E Making up Numbers A History of Invention in Mathematics KKEHARD EKKEHARD KOPP Making up Numbers: A History of Inventi on in Mathemati cs off ers a detailed but K Making up Numbers accessible account of a wide range of mathema� cal ideas. Star� ng with elementary OPP concepts, it leads the reader towards aspects of current mathema� cal research. A History of Invention in Mathematics Ekkehard Kopp adopts a chronological framework to demonstrate that changes in our understanding of numbers have o� en relied on the breaking of long-held conven� ons, making way for new inven� ons that provide greater clarity and widen mathema� cal horizons. Viewed from this historical perspec� ve, mathema� cal abstrac� on emerges as neither mysterious nor immutable, but as a con� ngent, developing human ac� vity. Chapters are organised thema� cally to cover: wri� ng and solving equa� ons, geometric construc� on, coordinates and complex numbers, a� tudes to the use of ‘infi nity’ in mathema� cs, number systems, and evolving views of the role of M axioms. The narra� ve moves from Pythagorean insistence on posi� ve mul� ples to gradual acceptance of nega� ve, irra� onal and complex numbers as essen� al tools in quan� ta� ve analysis. AKING Making up Numbers will be of great interest to undergraduate and A-level students of mathema� cs, as well as secondary school teachers of the subject. By virtue of UP its detailed treatment of mathema� cal ideas, it will be of value to anyone seeking N to learn more about the development of the subject. UMBERS As with all Open Book publica� ons, this en� re book is available to read for free on the publisher’s website. Printed and digital edi� ons, together with supplementary digital material, can also be found at www.openbookpublishers.com Cover images from Wikimedia Commons. For image details see capti ons in the book. KKEHARD OPP Cover Design by Anna Gatti . E K book eebook and OA edi� ons also available www.openbookpublishers.com OBP https://www.openbookpublishers.com c 2020 Ekkehard Kopp This work is licensed under a Creative Commons Attribution 4.0 Interna- tional license (CC BY 4.0). This license allows you to share, copy, distribute and transmit the text; to adapt the text and to make commercial use of the text providing attribution is made to the authors (but not in any way that suggests that they endorse you or your use of the work). Attribution should include the following information: Ekkehard Kopp, Making Up Numbers: A History of Invention in Mathematics Cambridge, UK: Open Book Publishers, 2020, https://doi.org/10.11647/OBP.0236 In order to access detailed and updated information on the license, please visit https://www.openbookpublishers.com/product/1279#copyright Further details about CC BY licenses are available at https://creativecommons.org/licenses/by/4.0/ All external links were active at the time of publication unless otherwise stated and have been archived via the Internet Archive Wayback Machine at https://archive.org/web Updated digital material and resources associated with this volume are avail- able at https://www.openbookpublishers.com/product/1279#resources Every effort has been made to identify and contact copyright holders and any omission or error will be corrected if notification is made to the pub- lisher. For image details see captions in the book. ISBN Paperback: 978-1-80064-095-5 ISBN Hardback: 978-1-80064-096-2 ISBN Digital (PDF): 978-1-80064-097-9 DOI: 10.11647/OBP.0236 Cover images from Wikimedia Commons. For image details see captions in the book. Cover design by Anna Gatti. Prologue: Naming Numbers In the mathematics I can report no deficience, except it be that men do not sufficiently understand this excellent use of the pure mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For if the wit be too dull, they sharpen it; if too wandering, they fix it; if too inherent in the sense, they abstract it. Sir Francis Bacon, The Advancement of Learning, 1605 When I was very young I asked my father: ’What is the largest number you know?’ and he answered ’octillion’. At the time I diid not know any compact notation for writing large numbers, such as writing them in powers of 10, but I soon decided that an octillion, whatever it might look like, must be too small to be the largest number. After all, if you add 1 to it you get a bigger one! The obvious answer, I decided, was to count to infinity, or at least far enough to find a number for which I would need to invent a new name. I resolved to try this in bed that night, but sleep overcame me soon after the 12; 000th sheep. But I was now clear that there is no such thing as ‘the largest whole number’. And that is something that strikes me as