Burger navigates number history

What do you do when a speaker begins a talk with the disclaimer that “most of the things you will hear are mostly true”? If the speaker is Ed Burger, professor of mathematics, you keep your ears pricked up for an entertaining and enlightening time.

Burger lectured animatedly in a series of talks entitled “Zero to Infinity” last Thursday and Friday. He spoke before a full house in Wege Auditorium for the fall series of the Sigma Xi lectures. Sigma Xi is an international science and engineering honor society, which sponsors lectures twice a year at the College.

Thursday’s lecture, titled “Great Moments in the History of Numbers,” gave the audience an overview of the evolution of numbers. “Isn’t it hard, giving the whole history of numbers in an hour? Not when you know as little about it as I do,” joked Burger, adding that he started a pointed study of the history of numbers earlier in the year.

Nevertheless, he dexterously navigated the historical landscape, beginning with pre-counting days when quantities were estimated by comparison. Burger introduced the idea of the “limit of four,” which describes an innate human ability to instantaneously tell how many objects are in a group, as long as the group contains four or fewer entities. He suggested that this limit is the source of the barred gate system of tallying, where every fifth mark is drawn across the preceding four to denote a bundle.

The early Sumerians, however, needed to deal with larger groups and consequently developed an accounting method using pebbles, which in Latin are called calculi. A Sumerian farmer who needed to count his flock could transfer a pebble into a basket for every sheep that walked past him in the morning, and reverse the process when the flock returned from grazing to ascertain whether any were lost.

According to Burger, two more sophisticated styles of reckoning developed from the calculi. First, tallying sticks: for business transactions, etchings that described the amounts involved in a transaction would be made on a stick that would then be split in half for each trading party to keep. When the British government finally abolished this method in 1834 in favor of strictly paper transactions, the fire that was started to destroy the accumulated heaps of tallying sticks inadvertently consumed the Houses of Parliament itself. “This is what happens when you try to destroy numbers,” Burger said. “Number theory shapes history.”

The second development was making imprints of the pebbles in soft clay – the first abstract symbols of quantity. Eventually, this evolved into making marks with a reed stylus, then the Babylonian cuneiform writing system with nail and dovetail shapes. “Writing came from number theory, so Shakespeare needs to thank the number theorists,” Burger said.

One crucial evolution in numerical systems, he noted, was the switch from additive number systems – like Roman numerals, where every number is represented by a corresponding quantity of symbols – to positional, or place-value, number systems. While positional numbering existed as early as Babylonian days, this change was controversial because the abacus relies on additive numbering, and because zero – a symbol which was viewed with suspicion – can so easily be exploited for fraud in the place-value system, as any modern check-writer can attest.

In addition to zero, other “counterintuitive” numbers that Burger spoke about were negative numbers and irrational numbers. The latter, the class of numbers that cannot be expressed as fractions (i.e. ratios) of integers, was the focus of his second lecture, “A Rational Approach to Irrationality.”

In this lecture Burger discussed and proved several theorems relating rational and irrational numbers, including Diophantine approximation – approximating real numbers with rational numbers and assessing the accuracy of the approximation – one of his specialties. “I want you to think of these proofs as works of art from a period you’re not familiar with – yet,” he said, to the less mathematically inclined in the audience.

Burger concluded both lectures by addressing the relationship between rational and irrational numbers, which he saw as a cosmic image of stars – rational numbers – dotting the backdrop of black space – irrational numbers. “By studying the synergy between these types of numbers, we move the frontiers forward,” he said.