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Ancient Egyptian multiplication
Ancient Egyptian multiplication
Ancient Egyptian multiplication
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In mathematics, ancient Egyptian multiplication (also known as Egyptian multiplication, Ethiopian multiplication, Russian multiplication, or peasant multiplication), one of two multiplication methods used by scribes, is a systematic method for multiplying two numbers that does not require the multiplication table, only the ability to multiply and divide by 2, and to add. It decomposes one of the multiplicands (preferably the smaller) into a set of numbers of powers of two and then creates a table of doublings of the second multiplicand by every value of the set which is summed up to give result of multiplication.

This method may be called mediation and duplation, where mediation means halving one number and duplation means doubling the other number. It is still used in some areas.[1]

The second Egyptian multiplication and division technique was known from the hieratic Moscow and Rhind Mathematical Papyri written in the seventeenth century B.C. by the scribe Ahmes.[2]

Although in ancient Egypt the concept of base 2 did not exist, the algorithm is essentially the same algorithm as long multiplication after the multiplier and multiplicand are converted to binary. The method as interpreted by conversion to binary is therefore still in wide use today as implemented by binary multiplier circuits in modern computer processors.[1]

Method

[edit]

The ancient Egyptians had laid out tables of a great number of powers of two, rather than recalculating them each time. To decompose a number, they identified the powers of two which make it up. The Egyptians knew empirically that a given power of two would only appear once in a number. For the decomposition, they proceeded methodically; they would initially find the largest power of two less than or equal to the number in question, subtract it out and repeat until nothing remained. (The Egyptians did not make use of the number zero in mathematics.)

After the decomposition of the first multiplicand, the person would construct a table of powers of two times the second multiplicand (generally the smaller) from one up to the largest power of two found during the decomposition.

The result is obtained by adding the numbers from the second column for which the corresponding power of two makes up part of the decomposition of the first multiplicand.[1]

Because mathematically speaking, multiplication of natural numbers is just "exponentiation in the additive monoid", this multiplication method can also be recognised as a special case of the Square and multiply algorithm for exponentiation.

Example

[edit]

25 × 7 = ?

Decomposition of the number 25:

The largest power of two less than or equal to 25 is 16: 25 − 16 = 9.
The largest power of two less than or equal to 9 is 8: 9 − 8 = 1.
The largest power of two less than or equal to 1 is 1: 1 − 1 = 0.
25 is thus the sum of: 16, 8 and 1.

The largest power of two is 16 and the second multiplicand is 7.

1 7
2 14
4 28
8 56
16 112

As 25 = 16 + 8 + 1, the corresponding multiples of 7 are added to get 25 × 7 = 112 + 56 + 7 = 175.

Russian peasant multiplication

[edit]

In the Russian peasant method, the powers of two in the decomposition of the multiplicand are found by writing it on the left and progressively halving the left column, discarding any remainder, until the value is 1 (or −1, in which case the eventual sum is negated), while doubling the right column as before. Lines with even numbers on the left column are struck out, and the remaining numbers on the right are added together.[3]

Example

[edit]

238 × 13 = ?

13 238
6   (remainder discarded) 476   ( = 238 x 2 )
3 952   ( = 476 x 2 )
1   1904   ( = 952 x 2 )
     
13 +238
6 476
3 +952
1 +1904
3094
   

See also

[edit]

References

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Ancient Egyptian multiplication

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from Grokipedia
Ancient Egyptian multiplication, also known as the duplication or dyadic method, was an additive arithmetic technique employed by scribes in to compute products by decomposing one factor into powers of two and summing corresponding multiples of the other factor through repeated doubling, without relying on tables. This method, documented primarily in the (RMP)—a scribal copy from around 1650 BCE attributed to —facilitated practical calculations for administration, land measurement, and resource allocation in a society where served utilitarian purposes rather than abstract theory. The core procedure involves creating two columns: the left starting at 1 and doubling iteratively (powers of two), while the right doubles the multiplicand accordingly. The multiplier is then broken into a sum of these powers of two, and the matching entries from the right column are added to obtain the product. This binary foundation made the technique efficient, as supported easy doubling via their script, and it extended seamlessly to division by reversing the process—halving instead of doubling—and to handling unit fractions, which were the norm in Egyptian arithmetic. Notable for its simplicity and reliance solely on addition and multiplication by 2, the method is employed in many of the RMP's approximately 84 problems, including those on geometric volumes and proportions, and was analyzed in early modern scholarship by figures like August Eisenlohr (1877) and Otto Neugebauer (1926), who highlighted its additive purity and links to fraction decompositions like the 2/n table. Unlike later Greek or Babylonian systems, it avoided but proved robust for the era's decimal-based, non-place-value numerals, influencing understandings of pre-Hellenistic .

