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The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the number seven would be represented by /////. Tally marks represent one such system still in common use.
- Overview
- Numeral systems
- Simple grouping systems
- Multiplicative grouping systems
- Ciphered numeral systems
- Positional numeral systems
It appears that the primitive numerals were |, ||, |||, and so on, as found in Egypt and the Grecian lands, or ―, =, ≡, and so on, as found in early records in East Asia, each going as far as the simple needs of people required. As life became more complicated, the need for group numbers became apparent, and it was only a small step from the simple...
It appears that the primitive numerals were |, ||, |||, and so on, as found in Egypt and the Grecian lands, or ―, =, ≡, and so on, as found in early records in East Asia, each going as far as the simple needs of people required. As life became more complicated, the need for group numbers became apparent, and it was only a small step from the simple...
In its pure form a simple grouping system is an assignment of special names to the small numbers, the base b, and its powers b2, b3, and so on, up to a power bk large enough to represent all numbers actually required in use. The intermediate numbers are then formed by addition, each symbol being repeated the required number of times, just as 23 is written XXIII in Roman numerals.
The earliest example of this kind of system is the scheme encountered in hieroglyphs, which the Egyptians used for writing on stone. (Two later Egyptian systems, the hieratic and demotic, which were used for writing on clay or papyrus, will be considered below; they are not simple grouping systems.) The number 258,458 written in hieroglyphics appears in the figure. Numbers of this size actually occur in extant records concerning royal estates and may have been commonplace in the logistics and engineering of the great pyramids.
In multiplicative systems, special names are given not only to 1, b, b2, and so on but also to the numbers 2, 3, …, b − 1; the symbols of this second set are then used in place of repetitions of the first set. Thus, if 1, 2, 3, …, 9 are designated in the usual way but 10, 100, and 1,000 are replaced by X, C, and M, respectively, then in a multiplic...
In ciphered systems, names are given not only to 1 and the powers of the base b but also to the multiples of these powers. Thus, starting from the artificial example given above for a multiplicative grouping system, one can obtain a ciphered system if unrelated names are given to the numbers 1, 2, …, 9; X, 2X, …, 9X; C, 2C, …, 9C; M, 2M, …, 9M. This requires memorizing many different symbols, but it results in a very compact notation.
The first ciphered system seems to have been the Egyptian hieratic (literally “priestly”) numerals, so called because the priests were presumably the ones who had the time and learning required to develop this shorthand outgrowth of the earlier hieroglyphic numerals. An Egyptian arithmetical work on papyrus, employing hieratic numerals, was found in Egypt about 1855; known after the name of its purchaser as the Rhind papyrus, it provides the chief source of information about this numeral system. There was a still later Egyptian system, the demotic, which was also a ciphered system.
Click Here to see full-size tableAs early as the 3rd century bce, a second system of numerals, paralleling the Attic numerals, came into use in Greece that was better adapted to the theory of numbers, though it was more difficult for the trading classes to comprehend. These Ionic, or alphabetical, numerals, were simply a cipher system in which nine Greek letters were assigned to the numbers 1–9, nine more to the numbers 10, …, 90, and nine more to 100, …, 900. Thousands were often indicated by placing a bar at the left of the corresponding numeral.
Such numeral forms were not particularly difficult for computing purposes once the operator was able automatically to recall the meaning of each. Only the capital letters were used in this ancient numeral system, the lowercase letters being a relatively modern invention.
The decimal number system is an example of a positional system, in which, after the base b has been adopted, the digits 1, 2, …, b − 1 are given special names, and all larger numbers are written as sequences of these digits. It is the only one of the systems that can be used for describing large numbers, since each of the other kinds gives special names to various numbers larger than b, and an infinite number of names would be required for all the numbers. The success of the positional system depends on the fact that, for an arbitrary base b, every number N can be written in a unique fashion in the form N = anbn + an − 1bn − 1 + ⋯ + a1b + a0 where an, an − 1, …, a0 are digits; i.e., numbers from the group 0, 1, …, b − 1. Then N to the base b can be represented by the sequence of symbols anan − 1…a1a0. It was this principle which was used in the multiplicative grouping systems, and the relation between the two kinds of systems is immediately seen from the earlier noted equivalence between 7,392 and 7M3C9X2; the positional system derives from the multiplicative simply by omitting the names of the powers b, b2, and so on and by depending on the position of the digits to supply this information. It is then necessary, however, to use some symbol for zero to indicate any missing powers of the base; otherwise 792 could mean, for example, either 7M9X2 (i.e., 7,092) or 7C9X2 (792).
The Babylonians developed (c. 3000–2000 bce) a positional system with base 60—a sexagesimal system. With such a large base, it would have been awkward to have unrelated names for the digits 0, 1, …, 59, so a simple grouping system to base 10 was used for these numbers, as shown in the figure.
In addition to being somewhat cumbersome because of the large base chosen, the Babylonian system suffered until very late from the lack of a zero symbol; the resulting ambiguities may well have bothered the Babylonians as much as later translators.
In the course of early Spanish expeditions into Yucatan, it was discovered that the Maya, at an early but still undated time, had a well-developed positional system, complete with zero. It seems to have been used primarily for the calendar rather than for commercial or other computation; this is reflected in the fact that, although the base is 20, the third digit from the end signifies multiples not of 202 but of 18 × 20, thus giving their year a simple number of days. The digits 0, 1, …, 19 are, as in the Babylonian, formed by a simple grouping system, in this case to base 5; the groups were written vertically.
Neither the Mayan nor the Babylonian system was ideally suited to arithmetical computations, because the digits—the numbers less than 20 or 60—were not represented by single symbols. The complete development of this idea must be attributed to the Hindus, who also were the first to use zero in the modern way. As was mentioned earlier, some symbol is required in positional number systems to mark the place of a power of the base not actually occurring. This was indicated by the Hindus by a dot or small circle, which was given the name sunya, the Sanskrit word for “vacant.” This was translated into the Arabic ṣifr about 800 ce with the meaning kept intact, and the latter was transliterated into Latin about 1200, the sound being retained but the meaning ignored. Subsequent changes have led to the modern cipher and zero.
A symbol for zero appeared in the Babylonian system about the 3rd century bce. However, it was not used consistently and apparently served to hold only interior places, never final places, so that it was impossible to distinguish between 77 and 7,700, except by the context.
All known numeral systems developed before the Babylonian numerals are non-positional, [67] as are many developed later, such as the Roman numerals. The French Cistercian monks created their own numeral system.
May 27, 2024 · A number system, or the numeral system, is a mathematical way of representing a set of values using digits or symbols. It uniquely represents a number and helps perform mathematical operations: addition, subtraction, multiplication, and division.
In the spirit of keeping things simple, it’s the simplest number system that has the concept of “ticking over”. Unary, where we just write 1, 11, 111… just goes on forever. Binary, with two options (1 and 0) looks like this:
The base 1 number system is called the unary numeral system and is the simplest numeral system to represent natural numbers.
Decimal Numeral System. Hex Numeral System. Numeral System Conversion Table. b - numeral system base. dn - the n-th digit. n - can start from negative number if the number has a fraction part. N+1 - the number of digits. Binary Numeral System - Base-2. Binary numbers uses only 0 and 1 digits. B denotes binary prefix. Examples:
