676,498,390. Notice 23,958,233 is along the top of the lattice and 5,830 is along the right side. The products fill the lattice and the sum of those products (on the diagonal) are along the left and bottom sides. Then those sums are totaled as shown.

     2   3   9   5   8   2   3   3   +---+---+---+---+---+---+---+---+-   |1 /|1 /|4 /|2 /|4 /|1 /|1 /|1 /|   | / | / | / | / | / | / | / | / | 5 01|/ 0|/ 5|/ 5|/ 5|/ 0|/ 0|/ 5|/ 5|   +---+---+---+---+---+---+---+---+-   |1 /|2 /|7 /|4 /|6 /|1 /|2 /|2 /|   | / | / | / | / | / | / | / | / | 8 02|/ 6|/ 4|/ 2|/ 0|/ 4|/ 6|/ 4|/ 4|   +---+---+---+---+---+---+---+---+-   |0 /|0 /|2 /|1 /|2 /|0 /|0 /|0 /|   | / | / | / | / | / | / | / | / | 3 17|/ 6|/ 9|/ 7|/ 5|/ 4|/ 6|/ 9|/ 9|   +---+---+---+---+---+---+---+---+-   |0 /|0 /|0 /|0 /|0 /|0 /|0 /|0 /|   | / | / | / | / | / | / | / | / | 0 24|/ 0|/ 0|/ 0|/ 0|/ 0|/ 0|/ 0|/ 0|   +---+---+---+---+---+---+---+---+-     26  15  13  18  17  13  09  00

 01 002 0017 00024 000026 0000015 00000013 000000018 0000000017 00000000013 000000000009 0000000000000 —————————————  139676498390

= 139,676,498,390

The binary method is also known as peasant multiplication, because it has been widely used by people who are classified as peasants and thus have not memorized themultiplication tables required for long multiplication.^{[5][failed verification]} The algorithm was in use in ancient Egypt.^[6] Its main advantages are that it can be taught quickly, requires no memorization, and can be performed using tokens, such aspoker chips, if paper and pencil aren't available. The disadvantage is that it takes more steps than long multiplication, so it can be unwieldy for large numbers.

On paper, write down in one column the numbers you get when you repeatedly halve the multiplier, ignoring the remainder; in a column beside it repeatedly double the multiplicand. Cross out each row in which the last digit of the first number is even, and add the remaining numbers in the second column to obtain the product.

This example uses peasant multiplication to multiply 11 by 3 to arrive at a result of 33.

Decimal:     Binary:11   3       1011  115    6       101  110212       1011001   24       1  11000    ——         ——————    33         100001

Describing the steps explicitly:

• 11 and 3 are written at the top
• 11 is halved (5.5) and 3 is doubled (6). The fractional portion is discarded (5.5 becomes 5).
• 5 is halved (2.5) and 6 is doubled (12). The fractional portion is discarded (2.5 becomes 2). The figure in the left column (2) iseven, so the figure in the right column (12) is discarded.
• 2 is halved (1) and 12 is doubled (24).
• All not-scratched-out values are summed: 3 + 6 + 24 = 33.

The method works because multiplication isdistributive, so:

A more complicated example, using the figures from the earlier examples (23,958,233 and 5,830):

Decimal:             Binary:583023958233       101101100011010110110110010010110110012915  47916466       101101100011  101101101100100101101100101457  95832932       10110110001  101101101100100101101100100728191665864       10110110001011011011001001011011001000364383331728       10110110010110110110010010110110010000182766663456       1011011010110110110010010110110010000091  1533326912       1011011  101101101100100101101100100000045  3066653824       101101  10110110110010010110110010000000226133307648       1011010110110110010010110110010000000011 12266615296       1011  10110110110010010110110010000000005  24533230592       101  10110110110010010110110010000000000249066461184       101011011011001001011011001000000000001  98132922368       11011011011001001011011001000000000000  ————————————          1022143253354344244353353243222210110 (before carry)  139676498390         10000010000101010111100011100111010110

