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Leibniz on Number Systems

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Handbook of the History and Philosophy of Mathematical Practice

Abstract

This chapter examines the pioneering work of Gottfried Wilhelm Leibniz (1646–1716) on various number systems, in particular binary, which he independently invented in the mid-to-late 1670s, and hexadecimal, which he invented in 1679. The chapter begins with the oft-debated question of who may have influenced Leibniz’s invention of binary, though as none of the proposed candidates is plausible, I suggest a different hypothesis that Leibniz initially developed binary notation as a tool to assist his investigations in mathematical problems that were exercising him at the time, namely those concerning the divisibility of composite numbers, primality, and perfect numbers. The chapter then explores Leibniz’s development of binary, his little-known work on binary fractions and expansions, his use of binary as a symbol for creation in his philosophical-theology, and his response to the suggestion that there was a correlation between binary numeration and the hexagrams of the ancient Chinese divinatory text, the Yijing. The chapter then focuses on Leibniz’s work on other number systems, in particular his invention and exploration of hexadecimal as well as his work on duodecimal. The chapter concludes by revealing a hitherto unknown practical application of binary that Leibniz devised in the last year of his life.

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Notes

  1. 1.

    See Couturat (1901, 473); Zacher (1973, 9–33); Tropfke (1980, 12); and Ingaliso (2017, 111–112).

  2. 2.

    In 1711, decades after devising binary and only 5 years before his death, Leibniz (1768, V: 418) wrote: “Regarding Caramuel’s Mathesis biceps [Old] and New, for which he asks ten thalers, I am unable to judge well because I have not yet seen it, and I fear it may contain vain subtleties, which is not unusual for Caramuel.”

  3. 3.

    This work yielded his first publication in mathematics in February 1678, namely, the short journal article “A new observation about the way of testing whether a number is prime”; see Leibniz 1678/http://www.leibniz-translations.com/prime.htm. For details of Leibniz’s work on primes, see Mahnke (1912–1913).

  4. 4.

    In the manuscript entitled “Formarum reductio ad simplices,” written 12 September 1680: “Therefore, 2z–1 – 1 will be divisible by z, if z is prime” (LH 35, 3 A 4 Bl. 14r). During these investigations, Leibniz also glimpsed Wilson’s theorem: (n – 2)! ≡ 1 (mod n), or in Leibniz’s words: “The product of continuous [integers] up to the number which anteprecedes the given integer, when divided by the given integer, leaves 1, if the given integer is prime. If the given integer is derivative, it will leave a number which, since it has a common measure with the given integer, is greater than one” (LH 35 3 B 11 Bl. 21r). However, when testing his articulation of the theorem, Leibniz made a miscalculation that led him to add the false statement “(or the complement of 1)” after “leaves 1.”

  5. 5.

    In a slightly later manuscript, from 1679 (LH 35, 8, 30 Bl. 148), Leibniz again uses binary notation to illustrate his (still immature) formulation of a prime number theorem, namely, 2n – 1: “Let = 111111, = 1111, = 111, = 11. = A, = B. Now and are prime among themselves, because their exponents are such, that is, their indices, or the numbers 2 and 3. Therefore A and B are not prime among themselves and necessarily will become A = C and B = C, and will become = C. Zz – 1 is divisible by Z if Z is prime, and I have demonstrated this as follows: 22 – 1 by 3 and 24 – 1 by 5, therefore 1111 is divisible by 5 and 11 by 3 and 111111 by 7. But 11111 cannot be divided by 6, for since 11 can be divided by 3, 11111 and 11 have a common divisor, yet they are prime among themselves. And hence we have the sought-for demonstration of a reciprocal property of a prime number.”

  6. 6.

    Euclid, Elements, IX.36. That is, if p is a positive integer and 2p – 1 is prime, then 2p-1 (2p − 1) is perfect.

  7. 7.

    Perhaps because of this, in another manuscript on the subject, probably written in 1678, Leibniz sought to demonstrate perfect numbers using a mixture of binary numeration and algebra based thereon (so unrelated to Pauli’s algebra), eventually reaching the conclusion “22z+1 – 2z will be a perfect number if 2z+1 – 1 is prime. Likewise, 2z–1 – 1 will be divisible by z, if z is prime” (LH 35, 3 B 17 Bl. 1).

  8. 8.

    Leibniz explicitly acknowledged these features, at least in the 1680s onward. For example: “The very last digits of a number of the double geometric progression can be easily obtained like this: if 1 is subtracted from it, then it is written in binary: etc.1111111, that is, 1 + 2 + 4 + 8 + 16 + 32 etc.” (LH 35, 3 B 11 Bl. 10r; cf. LH 35, 8, 30 Bl. 75; LH 35, 13, 3 Bl. 33; LH 35, 15, 5 Bl. 10r; and LH 35, 3 B 5 Bl. 51r).

