In two articles [BaiBor2011a; BaiBor2011b], two earlier blog posts [BlogA; BlogB] and a *Conversation* piece, we have examined the discovery and development of our modern system of decimal arithmetic with zero, which discovery we believe to be among the greatest of all historical mathematical achievements. It is certainly nontrivial, as evidenced by the fact that it escaped even Archimedes, that extraordinary genius of the ~300 BCE Greek culture who anticipated much of modern mathematics, including numerical analysis and calculus. And the impact of this ingenious discovery in our modern computer-oriented society cannot be overstated.

One key aspect of this history is the discovery of zero, that mysterious but important entity that we now take for granted. Zero actually has two functions: one as a symbol meaning null value, and also as a placeholder, as for instance when we append zero to 1234 to produce 12340, thus producing a number ten times the original.

In an informative and entertainingly written article *New Scientist* article [Webb2011], author Richard Webb explores the intriguing fact that whereas the general notion of positional notation, at least base 60 if not decimal, was known as early as 3000 BCE in ancient Mesopotamia, nearly 2000 years elapsed before their scientists and engineers started to use an actual symbol for what we now know as zero. Greek mathematicians, for instance, refused to use or acknowledge the notion of zero on philosophical grounds, since zero does not correspond to any physical object.

In addition to the history mentioned in Webb’s article, we cited research in our article [BaiBor2011b] indicating that zero (in both senses) may have been first found in ancient India at about the same time — roughly 300 BCE.

Of course, with the pervasive exchange of information between ancient cultures, we can never be sure who was the very first to make this crucial discovery. But we do know that after this time (300 BCE) both Mesopotamian and especially Indian mathematicians began to perform prodigious calculations. Some Indian mathematician, for instance, calculated the square root of 10 to 12-digit accuracy.

Perhaps some day we will finally learn the identify of the still-unknown mathematicians of the ancient world who first began to use and understand the full scope of our modern system of positional decimal arithmetic with zero. If we do, we surely must grant to him or her the same degree of recognition and reverence that we now grant to geniuses such as Archimedes, Newton and Einstein. Until then, … now where did I put my calculator…?

For additional details, see [BaiBor2011b], and the other articles and posts in the references below. Also of interest is a *New Scientist* article by Ian Stewart, the well-known British mathematician, on how discrete mathematics is founded on zero [Stewart2011].

### References

- [BaiBor2011a] David H. Bailey and Jonathan M. Borwein, “The Greatest Mathematical Discovery?”, manuscript, May 2011, available at Online article.
- [BaiBor2011b] David H. Bailey and Jonathan M. Borwein, “Ancient Indian Square Roots: An Exercise in Forensic Paleo-Mathematics”,
*American Mathematical Monthly*, to appear, Nov 2011, available at Online article. - [BlogA] “The Greatest Mathematical Discovery”, blog post, Blog post.
- [BlogB] “Ancient Indian square roots”, blog post, Blog post.
- [BorBai2011] Jonathan Borwein and David H. Bailey, “Magic numbers: The beauty of decimal notation,”
*The Conversation*, 23 Aug 2011, available at Online article. - [Stewart2011] Ian Stewart, “Nothingness: Mathematics starts with an empty set,”
*New Scientist*, 21 Nov 2011, available at Online article. - [Webb2011] Richard Webb, “Nothingness: Zero, the number they tried to ban,”
*New Scientist*, 21 Nov, 2011, available at Online article.