This formula can in some cases be used, to make multiplication tasks easier to complete:

${\displaystyle \frac{{\left(x+y\right)}^2}{4}-\frac{{\left(x-y\right)}^2}{4}=\frac{1}{4}\left(\left(x^2+2xy+y^2\right)-\left(x^2-2xy+y^2\right)\right)=\frac{1}{4}\left(4xy\right)=xy.}$

In the case where${\displaystyle x}$ and${\displaystyle y}$ are integers, we have that

${\displaystyle (x+y{)}^2\equiv (x-y{)}^{2}mod4}$

because${\displaystyle x+y}$ and${\displaystyle x-y}$ are either both even or both odd. This means that

and it's sufficient to (pre-)compute the integral part of squares divided by 4 like in the following example.

Below is a lookup table of quarter squares with the remainder discarded for the digits 0 through 18; this allows for the multiplication of numbers up to9×9.

If, for example, you wanted to multiply 9 by 3, you observe that the sum and difference are 12 and 6 respectively. Looking both those values up on the table yields 36 and 9, the difference of which is 27, which is the product of 9 and 3.

In prehistoric time, quarter square multiplication involvedfloor function; that some sources^[7][8] attribute toBabylonian mathematics (2000–1600 BC).

Antoine Voisin published a table of quarter squares from 1 to 1000 in 1817 as an aid in multiplication. A larger table of quarter squares from 1 to 100000 was published by Samuel Laundy in 1856,^[9] and a table from 1 to 200000 by Joseph Blater in 1888.^[10]

Quarter square multipliers were used inanalog computers to form ananalog signal that was the product of two analog input signals. In this application, the sum and difference of two inputvoltages are formed usingoperational amplifiers. The square of each of these is approximated usingpiecewise linear circuits. Finally the difference of the two squares is formed and scaled by a factor of one fourth using yet another operational amplifier.

In 1980, Everett L. Johnson proposed using the quarter square method in adigital multiplier.^[11] To form the product of two 8-bit integers, for example, the digital device forms the sum and difference, looks both quantities up in a table of squares, takes the difference of the results, and divides by four by shifting two bits to the right. For 8-bit integers the table of quarter squares will have 2⁹−1=511 entries (one entry for the full range 0..510 of possible sums, the differences using only the first 256 entries in range 0..255) or 2⁹−1=511 entries (using for negative differences the technique of 2-complements and 9-bit masking, which avoids testing the sign of differences), each entry being 16-bit wide (the entry values are from (0²/4)=0 to (510²/4)=65025).

The quarter square multiplier technique has benefited 8-bit systems that do not have any support for a hardware multiplier. Charles Putney implemented this for the6502.^[12]

Unsolved problem in computer science

What is the fastest algorithm for multiplication of two${\displaystyle n}$-digit numbers?

More unsolved problems in computer science

A line of research intheoretical computer science is about the number of single-bit arithmetic operations necessary to multiply two${\displaystyle n}$-bit integers. This is known as thecomputational complexity of multiplication. Usual algorithms done by hand have asymptotic complexity of${\displaystyle O(n^2)}$, but in 1960Anatoly Karatsuba discovered that better complexity was possible (with theKaratsuba algorithm).^[13]

Currently, the algorithm with the best computational complexity is a 2019 algorithm ofDavid Harvey andJoris van der Hoeven, which uses the strategies of usingnumber-theoretic transforms introduced with theSchönhage–Strassen algorithm to multiply integers using only${\displaystyle O(n\log n)}$ operations.^[14] This is conjectured to be the best possible algorithm, but lower bounds of${\displaystyle \mathrm{\Omega }(n\log n)}$ are not known.

Karatsuba multiplication is an O(n^log₂3) ≈ O(n^1.585) divide and conquer algorithm, that uses recursion to merge subcalculations.

