Alex's Adventures in Numberland - Alex Bellos [179]
The harmonic series, therefore, is bigger than + + + + + …, which is infinity times a half, which is infinity. So the harmonic series is bigger than infinity; in other words, it is infinite.
Appendix Six
The continued fraction is a strange type of fraction constructed by an infinite process of additions and divisions.
When phi is expressed as a continued fraction it looks like this:
To understand how this works, let’s take the fraction line by line and see that it closes in on phi:
And so on.
Continued fractions provide mathematicians with a way of rating how irrational a number might be. Since the expression for phi contains only 1s, it is the ‘purest’ continued fraction that there is, and hence is considered the ‘most irrational’ number.
Notes on Chapters
During the writing of this book there were four tomes that were always on my desk, and whose contribution cannot be isolated to any individual chapter. Martin Gardner remains peerless in p
opular maths for his erudition, wit and clarity. Tobias Dantzig’s Number is a classic about the cultural evolution of mathematics. Both the Ifrah and the Cajori are painstakingly well researched and endlessly fascinating.
Cajori, F., A History of Mathematical Notations, Dover, 1993 (facsimile of original by Open Court, Illinois, 1928/9)
Dantzig, T., Number, Plume, New York, 2007 (originally Macmillan, 1930)
Gardner, M., Mathematical Games: The Entire Collection of His Scientific American Columns, Mathematical Association of America, 2005
Ifrah, G., The Universal History of Numbers, John Wiley, New York, 2000
CHAPTER ZERO
This chapter is based on conversations in London with Brian Butterworth, and in Paris with Stanislas Dehaene and Pierre Pica. At University College London I was screened for dyscalculia by Teresa Iuculano and Marinella Cappelletti, with a computer program now used in schools in the UK. I’m not dyscalculic, which is probably no great surprise. If you would like to help support the Munduruku’s protection of their traditional education and environment, donations can be sent to: The Munduruku Fund, The Arrow Rainforest Foundation, 5 Southridge Place, London SW20 8JQ, United Kingdom. More details can be found on: www.thearrowrainforestfoundation.com
Butterworth, B., The Mathematical Brain, Macmillan, London, 1999
Dehaene, S., The Number Sense, Oxford University Press, Oxford, 1997
Matzusawa, T. (ed.), Primate Origins of Human Cognition and Behavior, Springer, Tokyo, 2001
Angier, N., ‘Gut Instinct’s Surprising Role in Math’, New York Times, 2008
Dehaene, S., Izard, V., Spelke, E., and Pica, P., ‘Log or Linear?’, Science, 2008
Inoue, S., and Matsuzawa, T., ‘Working memory of numerals in chimpanzees’, Current Biology, 2007
Pica, P., Lerner, C., Izard, V., and Dehaene, S., ‘Exact and Appropriate Arithmetic in an Amazonian Indigene Group’, Science, 2004
Siegler, R.S., and Booth, J.L., ‘Development of Numerical Estimation in Young Children’, Child Development, 2004
CHAPTER ONE
Anyone wanting more information about the joys of base 12 can reach the Dozenal Society of America at contact@Dozenal.org, or 5106 Hampton Avenue Suite 205, Saint Louis, Missouri 63109-3115, USA. Little Twelvetoes is a classic of Schoolhouse Rock!, a series of musical cartoons about maths, science and grammar from the 1970s that can all be seen on the internet. My entry into the abacus world was only made possible by Kouzi Suzuki, a one-man soroban evangelist, who met me at a Tokyo rail station dressed up as Sherlock Holmes.
Andrews, F.E., New Numbers, Faber & Faber, London, 1936
Duodecimal Society of America, Inc., Manual of the Dozen System, Duodecimal Society of America, New York, 1960
Elbrow, Rear-Admiral G., The New English System of Money, Weights and Measures and of Arithmetic, P.S. King & Son, London, 1913
Essig, J.,