Zero

Summary: The concept of zero took time to be accepted and was explicitly rejected when first introduced to Greek and Roman culture.

Numbers initially served to count property, such as livestock. The numbers needed to count 1, 2, 3, 4,…became known as “counting” or “natural” numbers. The number zero is not found among these because one cannot count zero objects. Early civilizations existing over millennia used numbers only to count and so had no need for zero. The word “zero” has various linguistic origins: the French zéro and Venetian zero, which likely evolved from the Italian zefiro. This word came in turn from Arabic sifr, meaning “zero or nothing,” derived from word safira, meaning “it was empty.”

Early Development

The ancient Babylonians first introduced zero. With a base-60 system and initially two symbols (a wedge to represent “1” and a double wedge to represent 10), the Babylonians left empty spaces between groups of symbols. The fact that the spaces were not standardized in length made it difficult at times to distinguish between numbers because place value could not always be determined. To remedy this situation, the Babylonians developed zero but the zero was not a number in and of itself. It was rather a placeholder used to denote place values that had been skipped.

Independently and across the ocean, the Mayans developed a base-20 number system that included zero. Here, zero was used as a number to mean the absence of something. Zero also appeared in the Mayans’ calendar. There was a year zero, and each month had a day zero in it as well. Because of the vast distance between the Mayans and the old world, Mayans’ use and understanding of zero did not spread to these other areas.

Rejection by the Greeks and Romans

Despite the Babylonians use of zero, the Greeks and Romans initially rejected its use. Zero was considered dangerous spiritually as it represented the opposite of god and unity. It was associated with the void and chaos. Mathematically, zero presented many dilemmas. While any of the natural numbers (1, 2, 3, 4,…) when added to itself yields a larger number, zero added to itself does not. This characteristic violated Archimedes’s principal that repeatedly adding a number to itself tends to a sum that is infinitely large. Additionally, a natural number plus any other natural number yields a sum larger than the initial natural number but again zero added to a natural number does not yield a number larger than the original natural number. Finally, multiplication of any number by zero yields zero and division by zero was outside the acceptable norms for these civilizations. The Greeks, known for geometry, often associated geometric figures to the natural numbers but zero could be associated with no figure. They preferred to reject zero as a number altogether.

Zero in India

Indian mathematicians in the fifth century c.e. took ideas from the Babylonians, including the concept of zero. They treated zero as a number that was found in the number line between -1 and 1. They also introduced negative numbers and, in 700, Brahmagupta introduced the idea that 1/0=∞. Thus, infinity and unity depend upon the void and chaos. This idea was troubling to many civilizations, and the Hindu-Arabic numerals commonly used through the twenty-first century were not fully accepted until Leonardo de Pisa (also known as Fibonacci) introduced them to the Western world in his 1202 work Liber Abaci. One of the earliest recorded references to the mathematical impossibility of assigning a value to 1/0 occurred in George Berkeley’s 1734 work The Analyst, which criticizes the foundations of calculus.

Calendars

Zero also caused confusion with the calendar system. Dionysius’ calendar, created in 525 c.e., introduced the notation of BC and AD. However, it did not include a year zero. Thus, 1 BC is followed by 1 AD. This omission of zero causes confusion into the twenty-first century. Consider a person born in 1 AD. This person would have to go through 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 to have lived 10 years, and a new decade would begin at the end of this first decade (10 years). That is, it would begin in 11. Thus, the next decade would begin in 21. The first century would end in 100, and the new one would begin in 101. Thus, the twenty-first century technically began in 2001, not in 2000 when most everyone celebrated it. This confusion rears its head at the start of every decade and century all a result of the omission of a year zero.

Division by Zero

One way in which mathematicians interpret division by zero is to reframe division in terms of other arithmetic operations. Using standard rules for arithmetic, division by zero is undefined, since division is defined to be the inverse operation of multiplication. While division by zero cannot reasonably be resolved with real numbers and integers, it can be defined using other algebraic structures or analytical extensions.

Zero in the Physical Sciences

Zero is an important value for many physical quantities or measurements. In some cases, zero means “nothing” or an absence of the characteristic, such as in most units of length and mass. However, in some cases, zero represents an arbitrarily chosen starting point for counting or measuring, such as in the Fahrenheit and Celsius temperature scales (though on the Kelvin scale, zero, or absolute zero, is the coldest possible temperature that matter can reach).

Other more advanced examples can be found in chemistry and physics. Zero-point energy is the lowest possible energy that a quantum mechanical physical system may possess. This energy level is called the “ground state” of the system and is important for investigating concepts such as entropy and perfect crystal lattices. Professor Andreas von Antropoff introduced the term “neutronium” for theoretical matter made solely of neutrons. As early as 1926, he redefined the periodic table with the atomic number zero, rather than the standard hydrogen (Atomic Number 1) in the initial position. More recent investigations suggest that the hypothesized element tetraneutron, a stable cluster of four neutrons with no protons or electrons, could have this atomic number zero.

Zero and Computers

In 1997, the naval vessel USS Yorktown’s propulsion system was brought to a dead stop by a computer network failure resulting from an attempt to divide by zero. Mathematical operations like these are problematic for computers, leading to various methods to avoid errors. The floating-point standard used in most modern computer processors has two distinct zeroes: a +1 (positive zero) and a -0 (negative zero). They are considered equal in numerical comparisons but some mathematical operations will have different results depending on which zero is used. For example, 1/-0 yields negative infinity, while 1/+1 gives positive infinity, though a “divide by zero” warning is usually issued in either case. Integer division by zero is usually handled differently from floating point, as there is no integer representation for the answer. Some processors generate an exception for integer division by zero, although others will simply generate an incorrect result for the division.

Bibliography

Ifrah, Georges, and Lowell Bair. From One to Zero: A Universal History of Numbers. New York: Penguin, 1987.

Kaplan, Robert. The Nothing That Is: A Natural History of Zero. Oxford, England: Oxford University Press, 2000.

Seife, Charles. Zero: The Biography of a Dangerous Idea. New York: Penguin Books, 2000.