By rewriting the formula, one makes it possible to do sub calculations / recursion. By doing recursion, one can solve this in a fast manner.

Let${\displaystyle x}$ and${\displaystyle y}$ be represented as${\displaystyle n}$-digit strings in some base${\displaystyle B}$. For any positive integer${\displaystyle m}$ less than${\displaystyle n}$, one can write the two given numbers as

${\displaystyle x=x_1{B}^m+x_0,}$

${\displaystyle y=y_1{B}^m+y_0,}$

where${\displaystyle x_0}$ and${\displaystyle y_0}$ are less than${\displaystyle B^m}$. The product is then

where

${\displaystyle z_2=x_1{y}_1,}$

${\displaystyle z_1=x_1{y}_0+x_0{y}_1,}$

${\displaystyle z_0=x_0{y}_0.}$

These formulae require four multiplications and were known toCharles Babbage.^[15] Karatsuba observed that${\displaystyle xy}$ can be computed in only three multiplications, at the cost of a few extra additions. With${\displaystyle z_0}$ and${\displaystyle z_2}$ as before one can observe that

Because of the overhead of recursion, Karatsuba's multiplication is slower than long multiplication for small values ofn; typical implementations therefore switch to long multiplication for small values ofn.

By exploring patterns after expansion, one see following:

Each summand is associated to a unique binary number from 0 to${\displaystyle 2^{N+1}-1}$, for example${\displaystyle a_1{x}_1{y}_1{z}_1⟷1111,a_1{x}_0{y}_1{z}_0⟷1010}$ etc. Furthermore; B is powered to number of 1, in this binary string, multiplied with m.

If we express this in fewer terms, we get:

$${\displaystyle \prod _{j=1}^N(x_{j,1}{B}^m+x_{j,0})=\sum _{i=1}^{2^{N+1}-1}\prod _{j=1}^N{x}_{j,c(i,j)}{B}^{m\sum _{j=1}^{N}c(i,j)}=\sum _{j=0}^N{z}_j{B}^{jm}}$$, where${\displaystyle c(i,j)}$ means digit in number i at position j. Notice that${\displaystyle c(i,j)\in \{0,1\}}$

Karatsuba's algorithm was the first known algorithm for multiplication that is asymptotically faster than long multiplication,^[16] and can thus be viewed as the starting point for the theory of fast multiplications.

Another method of multiplication is called Toom–Cook or Toom-3. The Toom–Cook method splits each number to be multiplied into multiple parts. The Toom–Cook method is one of the generalizations of the Karatsuba method. A three-way Toom–Cook can do a size-3N multiplication for the cost of five size-N multiplications. This accelerates the operation by a factor of 9/5, while the Karatsuba method accelerates it by 4/3.

Although using more and more parts can reduce the time spent on recursive multiplications further, the overhead from additions and digit management also grows. For this reason, the method of Fourier transforms is typically faster for numbers with several thousand digits, and asymptotically faster for even larger numbers.

Every number in base B, can be written as a polynomial:

$${\displaystyle X=\sum _{i=0}^N{x}_i{B}^i}$$

Furthermore, multiplication of two numbers could be thought of as a product of two polynomials:

$${\displaystyle XY=(\sum _{i=0}^N{x}_i{B}^i)(\sum _{j=0}^N{y}_i{B}^j)}$$

Since the coefficient of⁠${\displaystyle B^k}$⁠ in the product is$${\displaystyle z_k=\sum _{(i,j):i+j=k}{x}_i{y}_j=\sum _{i=0}^k{x}_i{y}_{k-i},}$$one has a convolution, and one can use thefast Fourier transform (FFT):$${\displaystyle \hat{f}(XY)=\hat{f}(\sum _{i=0}^k{x}_i{y}_{k-i})=\hat{f}(X)\cdot \hat{f}(Y).}$$

Therefore, the multiplication is reduced to a FFT,⁠${\displaystyle N}$⁠ multiplications, and an inverse FFT. It results atime complexity ofO(n log(n) log(logn)).