quite profound. After all, our senses provide us with information that reflects the finiteness of our surroundings—even if they may seem forbiddingly large when you’re six years old—yet here we have a system of allocating names, or symbols, that is essentially without any limits. How does this system of numbering, ab- stracted by us as a collective mental construct, reflect the finite physical en- vironment? Or does our perception of the need for counting not originate in the observation of our immediate surroundings? Counting numbers were used for practical purposes in pre-historic times. Whether hunting or gathering, farming or trading, even in battle, no-one could readily escape the need to distinguish between ‘one’, ‘two’ and ‘many’: for example, when describing a pack of wolves, the day’s wild fruit pick- ings, a flock of sheep, sacks of corn offered for exchange, or the size of the enemy’s clan. Simple tallying, such as recording the number of any group with notches on a stick or a collection of pebbles, probably preceded the actual naming of numbers, but at some stage the need to invent verbal de- scriptions of the counting process became unavoidable. c Ekkehard Kopp CC BY 4.0 https://doi.org/10.11647/OBP.0236.12 2 PROLOGUE: NAMING NUMBERS Today, we have become so used to our decimal system of naming the numbers we use every day that questions about the origin of their names seldom enter our consciousness—we learn them at mother’s knee, at the same time as the alphabet. Usually we start by counting on the fingers of both hands. 1. Naming large numbers In any event, I might have had trouble working out what ‘octillion’ could mean, since even now there is no universal agreement about the names of various (fairly) large numbers! Everyone agrees that we call a thousand times a thousand a million. In the decimal system we ‘add a zero’ when- ever we multiply to 10 and we have a convenient shorthand notation, using, for example, 1000 = 10 × 10 × 10 = 103, where the exponent 3 simply shows how many times we multiplied by 10: Similarly, we write one million as 103 × 103 = 106. After that, however, different naming conventions emerge. If, like my father, you are German, or any continental European, you would (today) stick to the Latin origins of the terms we might use when multiplying a million times—which means that we add six zeroes each time: a billion is a ‘bi-million’, which is obtained by multiplying a million (106) by a million (106), so it becomes 106 × 106 = 1012; a trillion (tri-million) is a million billion (106 × 1012 = 1018), a quadrillion becomes a million trillion (106 × 1018 = 1024), and we continue via quintillion, sextillion, septillion—adding 6 to the exponent each time—to reach octillion as 1048. You could go on to nonillion (1054) and decillion (1060), although these terms reached the Oxford English Dictionary only relatively recently and the dictionary entry currently stops there. Not to be outdone, Wikipedia lists 100 such number names—each a million times the previous one—up to cen- tillion (10600). This continental European number naming scale is now known as the long scale. It certainly has a long history. One of the earliest consistent ac- counts of number names generated in this fashion occurs in a 1484 article Triparty en la science des nombres by the French mathematician Nicolas Chu- quet (c.1445-c.1488) who mentions number names very similar to the above, up to ‘nonyllion’, and continues: ‘and so on with others as far as you wish to go’. The American version of these number names is known as the short scale In this scale, having reached a million, we then create a new number name when we’ve reached a thousand times the previous one. In terms of powers 1. NAMING LARGE NUMBERS 3 Figure 1. 1021 pengo banknote1 of 10, in the short scale we create new names whenever we add 3 to the expo- nent. In this scale a billion is therefore a thousand million (103 × 106 = 109), although the Europeans confuse matters by calling 109 a milliard instead! Continuing up the short scale, we reach a trillion as a thousand (short- scale) billion (103 × 109 = 1012), so a short-scale trillion is the same num- ber as a long-scale billion. And so it goes on, confusing us all. Just for the record: repeatedly multiplying by 103 means that the short-scale octillion is a mere 1027—while in the long scale, 1027 becomes a thousand quadrillion, so perhaps it should really be a ‘quadrilliard’? Currently, the USA, UK and Canada—and with them most other Anglo- phone as well as Arabic-speaking countries—use the short scale, while most countries in Europe, plus most French, Spanish or Portuguese-speaking coun- tries elsewhere, prefer the long scale.