“Arithmetic” (Greek arithmos), the most elementary branch of mathematics, is the study of numbers and the traditional operations on them—addition, subtraction, multiplication, division, exponentiation and extraction of roots. Unfortunately for those of us with a short attention span, the language used to describe arithmetic operations is far more complicated than the operations themselves.
I’ve always been curious about this word because the pronunciation seems awkward. The noun “arithmetic” is pronounced a rith meh TIC, which seems awkward. But the adjective, as in “The problem is arithmetic,” is pronounced ar ith MET ic, which rolls off my tongue as smooth as silk. That said, number theory has a long history.
The prehistory is limited to a few artifacts, the best-known being the Ishango bone from central Africa, dating from somewhere between 20,000 and 18,000 BCE.
The earliest written records show the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BCE. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, descended from tally marks used for counting. In both cases, a decimal base was used, but not positional notation.
Early number systems that included positional notation were not decimal, such as the sexagesimal (base 60) system for Babylonian numerals, and the vigesimal (base 20) system for Maya numerals. Because of this place-value concept, the ability to reuse the same digits for different values contributed to simpler and more efficient methods of calculation.
Modern arithmetic starts with ancient Greece, much later than the Babylonian and Egyptian examples. Prior to the works of Euclid around 300 BCE, Greek studies in mathematics overlapped with philosophical and mystical beliefs. The ancient Greeks lacked a symbol for zero until the Hellenistic period.
The ancient Chinese had advanced arithmetic studies dating from the Shang Dynasty, from basic numbers to advanced algebra. They used a positional notation similar to that of the Greeks, and were the first to discover, understand, and apply negative numbers. This is explained in the Nine Chapters on the Mathematical Art, written in the 2nd century BCE.
The gradual development of the Hindu–Arabic numeral system independently devised the place-value concept and positional notation, which combined a decimal base and the use of zero. This allowed the system to consistently represent both large and small integers—an approach which eventually replaced all other systems.
Before the Renaissance, the tools used to assist in numeric calculations were various types of abaci. More recent tools include slide rules, nomograms and mechanical calculators, now supplanted by electronic calculators and computers.
During the last two centuries, various aids were developed to aid the manipulation of compound units, particularly in commercial applications. One example is the mechanical till, adapted in countries such as the UK to accommodate pounds, shillings, pennies and farthings, and Ready Reckoners—books that catalogued the results of various routine calculations.