The algorithm was invented byStrassen (1968). It was made practical and theoretical guarantees were provided in 1971 bySchönhage and Strassen resulting in theSchönhage–Strassen algorithm.^[17]

In 2007 theasymptotic complexity of integer multiplication was improved by the Swiss mathematicianMartin Fürer of Pennsylvania State University to$O(n\log n\cdot 2^{\mathrm{\Theta }({\log }^{\ast }(n))})$ using Fourier transforms overcomplex numbers,^[18] where log^* denotes theiterated logarithm. Anindya De, Chandan Saha, Piyush Kurur and Ramprasad Saptharishi gave a similar algorithm usingmodular arithmetic in 2008 achieving the same running time.^[19] In context of the above material, what these latter authors have achieved is to findN much less than 2^3k + 1, so thatZ/NZ has a (2m)th root of unity. This speeds up computation and reduces the time complexity. However, these latter algorithms are only faster than Schönhage–Strassen for impractically large inputs.

In 2014, Harvey,Joris van der Hoeven and Lecerf^[20] gave a new algorithm that achieves a running time of${\displaystyle O(n\log n\cdot 2^{3{\log }^{\ast }n})}$, making explicit the implied constant in the${\displaystyle O({\log }^{\ast }n)}$ exponent. They also proposed a variant of their algorithm which achieves${\displaystyle O(n\log n\cdot 2^{2{\log }^{\ast }n})}$ but whose validity relies on standard conjectures about the distribution ofMersenne primes. In 2016, Covanov and Thomé proposed an integer multiplication algorithm based on a generalization ofFermat primes that conjecturally achieves a complexity bound of${\displaystyle O(n\log n\cdot 2^{2{\log }^{\ast }n})}$. This matches the 2015 conditional result of Harvey, van der Hoeven, and Lecerf but uses a different algorithm and relies on a different conjecture.^[21] In 2018, Harvey and van der Hoeven used an approach based on the existence of short lattice vectors guaranteed byMinkowski's theorem to prove an unconditional complexity bound of${\displaystyle O(n\log n\cdot 2^{2{\log }^{\ast }n})}$.^[22]

In March 2019,David Harvey andJoris van der Hoeven announced their discovery of anO(n logn) multiplication algorithm.^[23] It was published in theAnnals of Mathematics in 2021.^[24] Because Schönhage and Strassen predicted thatn log(n) is the "best possible" result, Harvey said: "... our work is expected to be the end of the road for this problem, although we don't know yet how to prove this rigorously."^[25]

There is a trivial lower bound ofΩ(n) for multiplying twon-bit numbers on a single processor; no matching algorithm (on conventional machines, that is on Turing equivalent machines) nor any sharper lower bound is known. TheHartmanis–Stearns conjecture would imply that${\displaystyle O(n)}$ cannot be achieved. Multiplication lies outside ofAC⁰[p] for any primep, meaning there is no family of constant-depth, polynomial (or even subexponential) size circuits using AND, OR, NOT, and MOD_p gates that can compute a product. This follows from a constant-depth reduction of MOD_q to multiplication.^[26] Lower bounds for multiplication are also known for some classes ofbranching programs.^[27]

Complex multiplication normally involves four multiplications and two additions.

${\displaystyle (a+bi)(c+di)=(ac-bd)+(bc+ad)i.}$

Or

As observed by Peter Ungar in 1963, one can reduce the number of multiplications to three, using essentially the same computation asKaratsuba's algorithm.^[28] The product (a + bi) · (c + di) can be calculated in the following way.

k₁ =c · (a +b)

k₂ =a · (d −c)

k₃ =b · (c +d)

Real part =k₁ −k₃

Imaginary part =k₁ +k₂.

