Decimal in the context of Place-value notation

Babylonian astronomy was the study or recording of celestial objects during the early history of Mesopotamia. The numeral system used, sexagesimal, was based on 60, as opposed to ten in the modern decimal system. This system simplified the calculating and recording of unusually great and small numbers.

During the 8th and 7th centuries BC, Babylonian astronomers developed a new empirical approach to astronomy. They began studying and recording their belief system and philosophies dealing with an ideal nature of the universe and began employing an internal logic within their predictive planetary systems. This was an important contribution to astronomy and the philosophy of science, and some modern scholars have thus referred to this approach as a scientific revolution. This approach to astronomy was adopted and further developed in Greek and Hellenistic astrology. Classical Greek and Latin sources frequently use the term Chaldeans for the philosophers, who were considered as priest-scribes specializing in astronomical and other forms of divination. Babylonian astronomy paved the way for modern astrology and is responsible for its spread across the Graeco-Roman empire during the 2nd-century Hellenistic Period. The Babylonians used the sexagesimal system to trace the planets' transits, by dividing the 360 degree sky into 30 degrees, they assigned 12 zodiacal signs to the stars along the ecliptic.

Large numbers are numbers far larger than those encountered in everyday life, such as simple counting or financial transactions. These quantities appear prominently in mathematics, cosmology, cryptography, and statistical mechanics. Googology studies the naming conventions and properties of these immense numbers.

Since the customary decimal format of large numbers can be lengthy, other systems have been devised that allows for shorter representation. For example, a billion is represented as 13 characters (1,000,000,000) in decimal format, but is only 3 characters (10) when expressed in exponential format. A trillion is 17 characters in decimal, but only 4 (10) in exponential. Values that vary dramatically can be represented and compared graphically via logarithmic scale.

In mathematics, a rational number is a number that can be expressed as the quotient or fraction ⁠${\displaystyle {\tfrac {p}{q}}}$⁠ of two integers, a numerator p and a non-zero denominator q. For example, ⁠${\displaystyle {\tfrac {3}{7}}}$⁠ is a rational number, as is every integer (for example, ${\displaystyle -5={\tfrac {-5}{1}}}$). The set of all rational numbers is often referred to as "the rationals", and is closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations and therefore also calledthe field of rationals or the field of rational numbers. It is usually denoted by boldface Q, or blackboard bold ⁠${\displaystyle \mathbb {Q} .}$⁠

A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: 3/4 = 0.75), or eventually begins to repeat the same finite sequence of digits over and over (example: 9/44 = 0.20454545...). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see Repeating decimal § Extension to other bases).

The Coinage Act of 1792 (also known as the Mint Act; officially: An act establishing a mint, and regulating the Coins of the United States), passed by the United States Congress on April 2, 1792, created the United States dollar as the country's standard unit of money, established the United States Mint, and regulated the coinage of the United States. This act established the silver dollar as the unit of money in the United States, declared it to be lawful tender, and created a decimal system for U.S. currency.

By the Act, the Mint was to be situated at the seat of government of the United States. The five original officers of the U.S. Mint were a Director, an Assayer, a Chief Coiner, an Engraver, and a Treasurer (not the same as the secretary of the treasury). The Act allowed that one person could perform the functions of Chief Coiner and Engraver. The Assayer, Chief Coiner and Treasurer were required to post a $10,000 bond with the Secretary of the Treasury.

A numeral system is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.

The same sequence of symbols may represent different numbers in different numeral systems. For example, "11" represents the number eleven in the decimal or base-10 numeral system (today, the most common system globally), the number three in the binary or base-2 numeral system (used in modern computers), and the number two in the unary numeral system (used in tallying scores).

The Hindu–Arabic numeral system (also known as the Indo-Arabic numeral system, Hindu numeral system, and Arabic numeral system) is a positional base-ten numeral system for representing integers; its extension to non-integers is the decimal numeral system, which is presently the most common numeral system.

The system was invented between the 1st and 4th centuries by Indian mathematicians. By the 9th century, the system was adopted by Arabic mathematicians who extended it to include fractions. It became more widely known through the writings in Arabic of the Persian mathematician Al-Khwārizmī (On the Calculation with Hindu Numerals, c. 825) and Arab mathematician Al-Kindi (On the Use of the Hindu Numerals, c. 830). The system had spread to medieval Europe by the High Middle Ages, notably following Fibonacci's 13th century Liber Abaci; until the evolution of the printing press in the 15th century, use of the system in Europe was mainly confined to Northern Italy.

