Al-Uqlidisi's decimal-fraction methods

A half can be written as five tenths, and half of that as twenty-five hundredths. A book composed by Abu al-Hasan al-Uqlidisi at Damascus in341/952–953 contains an early direct treatment of such fractions within Indian place-value arithmetic ^{1 3}. This is not invention of Indian numerals or today's decimal point: it concerns operations on fractional parts and a sign preserving the units position, with historians disagreeing over conceptual completeness ². We present the procedures and evidence limits together, without making chronological precedence mean that all modern arithmetic came from one book.

Decimal fractions represent parts with denominators ten, a hundred, a thousand and beyond; in modern explanation, place value continues below one. Saidan describes repeated halving producing decimal results and a stroke over the units position that the author insists should be used ³. That sign is not the dot separating whole and fractional parts in our modern image ^{4 3}. Writing whole numbers in base ten does not by itself establish the same treatment of fractions; this distinction is central to the historical question ².

Al-Uqlidisi inherited Indian place-value arithmetic, sexagesimal reckoning and procedures involving fingers or a dust board; Saidan explains these tools and comparisons ⁵. Al-Khwarizmi's transmission of Indian arithmetic belongs to that earlier setting, not a claim that al-Uqlidisi invented the nine numerals or zero ⁶. Erasing and shifting digits formed part of some board procedures, but place value is not identical to erasure: an operation can change its procedure while preserving the numerical idea ⁷. He is not credited with inventing pens, paper or every earlier use of fractions.

The manuscript witness names the author and composition at Damascus in952–953; TDV supplies the341/952–953 pair, while the surviving copy is dated1157 in the biography ^{2 1}. These date different acts, composition and copying, not competing attempts to identify an invention day. As reported by the biography, the author's introduction describes travel, reading and learning from arithmeticians he met ². This is transmitted testimony, not an independently established itinerary or complete teacher list. Modern birth and death estimates cannot make a sparsely documented life into a certain year-by-year story.

Abu al-Hasan Ahmad ibn Ibrahim al-Uqlidisi is not the ancient Euclid. Sources connect his nisba with possibly copying Euclid or expertise in that material, not a certain documented occupation ^{1 8}. The Muslim basis is Saidan's explicit named classification quoted by MacTutor, not a name or city ². Sunni is separately the museum's labelled default for the Islamic Damascus milieu, not personal school testimony; the biographies read name no contrary personal affiliation ^{2 1}. His role here is improving inherited tools while preserving Indian arithmetic's contribution ⁷.

Saidan gives the repeated-halving sequence13,6.5,3.25,1.625,0.8125 and describes reversing it by successive multiplication by two ³. Our dot is modern explanation, not a redrawing of the author's handwriting. Below one, the unit does not change invisibly: eight thousand one hundred twenty-five ten-thousandths equals0.8125 in our notation. Retaining the units position makes those places legible rather than confusing small parts with a large whole number ³.

A second teaching comparison distinguishes value from representation: a quarter is twenty-five hundredths and also fifteen sixtieths. Equality does not make the writing systems identical. Each successive decimal place divides the previous unit by ten, while sexagesimal places use sixty. The question is therefore not only whether an answer is correct, but which scale supports further calculation. Saidan distinguishes expressing a root on the decimal scale from converting its result into sexagesimal notation ³. These comparative numbers are our modern explanation, not a transcription of a missing source formula.

Alignment matters more than a familiar-looking sign. In our teaching example,0.5 plus0.25 gives0.75: five tenths can be read as fifty hundredths, then combined with twenty-five hundredths. These numbers illustrate the place-value principle Saidan discusses; they are not claimed as a manuscript example ³. Not every zero is redundant:0.50 equals0.5, but0.05 does not. Saidan's reference to dropping final zeros when reversing by multiplication does not authorize deleting a zero that preserves an internal place ³.

MacTutor discusses increasing a quantity by its tenth five times; Saidan mentions starting a repeated operation at136, but its array is missing from the available reproduction ^{2 3}. Our modern calculation, not a transcription, gives149.6,164.56,181.016,199.1176,219.02936. Each step adds a tenth of the new result, not always a tenth of the original136, so it is not five additions of one constant amount. This teaching sequence was checked with exact integer arithmetic; it does not fill a source gap and then pretend the missing material was read.

Paper adaptation is related but distinct. The fourth part modifies procedures for ink and paper instead of shifting and erasing figures on a board ⁷. One can explain fractions without inventing their writing medium, or adapt a medium without completing a theory of fractions. Keeping those strands separate avoids crediting every subject in the book as one universal invention ².