This algorithm uses only three multiplications, rather than four, and five additions or subtractions rather than two. If a multiply is more expensive than three adds or subtracts, as when calculating by hand, then there is a gain in speed. On modern computers a multiply and an add can take about the same time so there may be no speed gain. There is a trade-off in that there may be some loss of precision when using floating point.

Forfast Fourier transforms (FFTs) (or anylinear transformation) the complex multiplies are by constant coefficientsc + di (calledtwiddle factors in FFTs), in which case two of the additions (d−c andc+d) can be precomputed. Hence, only three multiplies and three adds are required.^[29] However, trading off a multiplication for an addition in this way may no longer be beneficial with modernfloating-point units.^[30]

All the above multiplication algorithms can also be expanded to multiplypolynomials. Alternatively theKronecker substitution technique may be used to convert the problem of multiplying polynomials into a single binary multiplication.^[31]

Long multiplication methods can be generalised to allow the multiplication of algebraic formulae:

 14ac - 3ab + 2 multiplied by ac - ab + 1

 14ac  -3ab   2   ac   -ab   1 ———————————————————— 14a2c2  -3a2bc   2ac        -14a2bc         3 a2b2  -2ab                 14ac           -3ab   2 ——————————————————————————————————————— 14a2c2 -17a2bc   16ac  3a2b2    -5ab  +2 =======================================[32]

As a further example of column based multiplication, consider multiplying 23 long tons (t), 12 hundredweight (cwt) and 2 quarters (qtr) by 47. This example usesavoirdupois measures: 1 t = 20 cwt, 1 cwt = 4 qtr.

    t    cwt  qtr   23     12    2               47 x ————————————————  141     94   94  940    470   29     23 ———————————————— 1110    587   94 ———————————————— 1110      7    2 =================  Answer: 1110 ton 7 cwt 2 qtr

First multiply the quarters by 47, the result 94 is written into the first workspace. Next, multiply cwt 12*47 = (2 + 10)*47 but don't add up the partial results (94, 470) yet. Likewise multiply 23 by 47 yielding (141, 940). The quarters column is totaled and the result placed in the second workspace (a trivial move in this case). 94 quarters is 23 cwt and 2 qtr, so place the 2 in the answer and put the 23 in the next column left. Now add up the three entries in the cwt column giving 587. This is 29 t 7 cwt, so write the 7 into the answer and the 29 in the column to the left. Now add up the tons column. There is no adjustment to make, so the result is just copied down.

The same layout and methods can be used for any traditional measurements and non-decimal currencies such as the old British£sd system.

• Binary multiplier
• Dadda multiplier
• Division algorithm
• Horner scheme for evaluating of a polynomial
• Logarithm
• Matrix multiplication algorithm
• Mental calculation
• Number-theoretic transform
• Prosthaphaeresis
• Slide rule
• Trachtenberg system
• Residue number system § Multiplication for another fast multiplication algorithm, specially efficient when many operations are done in sequence, such as in linear algebra
• Wallace tree

• Warren Jr., Henry S. (2013).Hacker's Delight (2 ed.).Addison Wesley -Pearson Education, Inc.ISBN 978-0-321-84268-8.
• Savard, John J. G. (2018) [2006]."Advanced Arithmetic Techniques".quadibloc.Archived from the original on 2018-07-03. Retrieved2018-07-16.
• Johansson, Kenny (2008).Low Power and Low Complexity Shift-and-Add Based Computations(PDF) (Dissertation thesis). Linköping Studies in Science and Technology (1 ed.). Linköping, Sweden: Department of Electrical Engineering,Linköping University.ISBN 978-91-7393-836-5.ISSN 0345-7524. No. 1201.Archived(PDF) from the original on 2017-08-13. Retrieved2021-08-23. (x+268 pages)

• The Many Ways of Arithmetic in UCSMP Everyday Mathematics
• A Powerpoint presentation about ancient mathematics
• Lattice Multiplication Flash Video

• Multiplication Algorithms used by GMP

• Computer arithmetic algorithms
• Multiplication

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