Digitization is the process of converting information into a digital (i.e. computer-readable) format. The result is the representation of an object, image, sound, document, or signal (usually an analog signal) obtained by generating a series of numbers that describe a discrete set of points or samples. The result is called digital representation or, more specifically, a digital image, for the object, and digital form, for the signal. In modern practice, the digitized data is in the form of binary numbers, which facilitates processing by digital computers and other operations, but digitizing simply means "the conversion of analog source material into a numerical format"; the decimal or any other number system can be used instead.

Digitization is of crucial importance to data processing, storage, and transmission, because it "allows information of all kinds in all formats to be carried with the same efficiency and also intermingled." Though analog data is typically more stable, digital data has the potential to be more easily shared and accessed and, in theory, can be propagated indefinitely without generation loss, provided it is migrated to new, stable formats as needed. This potential has led to institutional digitization projects designed to improve access and the rapid growth of the digital preservation field.

Positional notation, also known as place-value notation, positional numeral system, or simply place value, usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early numeral systems, such as Roman numerals, a digit has only one value: I means one, X means ten and C a hundred (however, the values may be modified when combined). In modern positional systems, such as the decimal system, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string.

The Babylonian numeral system, base 60, was the first positional system to be developed, and its influence is present today in the way time and angles are counted in tallies related to 60, such as 60 minutes in an hour and 360 degrees in a circle. The Inca used knots tied in a decimal positional system to store numbers and other values in quipu cords.

In a positional numeral system, the radix (pl. radices) or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal system (the most common system in use today) the radix is ten, because it uses the ten digits from 0 through 9.

In any standard positional numeral system, a number is conventionally written as (x)_y with x as the string of digits and y as its base. For base ten, the subscript is usually assumed and omitted (together with the enclosing parentheses), as it is the most common way to express value. For example, (100)₁₀ is equivalent to 100 (the decimal system is implied in the latter) and represents the number one hundred, while (100)₂ (in the binary system with base 2) represents the number four.

Hexadecimal (hex for short) is a positional numeral system for representing a numeric value as base 16. For the most common convention, a digit is represented as "0" to "9" like for decimal and as a letter of the alphabet from "A" to "F" (either upper or lower case) for the digits with decimal value 10 to 15.

As typical computer hardware is binary in nature and that hex is power of 2, the hex representation is often used in computing as a dense representation of binary information. A hex digit represents 4 contiguous bits –known as a nibble. An 8-bit byte is two hex digits, such as 2C.

0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and complex numbers, as well as other algebraic structures. Multiplying any number by 0 results in 0, and consequently dividing by 0 is generally considered to be undefined in arithmetic.

As a numerical digit, 0 plays a crucial role in decimal notation: it indicates that the power of ten corresponding to the place containing a 0 does not contribute to the total. For example, "205" in decimal means two hundreds, no tens, and five ones. The same principle applies in place-value notations that uses a base other than ten, such as binary and hexadecimal. The modern use of 0 in this manner derives from Indian mathematics that was transmitted to Europe via medieval Islamic mathematicians and popularized by Fibonacci. It was independently used by the Maya.

Decimalisation or decimalization (see spelling differences) is the conversion of a system of currency or of weights and measures to units related by powers of 10.

Most countries have decimalised their currencies, converting them from non-decimal sub-units to a decimal system, with one basic currency unit and sub-units that are valued relative to the basic unit by a power of 10, most commonly 100 and exceptionally 1,000, and sometimes at the same time, changing the name of the currency and/or the conversion rate to the new currency.

Decimal time is the representation of the time of day using units which are decimally related. This term is often used specifically to refer to the French Republican calendar time system used from 1794 to 1800, during the French Revolution, which divided the day into 10 decimal hours, each decimal hour into 100 decimal minutes and each decimal minute into 100 decimal seconds (100,000 decimal seconds per day), as opposed to the more familiar standard time, which divides the day into 24 hours, each hour into 60 minutes and each minute into 60 seconds (86,400 SI seconds per day).

The main advantage of a decimal time system is that, since the base used to divide the time is the same as the one used to represent it, the representation of hours, minutes and seconds can be handled as a unified value. Therefore, it becomes simpler to interpret a timestamp and to perform conversions. For instance, 12345 is 1 decimal hour, 23 decimal minutes, and 45 decimal seconds, or 1.2345 decimal hours, or 123.45 decimal minutes or 12345 decimal seconds; 3 hours is 300 minutes or 30,000 seconds.This property also makes it straightforward to represent a timestamp as a fractional day, so that 2025-12-02.54321 can be interpreted as five decimal hours, 43 decimal minutes and 21 decimal seconds after the start of that day, or a fraction of 0.54321 (54.321%) through that day (which is shortly after traditional 13:00). It also adjusts well to digital time representation using epochs, in that the internal time representation can be used directly both for computation and for user-facing display.

The metric system is a system of measurement that standardises a set of base units and a nomenclature for describing relatively large and small quantities via decimal-based multiplicative unit prefixes. Though the rules governing the metric system have changed over time, the modern definition, the International System of Units (SI), defines the metric prefixes and seven base units: metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), and candela (cd).

An SI derived unit is a named combination of base units such as hertz (cycles per second), newton (kg⋅m/s), and tesla (1 kg⋅s⋅A) and in the case of Celsius a shifted scale from Kelvin. Certain units have been officially accepted for use with the SI. Some of these are decimalised, like the litre and electronvolt, and are considered "metric". Others, like the astronomical unit are not. Ancient non-metric but SI-accepted multiples of time, minute and hour, are base 60 (sexagesimal). Similarly, the angular measure degree and submultiples, arcminute, and arcsecond, are also sexagesimal and SI-accepted.

Babylonian mathematics (also known as Assyro-Babylonian mathematics) is the mathematics developed or practiced by the people of Mesopotamia, as attested by sources mainly surviving from the Old Babylonian period (1830–1531 BC) to the Seleucid from the last three or four centuries BC. With respect to content, there is scarcely any difference between the two groups of texts. Babylonian mathematics remained constant, in character and content, for over a millennium.

In contrast to the scarcity of sources in Ancient Egyptian mathematics, knowledge of Babylonian mathematics is derived from hundreds of clay tablets unearthed since the 1850s. Written in cuneiform, tablets were inscribed while the clay was moist, and baked hard in an oven or by the heat of the sun. The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics that include fractions, algebra, quadratic and cubic equations and the Pythagorean theorem. The Babylonian tablet YBC 7289 gives an approximation of ${\displaystyle {\sqrt {2}}}$ accurate to three significant sexagesimal digits (about six significant decimal digits).

Chinese units of measurement, known in Chinese as the shìzhì ("market system"), are the traditional units of measurement of the Han Chinese. Although Chinese numerals have been decimal (base-10) since the Shang, several Chinese measures use hexadecimal (base-16). Local applications have varied, but the Chinese dynasties usually proclaimed standard measurements and recorded their predecessor's systems in their histories.

In the present day, the People's Republic of China maintains some customary units based upon the market units but standardized to round values in the metric system, for example the common jin or catty of exactly 500 g. The Chinese name for most metric units is based on that of the closest traditional unit; when confusion might arise, the word "market" (市, shì) is used to specify the traditional unit and "common" or "public" (公, gōng) is used for the metric value. Taiwan, like Korea, saw its traditional units standardized to Japanese values and their conversion to a metric basis, such as the Taiwanese ping of about 3.306 m based on the square ken. The Hong Kong SAR continues to use its traditional units, now legally defined based on a local equation with metric units. For instance, the Hong Kong catty is precisely 604.78982 g.

The Arabic numerals are ten symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) used for writing numbers. The term often also implies a positional notation number with a decimal base, in particular when contrasted with Roman numerals. However, the symbols are also used to write numbers in other bases, as well as non-numerical information such as trademarks or license plate identifiers.

They are also called Western Arabic numerals, Western digits, European digits, ASCII digits, Latin digits or Ghubār numerals to differentiate them from other types of digits. Hindu–Arabic numerals is used due to positional notation (but not these digits) originating in India. The Oxford English Dictionary uses lowercase Arabic numerals while using the fully capitalized term Arabic Numerals for Eastern Arabic numerals. In contemporary society, the terms digits, numbers, and numerals often implies only these symbols, although it can only be inferred from context.