Historical Background

Origins and Evidence

The earliest evidence of systematic mathematical practices in , including techniques, emerges during the Middle Kingdom period (c. 2055–1650 BCE), a time of significant administrative and scribal development that preserved numerical methods for practical . While fragmentary inscriptions from suggest basic arithmetic, the surviving papyri documenting date to the late Middle Kingdom and Second Intermediate Period, reflecting techniques likely in use since at least the 12th Dynasty (c. 1991–1802 BCE). These documents were often copied by scribes for educational purposes, indicating a continuity of knowledge from earlier compositions. The , acquired by Scottish antiquarian Alexander Henry Rhind in 1858 and now housed primarily in the , provides the most extensive evidence of ancient Egyptian . Copied around 1650 BCE by the scribe Ahmose during the period (Second Intermediate Period), it is based on an original text from approximately 1850 BCE in the Middle Kingdom, containing 87 problems that employ for everyday calculations. For instance, Problem 50 demonstrates the method in computing the area of a circular field with a of 9 khet (an Egyptian land measure), approximating the value of π as 256/81 through successive doublings and additions to yield the field's extent. Similarly, Problem 79 uses to tally resources in a : starting with 7 houses, each containing 7 cats that catch 7 mice, each mouse eating 7 ears of grain yielding 7 heqat (a ), resulting in a total of 19,607 (summing houses, cats, mice, ears, and heqat of grain across the ). These examples illustrate the method's application in and , essential for taxation and . Complementing the Rhind Papyrus, the (also known as the Golenishchev Papyrus), discovered in and held in the , offers additional insights from the Middle Kingdom. Dated to around 1850 BCE during the 13th Dynasty, this shorter document of 25 problems focuses more on but integrates in practical contexts. Problem 14, for example, calculates the volume of a of a (a truncated ) with base side 4, top side 2, and height 6, using the multiplication technique to find areas (4×4=16 for the base, 2×2=4 for the top) and then combining them with the average side product (4×2=8) before multiplying by one-third of the height to arrive at 56 cubic units—likely for assessing or architectural volumes. Such problems highlight the method's role in and storage, underscoring its utility in state projects.

Role in Ancient Egyptian Mathematics

The ancient Egyptian numeral system was decimal in nature, employing distinct hieroglyphs or hieratic symbols for powers of ten—such as a single stroke for 1, a heel bone for 10, a coiled rope for 100, and a lotus flower for 1,000—allowing representation of numbers through repetition and grouping without a concept of place value or a symbol for zero. This additive structure facilitated straightforward addition and subtraction by tallying or removing symbols, but multiplication was adapted to rely solely on these operations, using repeated doubling to generate multiples and selective addition to compute products, thereby integrating seamlessly into the existing arithmetic framework without requiring positional notation. Evidence of this approach appears in papyri like the Rhind Mathematical Papyrus, where such methods underpin practical calculations. In relation to division and handling, Egyptian played a key role in resolving problems involving unit fractions, which dominated their fractional arithmetic; for instance, to express a like 2/5, scribes decomposed it into sums of distinct unit fractions such as 1/3 + 1/15, often employing techniques to verify or derive these representations through scaling and summation. Division was typically performed inversely by finding how many times a doubled or halved to match the , mirroring the process and enabling fair apportionment in administrative contexts, while the emphasis on unit fractions ensured all rationals were expressed as finite sums without compound numerators beyond rare cases like 2/3. This integration supported the solution of equations and proportions central to Egyptian problem-solving, where by unit fractions helped equate quantities in or distribution. Despite its utility, the multiplication method exhibited limitations, particularly its reliance on iterative doubling and addition, which became increasingly laborious for very large numbers due to the need for numerous steps proportional to the binary logarithm of the multiplier, though still more efficient than naive repetition. The absence of zero and place-value notation further constrained scalability, making the system cumbersome for abstract or high-precision computations beyond practical scales. Nonetheless, this approach was well-suited to the hieratic script—a cursive, abbreviated form of hieroglyphs used by scribes for daily administrative tasks—allowing compact notation on papyrus for calculations in taxation, grain accounting, and labor allocation, where numbers rarely exceeded thousands and the method's step-by-step nature aligned with the repetitive, verifiable record-keeping essential to Egyptian bureaucracy.

