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“Fibonacci Expansions and ‘F-adic’ integers,” The Fibonacci Quarterly, v. 34, 1996

“On the Sums of Digits of Fibonacci Numbers,” The Fibonacci Quarterly, v. 35, 1996..

A Modification of Shanks’ Baby-Step Giant-Step Algorithm,” Mathematics of Computation, v. 69, 2000

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Various forms of numerals and counting systems have developed throughout history, although Hindu-Arabic numerals are almost universally used today. Below I describe some of the most commonly-used ones.

Egyptian Numerals
The ancient Egyptians used a decimal positional system written in heiroglyphs. They had different symbols for the numbers 1, 10, 100, 1000, 10000, 100000, and 1000000. They also had symbols for fractions.

Babylonian Numerals
The Babylonian numeral system, used from roughly 1900 to 1600 BC, consisted of cuneiform symbols etched in clay. The Babylonians used a sexagesimal (base-60) counting system. Our current system of measuring time in hours, minutes, and seconds is a relic of this system. The system did not include a symbol for zero, but instead used a blank space, which often lead to confusion. Nevertheless, it was a very powerful system, which allowed the ancient Babylonians to perform some awesome mathematical calculations such as Pythagorean triples (well before Pythagoras) as well as an amazingly accurate computation of the square root of 2.

Roman Numerals
Like the Egyptians, the ancient Romans used a decimal positional system, but they also used symbols for 5, 50, 500, etc. Roman numerals are still used for certain purposes and most children learn them.

Maya Numerals
The Maya culture, which fluorished in present-day southern Mexico and Central America from around 250-900, used a base-20 system with bars representing 5, dots representing 1, and a special symbol for zero. The system wasn't strictly base-20 however, since the second set of bars and dots did not exceed 17.

Alphabetic Numerals
Many cultures used letters to represent numbers, including the ancient Greeks and Hebrews. The 24 Greek letters plus three additional symbols are used to represent numbers from 1 to 900, which can be added together to obtain all numbers from 1 to 999. Similarly, 27 Hebrew letters (22 letters plus five Sophid forms) are used in the Hebrew system.

Hindu-Arabic Numerals
Our modern numeral system comes from India from approximately 400 BC to 400 AD. This system was brought to the Middle East in the 9th century and finally to Europe in the 10th century. It has two key advantages over other numeral systems. Perhaps most importantly, like the Maya system, it has a symbol for zero, but unlike the Maya system, it uses a constant base, namely base 10. This makes it easy to perform arithemetic with this system, multiplication and division in particular.