Positional notation

Positional notation, also known as place-value notation, is the property of a numeral system that the value represented by each symbol in a written numeral depends not only on its appearance but also on its position. Each symbol fits in a specific place or position, representing a power of a fixed base. The most common numeral system used today, the Hindu–Arabic numeral system, is a positional system in base ten; each of ten numerical digits is a distinct symbol representing one of the numbers zero through nine, and in the context of the full numeral, each symbol's value is the digit multiplied by a power of ten. Most early numeral systems, such as Roman numerals, are essentially based on the additive principle: each symbol type represents one fixed value, and the value of a numeral is the sum of the values of the separate symbols. For example, the Roman numeral has two copies of the symbol meaning , two copies of meaning , one meaning , and three copies of meaning , so overall represents the number ; by comparison, the equivalent Hindu–Arabic numeral, , consists of the symbol representing , another symbol representing , and finally an representing .…

Positional notation, also known as place-value notation, is the property of a numeral system that the value represented by each symbol in a written numeral depends not only on its appearance but also on its position. Each symbol fits in a specific place or position, representing a power of a fixed base. The most common numeral system used today, the Hindu–Arabic numeral system, is a positional system in base ten; each of ten numerical digits is a distinct symbol representing one of the numbers zero through nine, and in the context of the full numeral, each symbol's value is the digit multiplied by a power of ten. Most early numeral systems, such as Roman numerals, are essentially based on the additive principle: each symbol type represents one fixed value, and the value of a numeral is the sum of the values of the separate symbols. For example, the Roman numeral has two copies of the symbol meaning , two copies of meaning , one meaning , and three copies of meaning , so overall represents the number ; by comparison, the equivalent Hindu–Arabic numeral, , consists of the symbol representing , another symbol representing , and finally an representing . 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. The binary numeral system (base two) is used in almost all computers and electronic devices because it is easier to implement efficiently in electronic circuits. Systems with negative base, complex base or negative digits have been described. Most of them do not require a minus sign for designating negative numbers. The use of a radix point (decimal point in base ten), extends to include fractions and allows the representation of any real number with arbitrary accuracy. With positional notation, arithmetical computations are much simpler than with any older numeral system; this led to the rapid spread of the notation when it was introduced in western Europe. Upper row = vertical form Lower row = horizontal form|class=skin-invert-image Counting rods and most abacuses have been used to represent numbers in a positional numeral system. With counting rods or abacus to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables (as does positional notation) and could produce practical results quickly. The oldest extant positional notation system is either that of Chinese rod numerals, used from at least the early 8th century, or perhaps Khmer numerals, showing possible usages of positional-numbers in the 7th century. Khmer numerals and other Indian numerals originate with the Brahmi numerals of about the 3rd century BC, which symbols were, at the time, not used positionally. Medieval Indian numerals are positional, as are the derived Arabic numerals, recorded from the 10th century. After the French Revolution (1789–1799), the new French government promoted the extension of the decimal system. Some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world. In mathematical numeral systems the radix is usually the number of unique digits, including zero, that a positional numeral system uses to represent numbers. In some cases, such as with a negative base, the radix is the absolute value r=|b| of the base . For example, for the decimal system the radix (and base) is ten, because it uses the ten digits from 0 through 9. When a number "hits" 9, the next number will not be another different symbol, but a "1" followed by a "0". In binary, the radix is two, since after it hits "1", instead of "2" or another written symbol, it jumps straight to "10", followed by "11" and "100". The highest symbol of a positional numeral system usually has the value one less than the value of the radix of that numeral system. The standard positional numeral systems differ from one another only in the base they use. The radix is an integer that is greater than 1, since a radix of zero would not have any digits, and a radix of 1 would only have the zero digit. Negative bases are rarely used. In a system with more than |b| unique digits, numbers may have many different possible representations. It is important that the radix is finite, from which follows that the number of digits is quite low. Otherwise, the length of a numeral would not necessarily be logarithmic in its size. (In certain non-standard positional numeral systems, including bijective numeration, the definition of the base or