The Method

Procedure

The ancient Egyptian multiplication procedure, as attested in mathematical papyri such as the Rhind Papyrus (c. 1650 BCE), relies on iterative processes of halving one factor and doubling the other to compute the product without direct repeated addition beyond powers of two. To multiply two numbers, denoted here abstractly as the first factor mm and the second factor nn, the method begins by setting up a two-column notation: the left column starts with mm, and the right column starts with nn. This tabular format, depicted in script on the papyri, facilitates the alignment of corresponding values during the iterations. The core steps involve repeated mediation (halving) of the left-column value and duplation (doubling) of the right-column value until the left column reaches 1. At each iteration, if the current left-column value is even, it is halved exactly to produce the next entry, and the right-column value is doubled accordingly. If the left-column value is odd, the corresponding right-column value is retained for later summation, after which the odd value is adjusted by subtracting 1 (to make it even) and then halved, while the right column continues to double. This handling of even and odd cases ensures that only integer operations are performed, with the retained values capturing the contributions from the odd halvings. The process terminates when the left column arrives at 1, at which point the sum of all retained right-column values yields the product m×nm \times n. The emphasis on tabular notation in the papyri underscores the method's practical, visual organization, allowing scribes to track the iterative doublings and halvings systematically without algebraic symbols. This approach, requiring only knowledge of addition, doubling, and halving, reflects the additive foundation of Egyptian arithmetic and avoids tables beyond the simplest cases.

Basic Example

To illustrate the ancient Egyptian multiplication method, consider the calculation of 13 × 9, following the doubling and halving procedure documented in the . The process involves creating a table by repeatedly halving the first number (13) until reaching 1, while simultaneously doubling the second number (9), and selecting only the doubled values corresponding to odd halvings for summation. This yields the following table, akin to the tabular format implied in Egyptian papyri records:
Halving of 13Doubling of 9
13 (odd)9
6 (even)18
3 (odd)36
1 (odd)72
Summing the entries under the odd halvings gives 9 + 36 + 72 = 117. The result, 117, correctly verifies the product since 13 × 9 = 117, demonstrating the method's reliability for practical computations without memorized tables.

Mathematical Principles

Duplation and Mediation

Duplation, known historically as the process of repeated doubling, forms one of the foundational operations in ancient Egyptian multiplication. This technique involves successively adding a number to itself to generate powers of two, such as starting from 1 and producing 2, 4, 8, and so on, without requiring knowledge of higher multiplication tables. As described in translations of the (c. 1650 BCE), duplation allowed scribes to build a sequence of doubled values from the multiplicand, enabling the decomposition of the multiplier into a sum of these powers through simple . Mediation, the complementary operation of repeated halving, involves dividing a number by two successively, typically taking the integer part when dealing with odd numbers to maintain whole units, such as halving 9 to 4, then 2, and 1. In the context of Egyptian arithmetic, as evidenced in the same , mediation applies to the multiplier to identify which doubled terms from the duplation sequence should be selected and added, effectively handling the binary-like breakdown without explicit . This halving process discards fractional remainders initially, focusing on results that align with the method's additive nature. These operations, while not named "duplation" and "mediation" in the original hieratic texts, emerged in historical analyses and translations of Egyptian papyri, with influences traceable to Greek accounts of Egyptian computational prowess, such as Herodotus' observations on their arithmetic sophistication in the fifth century BCE. By relying solely on doubling, halving, and addition—skills presumed basic for scribes—the method obviated the need for memorized multiplication tables, making complex calculations accessible through practical repetition and summation. This approach is detailed in seminal translations like those by Peet (1923) and Gillings (1972), highlighting its efficiency for administrative and geometric tasks.