Saidan describes a long, uneven move away from the dust board, contrasting Ibn al-Banna in Morocco with al-Tusi in the east as evidence of different persistence ⁶. This is a historian's reading of later texts, not a school census or proof that a named reader directly borrowed al-Uqlidisi. The biography compares widely separated decimal-fraction treatments without documenting a reading chain from this manuscript to every later author ². Indian arithmetic's spread, changing calculation media and decimal-fraction history need not share one timeline or map.

The first part explains numerals, place value and operations; the second gathers earlier methods in Indian form; the third answers questions beginning “why” or “how” ⁷. This structure explains the biography's inference of experience teaching beginners, without naming a school or formal appointment ². Earlier sources are not an embarrassment to conceal: collecting them forms part of the author's own account ⁹. Our diagram represents arithmetic resources and procedures, not personal meetings with every earlier mathematician.

Studies compare al-Uqlidisi with al-Samawal, al-Kashi and Stevin, disagreeing over the stage at which the concept became complete ^{2 1}. Comparison is not proof that all read the same book. As reported by MacTutor, Rashed emphasizes the absence of an explicit extension of powers of ten through their reciprocals after defining the zero power ². This does not deny knowledge of the numeral zero: the criticism concerns a theoretical relationship between places. The comparison also does not date a separate post-conversion invention by al-Samawal.

The procedural effect concerns writing and calculating with parts of a unit in regular places; the biography stresses the practical value the author attributes to Indian arithmetic ². Declaring usefulness does not measure users. Modern examples involving a length or half a quantity are not presented as Damascus invoices or building records. Questions from learners do not prove that every merchant studied this book. The benefit claim remains attributed to the author; a documented application requires its own witness ².

This witness complicates a history beginning with Stevin and looking back only to al-Kashi: earlier treatment requires considering a concept's development across several texts ¹. Saidan sees conscious use of fractions and their sign, while Rashed distinguishes implicit procedures from explicit general theory ^{3 2}. We do not settle the disagreement with a heroic label. Agreement that an early step mattered does not require agreement that it supplied every later form or erased subsequent development's value.

The narrow claim is an early surviving direct treatment of decimal fractions and operations with a units sign, improving older arithmetic ³. MacTutor presents Saidan's interpretation of understanding and Rashed's more restrictive assessment, both retained as named perspectives ². “Earliest known” limits the surviving evidence, not everything once written and lost. Priority in handling decimal parts does not automatically extend to today's typographic dot, every division or root procedure, or every topic in the book ⁶.

GregWelch8's modern chart explains whole-number and fractional places, emphasizing the units column and English labels from millions to millionths ⁴. It is CC0 teaching material, not an old page or al-Uqlidisi's handwriting ¹⁰. The creator's description dates2023 and the current upload2025; neither dates a manuscript ¹¹. Its mirror analogy about English place names is modern pedagogy, not a historical theory attributed to the author. The chart helps explain position without proving a route to every modern calculator.

TDV identifies Yeni Cami802 and lists Saidan's1973 Arabic edition and1978 English translation ¹. Neither manuscript nor editions were independently inspected here. We read the Saidan study linked by MacTutor, a print of Encyclopedia.com rather than original dictionary pages. Visual inspection of its second page confirms that some arrays and formulae are actually missing, not merely absent from text extraction. They are not recreated and attributed to the source, and the print-file's page numbers are not used as original DSB pagination. Citations identify passages actually read.

Historical writing changes when uncertainty remains visible: sources compare the Euclid-related nisba, teaching experience suggested by beginners' questions and the little known about his life ^{2 1}. The work needs neither a fabricated biography nor an imagined portrait. Naming Indian resources and differing assessments prevents accumulated knowledge becoming exclusive cultural property. A visitor can appreciate a Damascus author while seeing numeral systems, working media and fractional concepts as connected stages rather than one invention moment ending every question.

Divide a strip into ten equal pieces: five make a half. Divide it into a hundred: twenty-five make a quarter. Today we write0.5 and0.25; our signs are modern, not a manuscript imitation. The historically relevant idea is that a digit's position tells us which part we count, and a units mark protects that reading ³. Try halving and doubling on paper, but a successful exercise alone cannot establish who first understood an idea. The coloured chart is modern teaching material, not an object from al-Uqlidisi's lifetime ¹².

• ¹³ — Biography, quoted Saidan classification and disagreement with Rashed; originals not independently read.
• ¹ — Composition341/952–953, manuscript, editions and career.
• ¹⁴ — Saidan's reproduced study; missing formulae not read.
• ¹⁵ — Modern chart, creator and licence.