  9. 9.

    A similar table is found in other manuscripts, such as LH 35, 4, 11 Bl. 10r, though this was likely written in 1681.

  10. 10.

    Another writing in this vein is printed in Strickland and Lewis (2022, 41–43).

  11. 11.

    Leibniz did not affix a date to them, and the paper contains no watermarks that could be used to determine a dating, so they have to be dated using internal evidence.

  12. 12.

    Leibniz also refers to “the binary progression, where only ones and 0 express a number,” in a manuscript on the construction of a universal language, tentatively dated to February 1678 (A VI 4, 68).

  13. 13.

    A third is outlined in Strickland and Lewis (2022, 40n2).

  14. 14.

    Although in one early writing he exclaims that binary will be “remarkable for periodic progressions in expressible quantities which are not whole or rational” (Strickland and Lewis 2022, 32).

  15. 15.

    For further details of Leibniz’s work on the binary expression of his circle formula, see Strickland (2023a).

  16. 16.

    Also: “Without doubt boundaries or limits are of the essence of creatures, but limits are something privative and consist in the denial of further progress.” Leibniz (2006, 38), cf. A I 15, 369; Leibniz (2006, 102–103); and Strickland (2014, 22).

  17. 17.

    Or as he puts it in a manuscript written perhaps in 1695 or possibly in 1702, “things are educed from God’s active power—as Julius Scaliger said—rather than from passive nothing. My invention of the composition of numbers from 0 and 1 would wonderfully support this doctrine” (LH 4, 3, 5e Bl. 5). See Scaliger (1557, fols. 16–17).

  18. 18.

    When Leibniz did come to write the Theodicy more than a decade later, the binary-creation analogy was not mentioned. The Theodicy was published in 1710. See Leibniz (1985).

  19. 19.

    There are other sequences of hexagrams apart from the two mentioned already, such as the Eight Palaces sequence, which predates the Fuxi sequence. These sequences have elicited many studies of their structure, interrelation, and history, though such matters are outside of the scope of this chapter.

  20. 20.

    For more information on Bouvet’s hypothesis and Leibniz’s reaction thereto, see Kempe (2022, 141–174).

  21. 21.

    For more information about the history of base 16, and Leibniz’s place therein, see Strickland and Jones (2023).

  22. 22.

    The same conversion of 100000 to 186u016 is also found in a contemporaneous text: LH 35, 8, 30 Bl. 148r.

  23. 23.

    Leibniz is probably thinking of Schwenter’s Deliciae physico-mathematicae [The Charms of Physico-Mathematics] of 1636, which was posthumously revised and expanded by Georg Philipp Harsdörffer (1607–1658) in 1651 and again in 1653. However, as far as I have been able to tell, in none of those works is there any mention of duodecimal, let alone any report of anyone endorsing it.

  24. 24.

    Leibniz actually wrote “1728,” but this is clearly a mistake as his example uses 1712.

  25. 25.

    After this example, Leibniz turned to the quaternary system, efficiently sketching out the steps required to convert the decimal value 1712 to 1223004 (LBr 705 Bl. 93r). He then did the same with binary, before turning to the periodicity of columns in binary.

  26. 26.

    For further details of Leibniz’s tactile binary clock, and an English translation of the manuscript, see Strickland (2023b).

  27. 27.

    Leibniz himself had devised a spring-driven pocket watch in the mid-1670s and continued to improve it even four decades later; see Leibniz 1675, and LH 38 Bl. 274–275.

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Acknowledgments

The author would like to express his gratitude to the Gerda Henkel Stiftung, Düsseldorf, for their award of a research scholarship (AZ 46/V/21), which made this chapter possible. The author would also like to thank Daniel J. Cook, Donald E. Knuth, and Harry R. Lewis for their helpful comments on an earlier version of this chapter; Siegmund Probst and Julia Weckend for their assistance in transcribing manuscripts and their illuminating discussion thereof; and Lisa Silva – @lisa_cs8 – for her illustration of Leibniz’s tactile clock.

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    Lloyd Strickland

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  1. Department of Mathematical Sciences, The University of Montana, Missoula, MT, USA

    Bharath Sriraman

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  1. Department of Mathematical Sciences, University of Montana, Missoula, MT, USA

    Bharath Sriraman

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Strickland, L. (2022). Leibniz on Number Systems. In: Sriraman, B. (eds) Handbook of the History and Philosophy of Mathematical Practice. Springer, Cham. https://doi.org/10.1007/978-3-030-19071-2_90-1

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  • DOI: https://doi.org/10.1007/978-3-030-19071-2_90-1

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