Binary Foundations

The Ancient Egyptian multiplication method, as documented in sources like the , fundamentally relies on binary decomposition, predating modern binary arithmetic by millennia. In this approach, the repeated halving of one number (the multiplier) serves to extract its binary representation: each odd result during halving indicates a '1' in the corresponding binary digit position, while even results correspond to '0's. Simultaneously, the doubling of the other number (the multiplicand) generates successive powers of 2, allowing the selection of those terms where the multiplier's binary digit is 1. This process effectively breaks down the multiplication into a sum of the multiplicand scaled by powers of 2, mirroring the structure of binary numeral systems. From a formulaic perspective, for positive integers aa and bb, the product a×ba \times b equals b×∑k=0mdk⋅2kb \times \sum_{k=0}^{m} d_k \cdot 2^k, where dk∈{0,1}d_k \in \{0, 1\} are the binary digits of aa, and mm is sufficiently large to cover all bits of aa. This summation selects only the terms where dk=1d_k = 1, corresponding to the doubled values retained in the Egyptian procedure. The method's efficacy stems from the fundamental identity that any positive integer nn can be uniquely expressed as n=∑k=0mdk⋅2kn = \sum_{k=0}^{m} d_k \cdot 2^k with dk∈{0,1}d_k \in \{0, 1\}, allowing multiplication by mm to distribute over the sum: n×m=m×∑k=0mdk⋅2k=∑k=0mdk⋅(m⋅2k)n \times m = m \times \sum_{k=0}^{m} d_k \cdot 2^k = \sum_{k=0}^{m} d_k \cdot (m \cdot 2^k). Thus, the Egyptian technique leverages this binary expansion to compute products efficiently without requiring direct knowledge of positional notation.

Comparisons and Variants

Russian Peasant Multiplication

The Russian peasant multiplication method is a duplication and mediation technique that proceeds by repeatedly halving one number (discarding any remainder) while doubling the other, then summing the doubled values corresponding to the odd halves until the first number reaches 1. This process is identical to the ancient Egyptian multiplication procedure. The name "" emerged in European mathematical , likely due to observations of its use among illiterate Russian peasants who relied on oral arithmetic traditions into the . Historical records suggest the designation may trace to descriptions in older Russian texts portraying the method as a practical tool for rural calculations, though definitive early sources remain elusive. To illustrate, consider multiplying 21 by 14 using the Russian peasant method. Begin with 21 and 14 in adjacent columns, then halve 21 (ignoring fractions) and double 14 iteratively:
Halves of 21Doubles of 14
21 (odd)14
10 (even)28
5 (odd)56
2 (even)112
1 (odd)224
Strike out the rows where the halved value is even, leaving the entries 21, 5, and 1. Sum the corresponding doubles: 14 + 56 + 224 = 294, which is the product 21 × 14. This tabular notation, common in modern explanations, differs slightly from ancient Egyptian hieratic markings but preserves the core operations.

Similar Ancient Techniques

In ancient , while the Nine Chapters on the Mathematical Art (Jiuzhang suanshu), a foundational mathematical text compiled around 200 BCE during the , primarily employed multiplication using for digit-by-digit calculations and precomputed tables, earlier texts like the Suan shu shu (c. 200 BCE) describe methods involving repeated doubling and halving for arithmetic operations, aligning with practical applications in and administration. In contrast, Mesopotamian mathematics from the Old Babylonian period (c. 2000–1600 BCE) emphasized precomputed tables for , inscribed on clay tablets, to handle operations in their (base-60) system. These tables listed products for pairs of digits up to 59, along with reciprocals and squares, allowing scribes to approximate or directly retrieve results for larger numbers by breaking them into components and adding partial products. Unlike the iterative doubling of Egyptian methods, this tabular system prioritized lookup efficiency over dynamic decomposition, reflecting the administrative demands of and astronomy but limiting flexibility for irregular computations. Ancient Indian texts, such as the Sulba Sutras (c. 800–200 BCE), focused primarily on geometric constructions for Vedic rituals, incorporating implicit through decompositions for calculating altar dimensions and approximations like √2 ≈ 1.4142. However, explicit techniques in these works leaned toward proportional parts and rather than binary-like iterative halving, with more structured decomposition methods emerging later in texts like the Bakhshali manuscript (c. 200–300 CE). The Russian peasant multiplication, likely rooted in ancient practices through oral traditions, serves as a later parallel to these ancient doubling-and-halving strategies across cultures.

