International Study Group on the Relations Between
HISTORY and PEDAGOGY of MATHEMATICS NEWSLETTER
No. 46, March 2001
An Affiliate of the International Commission on Mathematical Instruction
The Dangerous Hole of Zero
History makes a man wise is a common saying. By studying history we can know
the errors and mistakes committed in the past and save ourselves from
repeating them. According to P. S. Jones "One use of the history of
mathematics is to reveal to students come of the conceptual difficulties and
errors which have impeded progress". G. A. Miller even says "The teachers of
mathematics may frequently gain more from a clear exposition of failures
than from such an exposition of successes on the part of the eminent
mathematicians of the past".
In this brief note we mention the mistakes, gathered from a few earlier
works, in connection with some arithmetical operations involving the number
zero (now denoted by the hole "0").
1. The great Indian mathematician Brahmagupta (7th century AD) was the first
to give explicitly in his Brahmasphuta-Siddhanta (chapter XVIII), the
various rules involving zero (in arithmetical operations) but they also
include his statement that "zero divided by zero is zero". That is, 0 ÷ 0 =
0 which is not true in general.
2. The Ganitasara-sangraha (I, 49) of the Jaina mathematician Mahavira (9th
century) contains a ÷ 0 = a
3. On the other hand Sripati (11th century) in his Ganita-tilaka (rule 45)
gives a ÷ 0 = 0
4. Bhaskara II in his famous Lilavati (12th century) gives the wrong rule (a
x 0) = a. His commentator Ganesa (1545) remarks that the rule comes by cancelling zero
from the numerator and denominator!
5. Leonhard Euler (1707-1783) wrongly got
( … + 1/x3 + 1/x2 + 1/x + 1) + (x + x2 + x3 + …) = 0
by summing the two
G.P.s which cannot be simultaneously
convergent. | | 
Euler’s tomb, St. Petersburg, Russia |
6. It is said that William Emerson (c.1780) and
George Baron (1804) asserted
that 00 = 1. | The Emerson Arms in Emerson’s town
of Hurworth, Great Britain |
7. Martin Ohm (1828) gave the verbal equivalent of (a ÷ 0) x 0 = 0, if a != 0
8. A few years later is found the statement (by A. L. G. Demonville?) that
"nothing multiplied by nothing is one" i.e. 0 x 0 = 1
9. S. G. Abel (father of the famous N. H. Abel, 1802-1829) wrote a textbook
which contains 1 + 0 = 0
10. Surprisingly, even the great Shinivasa Ramanujan (1887-1920) got the
wrong result 12 + 22 + 32 + … = 0
The above examples show that there was a lack of understanding of the real
nature and behaviour of zero.
References
- R. C. Gupta: ‘Mathematical Lapses’ Ganita-Bharati 18, (1996) notes 1, 2, 4,
5
- R. C. Gupta: Ganita-Bharati 4, 121 note 3
- E. R. Hogan: Historia Mathematica 3, 405, 414 (1976) note 6
- Datta & Singh: History of Hindu Mathematics, vol 1. p246 note 7
- A. De Morgan: The Encyclopaedia of Eccentrics. La Salle, 1974, pp292-293
note 8
- A. Stubhaug: N. H. Abel and his Times. Springer, Berlin, 2000, pp89 and 135
note9
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