Decimal fractions in computation
The place-value decimal notation for fractional numbers (0.27 rather than 27/100): they can be handled exactly like integers by reading them as 'so many hundredths'. In the booklet De Thiende (Leiden, 1585; French version La Disme the same year) the Fleming Simon Stevin showed this procedure and proposed the decimal division of coinage, measures and weights.
Stevin did not invent decimal fractions. Al-Uqlidisi described them in 952/953, as did al-Samaw'al and al-Kashi; in Europe they had occurred in trigonometric tables. But it was after De Thiende that they entered everyday computation, and Stevin's notation, through Clavius and Napier, developed into the modern one.
Norton's English translation (1608) suggested to Jefferson the decimal coinage of the United States. One and the same computation fits both millions and small change.
Why this date
exact for the publication of De Thiende (Leiden, 1585). The idea itself is older: al-Uqlidisi (Damascus, 952/953; Saidan sees the idea itself in him, Rashed only a prehistory and takes al-Samaw'al as the first chapter), al-Samaw'al (1172), al-Kashi (1427), in Europe Bianchini (1440s, after Van Brummelen 2024; the paper is closed and has not been checked verbatim)
What it grew from
-
Positional decimal system and zero
prerequisite for · Confidence: high
Decimal fractions extend positional decimal notation to the right of the units place: Stevin reads 3.27 as 327 units of 0.01, and the earliest direct treatment of decimal fractions (al-Uqlidisi, 952/953) appears in a book on Indian arithmetic; without positional notation such a fraction is impossible.
Link sources
- Minnaert M. G. J. «Stevin, Simon», Dictionary of Scientific Biography (1976) Verified 20 Sep 2026
327 items of the unit 0.01
- O'Connor J. J., Robertson E. F. «Al-Uqlidisi», MacTutor History of Mathematics (1999) Verified 20 Sep 2026
the earliest known text offering a direct treatment of decimal fractions
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Minnaert M. G. J. «Stevin, Simon», Dictionary of Scientific Biography (1976) Verified 20 Sep 2026
-
Algebra as a distinct method
influenced · Confidence: medium
In the Arabic tradition decimal fractions were developed in al-Karaji's school to approximate roots in algebraic problems (al-Samaw'al, 1172), and Stevin, who issued De Thiende together with L'Arithmetique, extended uniform rules of operation to all numbers, roots included.
Dispute: The European line of decimal fractions did not run through algebra: according to the DSB (Minnaert), before Stevin they "had previously found only occasional use in trigonometric tables", and Van Brummelen (2024) found decimal fractional numeration and a decimal point in Bianchini's tables of the 1440s. No direct dependence of Stevin on Arabic authors has been shown; the Muslim Heritage claim that Stevin "learnt of the Arab methods" is unsupported by a source. Stevin's systematic work on decimal expansions of polynomial roots came after De Thiende (1594, Fowler), so in Stevin's own case the fractions precede the algebraic application.
Link sources
- O'Connor J. J., Robertson E. F. «Al-Kashi», MacTutor History of Mathematics (1999) Verified 20 Sep 2026
The usage of decimal fractions no longer for approaching algebraic real numbers, but for real numbers such as π
- Katz K. U., Katz M. G. «Stevin numbers and reality», Foundations of Science 17(2), 109-123 (2012) Verified 20 Sep 2026
The belief that all arithmetic operations, as well as extracting roots, etc., should follow the 'same' rules as the rationals, originates precisely with Stevin
- FSTC, «Decimal Arithmetic», Muslim Heritage (2007) Verified 20 Sep 2026
Al-Samawal (1172) considered decimal fractions in problems of division and extraction of roots.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- O'Connor J. J., Robertson E. F. «Al-Kashi», MacTutor History of Mathematics (1999) Verified 20 Sep 2026
What grew out of it
-
Logarithms and tables
influenced · Confidence: high
Napier learned decimal notation for fractions from Stevin and used it with the decimal point in the Constructio, but the 1614 tables themselves are built on integers (whole sine 10^7), so Stevin's fractions were not a physical prerequisite of logarithms.
Dispute: The influence runs both ways: Stevin gave Napier the notation, but according to Roegel it was Napier's logarithms that became the main vehicle of the decimal idea in practice. Moreover, Clavius already used a decimal point in a table of sines in 1593 (Cajori vol. 1, p. 324; Boyer), so Napier is not the only channel from Stevin to modern notation.
