Making up Numbers: A History of Invention in Mathematics - cover image


Ekkehard Kopp

Published On





  • English

Print Length

280 pages (x+268)


Paperback156 x 19 x 234 mm(6.14" x 0.76" x 9.21")
Hardback156 x 22 x 234 mm(6.14" x 0.88" x 9.21")


Paperback1174g (41.41oz)
Hardback1562g (55.10oz)

OCLC Number





  • PB
  • YQM
  • PBK


  • MAT000000
  • MAT015000
  • MAT027000


  • QA21


  • mathematical ideas
  • development of numbers
  • development number systems
  • negative numbers
  • irrationals numbers
  • complex numbers

Making up Numbers

A History of Invention in Mathematics

  • Ekkehard Kopp (author)
Making up Numbers: A History of Invention in Mathematics offers a detailed but accessible account of a wide range of mathematical ideas. Starting with elementary concepts, it leads the reader towards aspects of current mathematical research.

The book explains how conceptual hurdles in the development of numbers and number systems were overcome in the course of history, from Babylon to Classical Greece, from the Middle Ages to the Renaissance, and so to the nineteenth and twentieth centuries. The narrative moves from the Pythagorean insistence on positive multiples to the gradual acceptance of negative numbers, irrationals and complex numbers as essential tools in quantitative analysis.

Within this chronological framework, chapters are organised thematically, covering a variety of topics and contexts: writing and solving equations, geometric construction, coordinates and complex numbers, perceptions of ‘infinity’ and its permissible uses in mathematics, number systems, and evolving views of the role of axioms.

Through this approach, the author demonstrates that changes in our understanding of numbers have often relied on the breaking of long-held conventions to make way for new inventions at once providing greater clarity and widening mathematical horizons. Viewed from this historical perspective, mathematical abstraction emerges as neither mysterious nor immutable, but as a contingent, developing human activity.

Making up Numbers will be of great interest to undergraduate and A-level students of mathematics, as well as secondary school teachers of the subject. In virtue of its detailed treatment of mathematical ideas, it will be of value to anyone seeking to learn more about the development of the subject.


This book tells a lively story and is crammed with interesting, and often technical, mathematics, which the author nevertheless makes accessible to a wide readership.

Christopher Hollings

University of Oxford


This book examines the theoretical growth of mathematics from a historical perspective. Kopp (Univ. of Hull) offers a fascinating and enlightening presentation in which basic notions are evolved into advanced mathematical concepts. As shown here, abstraction becomes a natural result of mathematical development.

D. P. Turner, Faulkner University

Choice Connect (0009-4978), vol. 58, no. 10, 2021.

Additional Resources


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(pp. 257–258)
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Ekkehard Kopp

Professor Emeritus at University of Hull