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Large Numbers

Published online by Cambridge University Press:  03 November 2016

J.E. Littlewood
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Extract

The problem how to express very large numbers is discussed in ‘The Sand-Reckoner’ of Archimedes. Grains of sand being proverbially ‘innumerable’, Archimedes develops a scheme, the equivalent of a 10 n notation, in which the ‘Universe’, a sphere reaching to the Sun and calculated to have a diameter less than 1010 stadia, would contain, if filled with sand, fewer grains than ‘1000 units of the seventh order of numbers’, which is 1051. [A myriad-myriad is 108; this is taken as the base of what we should call exponents, and Archimedes contemplates 108 ‘periods’, each containing 108 ‘orders’ of numbers ; tlle final number in the scheme is 108. 1015.] The problem of expression is bound up with the invention of a suitable notation; Archimedes does not have our ab , with its potential extension to aa a . We return to this question at the end ; the subject is not exhausted.

Type
Research Article
Information
The Mathematical Gazette , Volume 32 , Issue 300 , July 1948 , pp. 163 - 171
Copyright
Copyright © The Mathematical Association 1948

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References

Page 165 of note * Note that it is pairs and not ordered pairs that are relevant.

Page 166 of note * I learn from Mr. H. A. Webb that in one of Blackburne’s games a position recurred for a second time, but with a pair of rooks interchanged. Each player expected to win, otherwise (as Blackburne said) a delicate point for decision would have arisen.

Page 167 of note * We may suppose that C (in the light of A’s performance!) does not suspect the position, and resigns in the ordinary way

Page 169 of note * See Math. Ann. 109 (1934), 661-667 ; Bull. Anzer Math. Soc. 1928, 84-56 ; Amer Math. Monthly, 43 (1036), 347-354.

Page 169 of note † It is acessible in a Ph.D. thesis deposited in the Cambridge University Library.

Page 169 of note ‡ See J. L. M. S., 8 (1933), 277-283.