Link sources
- Baron M. E. «Napier, John», Dictionary of Scientific Biography (via Encyclopedia.com) (1974) Verified 20 Sep 2026
Another achievement was the effective use he made of decimal notation (which he had learnt of from Stevin) in conjunction with the decimal point.
- O'Connor J. J., Robertson E. F. «Simon Stevin», MacTutor History of Mathematics (2004) Verified 20 Sep 2026
Stevin's notation was to be taken up by Clavius and Napier and it developed into that used today.
- Roegel D. «Napier's ideal construction of the logarithms», LOCOMAT project, LORIA (2012) Verified 20 Sep 2026
Napier was in fact the main introducer of decimal fractions into common practice and his new mathematical instrument became the best vehicle of the decimal idea.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Baron M. E. «Napier, John», Dictionary of Scientific Biography (via Encyclopedia.com) (1974) Verified 20 Sep 2026
Node sources
- Minnaert M. G. J. «Stevin, Simon», Dictionary of Scientific Biography (1976) Verified 20 Sep 2026
In De Thiende, a twenty-nine-page booklet published in 1585, Stevin introduced decimal fractions for general purposes and showed that operations could be performed as easily with such fractions as with integers. He eliminated all difficulties in handling decimal fractions by interpreting 3.27, for example, as 327 items of the unit 0.01. Decimal fractions had previously found only occasional use in trigonometric tables; although Stevin’s notation was somewhat unwieldy, his argument was convincing, and decimal fractions were soon generally adopted. At the end of the tract, Stevin went on to suggest that a decimal system should also be used for weights and measures, coinage, and divisions of the degree of arc.
- O'Connor J. J., Robertson E. F. «Simon Stevin», MacTutor History of Mathematics (2004) Verified 20 Sep 2026, quotation assembled from fragments
In 1585 he published La Thiende, a twenty-nine page booklet in which he presented an elementary and thorough account of decimal fractions. He wrote this small book for the benefit of:- ... stargazers, surveyors, carpet-makers, wine-gaugers, mint-masters and all kind of merchants. Although he did not invent decimals (they had been used by the Arabs and the Chinese long before Stevin’s time) he did introduce their use in mathematics in Europe. Stevin states that the universal introduction of decimal coinage, measures and weights would only be a matter of time.
- O'Connor J. J., Robertson E. F. «Simon Stevin», MacTutor History of Mathematics (2004) Verified 20 Sep 2026, quotation assembled from fragments
Robert Norton published an English translation of La Thiende in London in 1608. It was titled Disme, The Arts of Tenths or Decimal Arithmetike and it was this translation which inspired Thomas Jefferson to propose a decimal currency for the United States... Stevin’s notation was to be taken up by Clavius and Napier and it developed into that used today.
- O'Connor J. J., Robertson E. F. «Al-Uqlidisi», MacTutor History of Mathematics (1999) Verified 20 Sep 2026
The manuscript gives al-Uqlidisi’s full name on the front page as well as the information that he composed the text in Damascus in 952-53.
- O'Connor J. J., Robertson E. F. «Al-Uqlidisi», MacTutor History of Mathematics (1999) Verified 20 Sep 2026
Al-Uqlidisi’s work is historically important as it is the earliest known text offering a direct treatment of decimal fractions.
- O'Connor J. J., Robertson E. F. «Al-Uqlidisi», MacTutor History of Mathematics (1999) Verified 20 Sep 2026
At one time it was thought that Stevin was the first to propose decimal fractions. Further research showed that decimal fractions appeared in the work of al-Kashi, who was then credited with this extremely important contribution.
- Britannica, «La Thiende» (2024)
In 1585 Stevin published a small pamphlet, La Thiende (“The Tenth”), in which he presented an elementary and thorough account of decimal fractions and their daily use. Although he did not invent decimal fractions and his notation was rather unwieldy, he established their use in day-to-day mathematics. He declared that the universal introduction of decimal coinage, measures, and weights would be only a question of time. The same year he wrote La Disme (“The Decimal”) on the same subject.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong. "Assembled from fragments" means the words do not stand together on the page: we found each fragment separately and joined them with an ellipsis.
Further reading
Not evidence: these carry no quotation and support nothing we wrote. They are books and scans to read next.
- Sarton G. «The First Explanation of Decimal Fractions and Measures (1585). Together with a History of the Decimal Idea and a Facsimile (No. XVII) of Stevin's Disme», Isis 23(1), 153-244 (1935)
- Van Brummelen G. «Decimal fractional numeration and the decimal point in 15th-century Italy», Historia Mathematica (2024)