Positional notation
The Babylonian numeral system, base 60, was the first positional system to be developed. This ancient method influenced how time and angles are counted today. We still use 60 minutes in an hour and 360 degrees in a circle because of this early innovation. The Babylonians lacked a real zero initially. By about 700 BC, they indicated zero with a space or punctuation symbol between numerals. It served as a placeholder rather than a true zero. Numbers like 2 and 120 looked identical without context. Only context could differentiate them. Archimedes invented a decimal positional system based on 10^8 in his Sand Reckoner around 287, 212 BC. He used powers of ten to count grains of sand. Carl Gauss later lamented that science might have progressed faster if Archimedes had made the leap to modern decimal systems. Before positional notation became standard, simple additive systems were common. Roman numerals and Chinese numerals relied on sign-value notation. Accountants used abacuses or stone counters for arithmetic until positional notation arrived. The oldest extant positional notation system is either Chinese rod numerals from at least the early 8th century or Khmer numerals showing possible usages in the 7th century. Medieval Indian numerals became positional by the 10th century. After the French Revolution, the new government promoted extending the decimal system. Some pro-decimal efforts like decimal time failed. Others such as currency decimalisation spread widely out of France.
In mathematical numeral systems the radix is usually the number of unique digits including zero. For example, the decimal system uses ten digits from 0 through 9. When a number hits 9, the next number becomes 1 followed by 0. In binary, the radix is two since after hitting 1 it jumps straight to 10. The highest symbol of a positional numeral system has value one less than the base. A radix must be an integer greater than 1. Negative bases are rarely used. Positional systems work using exponentiation of the base. A digit's value equals the digit multiplied by its place value. Place values equal the base raised to the nth power where n counts other digits between a given digit and the radix point. If a digit sits left of the radix point then n is positive or zero. If it sits right of the radix point then n is negative. The notation can extend into negative exponents of the base b. Thereby the so-called radix point mostly . separates positions with non-negative from those with negative exponent. Numbers that are not integers use places beyond this point. For every position behind this point the exponent n decreases by 1. The power approaches 0 as you move further right. The number 2.35 equals 2 times 10^0 plus 3 times 10^-1 plus 5 times 10^-2. Every representation corresponds to exactly one real number. Every real number has at least one representation.
In computing, the binary base-2 octal base-8 and hexadecimal base-16 bases are most commonly used. Computers deal only with sequences of conventional zeroes and ones. It is easier in this sense to deal with powers of two. The hexadecimal system serves as shorthand for binary. Every 4 binary bits relate to one and only one hexadecimal digit. In hexadecimal the six digits after 9 are denoted by A through F. The octal numbering system also represents binary numbers. In this case the base is 8 and therefore only digits 0 through 7 are used. When converting from binary to octal every 3 bits relate to one and only one octal digit. Hexadecimal decimal octal and a wide variety of other bases have been used for binary-to-text encoding. Implementations of arbitrary-precision arithmetic rely on these systems too. For example, converting A10BHex to decimal yields 41227. Converting 0b11111001 binary to decimal gives 249. Horner's method can be used for base conversion using repeated multiplications. This approach shares computational complexity with repeated divisions. A number in positional notation functions like a polynomial where each digit acts as a coefficient. Coefficients can exceed one digit so an efficient way to convert bases involves converting each digit then evaluating the polynomial via Horner's method within the target base.
Base-12 systems called duodecimal or dozenal have been popular because multiplication and division are easier than in base-10. Twelve has many factors including one two three four and six. There is still a special word for dozen in English. Commerce developed a word for 12^2 called gross. The standard 12-hour clock emphasizes utility of this base. Prior to its conversion to decimal old British currency Pound Sterling partially used base-12. There were 12 pence in a shilling and 20 shillings in a pound. The Maya civilization and other civilizations of pre-Columbian Mesoamerica used base-20 vigesimal counting. Several North American tribes also employed this system. Evidence of base-20 counting appears in languages of central and western Africa. Remnants of a Gaulish base-20 system exist in French today. Sixty-five is soixante-cinq literally meaning sixty and five. Eighty-two is quatre-vingt-deux literally four twenty[s] and two. The Irish language used base-20 in the past with twenty being fichid. Welsh continues using base-20 particularly for ages dates and common phrases. The Inuit languages use base-20 too. Students from Kaktovik Alaska invented a base-20 numeral system in 1994. Danish numerals display similar base-20 structure. The Māori language has evidence of underlying base-20 systems seen in terms like Te Hokowhitu a Tu referring to seven 20s of Tu. A number of Australian Aboriginal languages employ binary or binary-like counting systems. Plainly it is based on the number of digits on a human hand.
Interesting properties exist when the base is not fixed or positive and when digit symbol sets denote negative values. Balanced ternary uses a base of 3 but the digit set includes -1 0 and 1 instead of 0 1 2. The symbol -1 has equivalent value minus one. Negation of a number forms easily by switching symbols. This system solves balance problems requiring minimal known counterweights. Weights of 1 3 9 up to 3^n units determine any unknown weight up to sum of those weights. An unknown weight W balanced with 3^1 on its pan and 1 plus 3^3 on other yields decimal 25 or 101 in balanced base-3. The factorial number system uses varying radix giving factorials as place values. These relate to Chinese remainder theorem and residue number system enumerations. This system effectively enumerates permutations. A derivative uses Towers of Hanoi puzzle configuration as counting system. Configuration puts into 1-to-1 correspondence with decimal count of step at which configuration occurs. Decimal equivalents show patterns for bases like -2 and factoroid systems. Negative bases rarely used yet offer theoretical value to computer scientists. Systems with complex base or negative digits have been described. Most do not require minus sign for designating negative numbers.
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Common questions
What was the first positional numeral system to be developed?
The Babylonian numeral system, base 60, was the first positional system to be developed. This ancient method influenced how time and angles are counted today.
When did Archimedes invent a decimal positional system based on 10^8 in his Sand Reckoner?
Archimedes invented a decimal positional system based on 10^8 in his Sand Reckoner around 287 BC to 212 BC. He used powers of ten to count grains of sand.
Which numeral systems were commonly used before positional notation became standard?
Roman numerals and Chinese numerals relied on sign-value notation before positional notation arrived. Accountants used abacuses or stone counters for arithmetic until positional notation arrived.
How does the binary base-2 octal base-8 and hexadecimal base-16 bases function in computing?
Computers deal only with sequences of conventional zeroes and ones because it is easier to deal with powers of two. The hexadecimal system serves as shorthand for binary where every 4 binary bits relate to one and only one hexadecimal digit.
Why have Base-12 systems called duodecimal or dozenal been popular historically?
Base-12 systems called duodecimal or dozenal have been popular because multiplication and division are easier than in base-10. Twelve has many factors including one two three four and six.