Numeral system in the context of Place-value notation

An ideogram or ideograph (from Greek idéa 'idea' + gráphō 'to write') is a symbol that is used within a given writing system to represent an idea or concept in a given language. (Ideograms are contrasted with phonograms, which indicate sounds of speech and thus are independent of any particular language.) Some ideograms are more arbitrary than others: some are only meaningful assuming preexisting familiarity with some convention; others more directly resemble their signifieds. Ideograms that represent physical objects by visually illustrating them are called pictograms.

• Numerals and mathematical symbols are ideograms, for example ⟨1⟩ 'one', ⟨2⟩ 'two', ⟨+⟩ 'plus', and ⟨=⟩ 'equals'.
• The ampersand ⟨&⟩ is used in many languages to represent the word and, originally a stylized ligature of the Latin word et.
• Other typographical examples include ⟨§⟩ 'section', ⟨€⟩ 'euro', ⟨£⟩ 'pound sterling', and ⟨©⟩ 'copyright'.

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.

A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers: 1, 2, 3, 4, 5, and so forth. Individual numbers can be represented in language with number words or by dedicated symbols called numerals; for example, "five" is a number word and "5" is the corresponding numeral. As only a limited list of symbols can be memorized, a numeral system is used to represent any number in an organized way. The most common representation is the Hindu–Arabic numeral system, which can display any non-negative integer using a combination of ten symbols, called numerical digits. Numerals can be used for counting (as with cardinal number of a collection or set), labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, a numeral is not clearly distinguished from the number that it represents.

In mathematics, the notion of number has been extended over the centuries to include zero (0), negative numbers, rational numbers such as one half ${\displaystyle \left({\tfrac {1}{2}}\right)}$, real numbers such as the square root of 2 ${\displaystyle \left({\sqrt {2}}\right)}$, and π, and complex numbers which extend the real numbers with a square root of −1, and its combinations with real numbers by adding or subtracting its multiples. Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, a term which may also refer to number theory, the study of the properties of numbers.

A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically 0 (zero) and 1 (one). A binary number may also refer to a rational number that has a finite representation in the binary numeral system, that is, the quotient of an integer by a power of two.

The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation.

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.

Dance notation is the symbolic representation of human dance movement and form, using methods such as graphic symbols and figures, path mapping, numerical systems, and letter and word notations. Several dance notation systems have been invented, many of which are designed to document specific types of dance while others have been developed with capturing the broader spectrum of human movement potential. A dance score is a recorded dance notation that describes a particular dance.

Sexagesimal, also known as base 60, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form—for measuring time, angles, and geographic coordinates.

The number 60, a superior highly composite number, has twelve divisors, namely 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, of which 2, 3, and 5 are prime numbers. With so many factors, many fractions involving sexagesimal numbers are simplified. For example, one hour can be divided evenly into sections of 30 minutes, 20 minutes, 15 minutes, 12 minutes, 10 minutes, 6 minutes, 5 minutes, 4 minutes, 3 minutes, 2 minutes, and 1 minute. 60 is the smallest number that is divisible by every number from 1 to 6; that is, it is the lowest common multiple of 1, 2, 3, 4, 5, and 6.

The decimal numeral system (also called the base-ten positional numeral system and denary /diːnəri/ or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (decimal fractions) of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as decimal notation.

A decimal numeral (also often just decimal or, less correctly, decimal number), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or as in 25.9703 or 3,1415).Decimal may also refer specifically to the digits after the decimal separator, such as in "3.14 is the approximation of π to two decimals".

The Mayan numeral system was the system to represent numbers and calendar dates in the Maya civilization. It was a vigesimal (base-20) positional numeral system. The numerals are made up of three symbols: zero (a shell), one (a dot) and five (a bar). For example, thirteen is written as three dots in a horizontal row above two horizontal bars; sometimes it is also written as three vertical dots to the left of two vertical bars. With these three symbols, each of the twenty vigesimal digits could be written.

Numbers after 19 were written vertically in powers of twenty. The Mayan used powers of twenty, just as the Hindu–Arabic numeral system uses powers of ten.

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.

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.