Significance

Applications in Egyptian Society

Ancient Egyptian multiplication, based on duplation and , played a crucial role in land surveying by enabling the calculation of field areas essential for and taxation after annual floods. Surveyors measured lengths and widths using knotted ropes and rods, then applied the method to multiply these dimensions for rectangular fields, yielding areas in setat (approximately 2,735 square meters). For instance, in the , problems involving geometric shapes like trapezoids (e.g., Problem 52) combined measurements through doubling and halving to determine areas, such as a field with a 20-khet , 6-khet base, and 4-khet cut side resulting in 100 setat. This practical application ensured accurate boundary re-establishment and land valuation, supporting the pharaonic economy. In administrative contexts, the technique facilitated tax assessments, grain distribution, and resource allocation across Egyptian society. Land taxes were levied based on computed field areas, with multiplication scaling measurements to quantify yields for royal or temple granaries. Grain and rations for workers and officials were calculated similarly; the Rhind Papyrus contains numerous problems on dividing provisions, such as Problems 1–6, which distribute loaves among 10 men using fractional multiples derived from doubling tables, and problems involving distribution of varying strengths (pesu), such as Problems 69–78. These examples demonstrate the method's scalability for practical quantities, ensuring equitable distribution in labor-intensive tasks like farming and . Problem 30 further illustrates of 100 loaves between groups, adaptable to beer or grain, highlighting its role in . For pyramid construction, Egyptian supported resource planning by computing volumes and material quantities needed for massive stone placements. The (Problem 14) calculates the volume of a truncated —relevant to pyramid frusta or granaries—with bases of 4 and 2 cubits and height 6 cubits, yielding 56 cubic cubits through multiplication: one-third times height times the sum of bases and of their product (1/3 × 6 × (16 + 4 + 8)). Such computations aided in estimating stone blocks, labor rations, and ramp dimensions, integrating with administrative distributions for workforce sustenance during projects like the . This application underscored the method's utility in monumental engineering, where precise scaling prevented resource shortages.

Modern Educational Value

Ancient Egyptian multiplication, also known as the method of doubling and adding, offers significant pedagogical benefits in contemporary by introducing students to binary concepts through repeated doubling, which mirrors binary representation without requiring advanced knowledge of base-2 systems. This approach strengthens addition skills by emphasizing selective summation of doubled values and fosters algorithmic thinking, as learners must systematically halve one factor while doubling the other to identify relevant terms, promoting step-by-step problem-solving over rote of tables. Educators value it for building conceptual understanding, particularly in non-positional numeral systems, where students explore how ancient methods rely on basic operations like duplication and rather than place value. In modern curricula, the technique integrates into STEM programs to bridge mathematics and , illustrating binary operations essential for programming and representation; for instance, it serves as a "CS-complete example" applicable across introductory courses (CS1 for problem-solving, CS2 for , and for proofs). This linkage highlights the method's binary foundations, where powers of two align with binary decomposition, helping students grasp foundational algorithms in coding without modern tools. Such applications appear in middle school activities that promote cultural awareness and through historical algorithms. While less efficient than long multiplication—requiring more steps and deemed slower by trainee teachers (e.g., 57.5% of students noted its prolonged process)—the Egyptian method remains valuable for teaching non-positional arithmetic and , encouraging appreciation for diverse mathematical traditions over speed alone. In the 21st century, it has seen revivals in mental math applications, such as dedicated and Windows tools that guide users through doubling steps interactively to build computational fluency. These digital resources, alongside classroom activities, underscore its role in engaging learners with intuitive, low-memorization strategies amid critiques of practicality for .

References

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