Decimal Notation
Decimal notation refers to the base-ten numeral system that is widely used across modern civilizations, particularly the Hindu-Arabic numeral system, which employs the digits 0 through 9. This system is notable for its positional notation, where the position of each digit determines its value—each position being ten times greater than the one to its right. For example, the number 123 represents one hundred, two tens, and three ones. The concept of zero as a true number, introduced in India around the fifth century, is a fundamental aspect of positional systems, enabling decimal fractions to be expressed in a compact form, such as 0.25 instead of 1/4.
While Arabic numerals are predominant in Europe and North America, other cultures have developed their own base-ten systems. For instance, China utilizes a nonpositional base-ten numeral system. The use of trailing zeroes in decimal notation can indicate the precision of a number, especially in scientific and financial contexts. Decimal notation also allows for the expression of rational numbers as terminating or repeating decimals, while irrational numbers, like pi, cannot be accurately represented in this way. Overall, decimal notation serves as a universal language for numerical representation, deeply integrated into various aspects of mathematics, commerce, and science.
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Decimal Notation
The decimal numeral system is the base-ten system used by most modern civilizations, and it is often used to refer specifically to the base-ten positional notation system (using 0–9 as numerals) of the Hindu-Arabic numeral system, used in the West. Decimal systems have developed independently in different parts of the world, for a natural reason: Counting on fingers and thumbs leads to groups of ten. (The word "digit" means both "numeral" and "finger" or "toe," a shared meaning also found in other languages.) While the Hindu-Arabic numeral system has become familiar in most of Asia, China also uses an indigenous base-ten numeral system that is nonpositional. In any event, the mathematics of decimal numeral systems are all the same, positionality affecting only the way numbers are written.
In Europe and North America, the symbols used for numerals are called Arabic numerals, though the same symbols are called Indian numerals in the Middle East, having been introduced to the Arab world by Muslim, Persian, and Arab mathematicians who studied in India near the end of the first millennium. One of the key features of this numeral system is the true zero, which is necessary in positional number systems and is one of the key concepts in modern mathematics; students are sometimes surprised to discover that not all number systems have a zero, and that zero was not introduced as a true number until India around the fifth century. Before this, even decimal numeral systems had used a non-number placeholder.
Overview
Arabic numerals are now the most common numerals in use in the world. They were introduced to Europe both in the form of Arabic mathematics texts (and Arabic translations of Greek and Roman mathematics texts) and on their own. European mathematicians like Fibonacci recognizing their utility. European trade and imperialism later spread the adopted numeral system along with other cultural exports.
Positionality means that numbers are written as a string of values entered into sequential positions such that each position’s value is ten times that of the one to its right. The number 10 literally indicates "one ten and zero ones," while 123 indicates "one hundred, two tens, and three ones." While names for numbers at lower values cloak this (we say "thirteen," not "ten and three"), the positional system shows through the seams when we speak of higher numbers, whether referring to 1982 as "one thousand, nine hundred[s], and eighty two" or "nineteen hundred[s] and eighty-two."
Positionality allows for an economy of symbols, using position rather than separate symbol sets to indicate orders of magnitude. Nonpositional numeral systems such as that of Roman numerals are called additive or sign-value notation systems. The additive system remained popular in Europe even after the introduction of the decimal system, because it worked well with the abacus, which was the height of arithmetic-performing technology at the time. This is one reason the abacus today remains more common in Asian countries.
The difference positionality makes is that in the Hindu-Arabic numeral system, decimal fractions are supported. That is, while it is valid in many applications to write fractions as 1/4 or 1/8, the positional system allows them to be written instead as 0.25 or 0.125—converting them to fractions with a denominator that is a power of ten; this is the standard form in most everyday cases (such as when listing prices or weights). It is less often used when referring to time, such that no one refers to themselves as 7.5 years old or to a movie as 2.25 hours long.
The decimal mark (which in both handwriting and typography is the same as the period or full stop) is used as a decimal separator to separate the ones position from the tenths position. It is generally left out when there are no values in the positions to its right, just as we do not write out 123 as 000123. But in some contexts it is conventional to write out the decimal separator and first two or three zeroes. This is true in financial contexts, for instance, where $1.00 is preferred to $1, and in some scientific or statistical contexts in which results are rounded to the nearest nth, in which writing out 1.00 or 1.000 may be preferred (to imply a result that has been rounded off from 1.0000x). In engineering and statistics, the number of these zeroes (called trailing zeroes) is generally taken as an indicator of the level of confidence in the accuracy of the number. In this case the difference between .08 and .08000 becomes quite significant, even though they both refer to the same numeric value.
The two parts of a decimal number are called the integer or integral part (left of the decimal separator) and the fractional part. For clarification, decimal numbers without integer parts (proper fractions) are conventionally written with a leading zero in the ones position (i.e., 0.25 or 0.125, even though .25 and .125 are also technically correct).
A rational number with a denominator whose prime factors include numbers other than 2 and 5 cannot be expressed as a terminating decimal fraction, and instead is an infinite repeating decimal fraction: 1/3, for instance, becomes 0.3333333… (repeating), while 1/81 becomes 0.012345679012… (where 012345679 repeats infinitely). In contrast, 1/2 is simply 0.5, and 1/5 is simply 0.2. Irrational numbers, which are real numbers that cannot be expressed as a ratio of integers, similarly cannot be expressed as a terminating or repeating decimal fraction. Pi is the most famous example.
Approximated as 3.14159, the digits of pi’s fractional part go on infinitely without ever settling into a repeating pattern. By the 1970s, the first one million digits of pi had been discovered, and in the twenty-first century, it had been computed to over 10 trillion digits. Contests to memorize digits of pi have resulted in a Guinness World Record of 67,890 digits, set by Chinese graduate student Lu Chao, who took 24 hours and 4 minutes to recite the digits with five-minute breaks for snacks.
Bibliography
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Krappraff, Jay. Beyond Measure: A Guided Tour Through Nature, Myth, and Number. River Edge: World Scientific, 2012.
Nahin, Paul J. An Imaginary Tale: The Story of [the Square Root of Minus One]. Princeton: Princeton UP, 2010.
Shapiro, Stewart. Thinking About Mathematics: The Philosophy of Mathematics. New York: Oxford UP, 2013.