the allowed digits deviates from the above.) In standard base-ten (decimal) positional notation, there are ten decimal digits and the number 5305_{\mathrm{dec}} = (5 \times 10^3) + (3 \times 10^2) + (0 \times 10^1) + (5 \times 10^0) . In standard base-sixteen (hexadecimal), there are the sixteen hexadecimal digits (0–9 and A–F) and the number 14\mathrm{B}9_{\mathrm{hex}} = (1 \times 16^3) + (4 \times 16^2) + (\mathrm{B} \times 16^1) + (9 \times 16^0) \qquad (= 5305_{\mathrm{dec}}) , where B represents the number eleven as a single symbol. In general, in base-b, there are b digits \{d_1,d_2,\dotsb,d_b\} =:D and the number (a_3 a_2 a_1 a_0)_b = (a_3 \times b^3) + (a_2 \times b^2) + (a_1 \times b^1) + (a_0 \times b^0) has \forall k \colon a_k \in D . Note that a_3 a_2 a_1 a_0 represents a sequence of digits, not multiplication. 465_{10} = 4\times 10^2 + 6\times 10^1 + 5\times 10^0 = 4\times 100 + 6\times 10 + 5\times 1 = 465_{10} If however, the number were in base 7, then it would equal: 465_{7} = 4\times 7^2 + 6\times 7^1 + 5\times 7^0 = 4\times 49 + 6\times 7 + 5\times 1 = 243_{10} 10 b = b for any base b, since 10 b = 1×b 1 + 0×b 0 . For example, 10 2 = 2; 10 3 = 3; 10 16 = 16 10 . Note that the last "16" is indicated to be in base 10. The base makes no difference for one-digit numerals. This concept can be demonstrated using a diagram. One object represents one unit. When the number of objects is equal to or greater than the base b, then a group of objects is created with b objects. When the number of these groups exceeds b, then a group of these groups of objects is created with b groups of b objects; and so on. Thus the same number in different bases will have different values: 241 in base 5: 2 groups of 5 2 (25) 4 groups of 5 1 group of 1 ooooo ooooo ooooo ooooo ooooo ooooo ooooo ooooo + + o ooooo ooooo ooooo ooooo ooooo ooooo 241 in base 8: 2 groups of 8 2 (64) 4 groups of 8 1 group of 1 oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo + + o oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo The notation can be further augmented by allowing a leading minus sign. This allows the representation of negative numbers. For a given base, every representation corresponds to exactly one real number and every real number has at least one representation. The representations of rational numbers are those representations that are finite, use the bar notation, or end with an infinitely repeating cycle of digits. A non-zero numeral with more than one digit position will mean a different number in a different number base, but in general, the digits will mean the same. For example, the base-8 numeral 23 8 contains two digits, "2" and "3", and with a base number (subscripted) "8". When converted to base-10, the 23 8 is equivalent to 19 10 , i.e. 23 8 = 19 10 . In our notation here, the subscript " 8 " of the numeral 23 8 is part of the numeral, but this may not always be the case. Imagine the numeral "23" as having an ambiguous base number. Then "23" could likely be any base, from base-4 up. In base-4, the "23" means 11 10 , i.e. 23 4 = 11 10 . In base-60, the "23" means the number 123 10 , i.e. 23 60 = 123 10 . The numeral "23" then, in this case, corresponds to the set of base-10 numbers {11, 13, 15, 17, 19, 21, 23, ..., 121, 123} while its digits "2" and "3" always retain their original meaning: the "2" means "two of", and the "3" means "three of". In certain applications when a numeral with a fixed number of positions needs to represent a greater number, a higher number-base with more digits per position can be used. A three-digit, decimal numeral can represent only up to 999. But if the number-base is increased to 11, say, by adding the digit "A", then the same three positions, maximized to "AAA", can represent a number as great as 1330. We could increase the number base again and assign "B" to 11, and so on (but there is also a possible encryption between number and digit in the number-digit-numeral hierarchy). A three-digit numeral "ZZZ" in base-60 could mean . If we use the entire collection of our alphanumerics we could ultimately serve a base-62 numeral system, but we remove two digits, uppercase "I" and uppercase "O", to reduce confusion with digits "1" and "0". We are left with a base-60, or sexagesimal numeral system utilizing 60 of the 62 standard alphanumerics. (But see Sexagesimal system below.) In general, the number of possible values that can be represented by a d digit number in base r is r^d . The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16). In binary only digits "0" and "1" are in the numerals. In the octal numerals, are the eight digits 0–7. Hex is 0–9 A–F, where the ten numerics retain their usual meaning, and the alphabetics correspond to values 10–15, for a total of sixteen digits. The numeral "10" is binary numeral "2", octal numeral "8", or hexadecimal numeral "16". 0xA10B/10 = Q: 0x101A, R: 7 (ones place) 0x101A/10 = Q: 0x19C, R: 2 (tens place) 0x19C/10 = Q: 0x29, R: 2 (hundreds place) 0x29/10 = Q: 0x4, R: 1 ... 4 When converting to a larger base (such as from binary to decimal), the remainder represents b_2 as a single digit, using digits from b_1 . For example: converting 0b11111001 (binary) to 249 (decimal): 0b11111001/10 = Q: 0b11000, R: 0b1001 (0b1001 = "9" for ones place) 0b11000/10 = Q: 0b10, R: 0b100 (0b100 = "4" for tens) 0b10/10 = Q: 0b0, R: 0b10 (0b10 = "2" for hundreds) For the fractional part, conversion can be done by taking digits after the radix point (the numerator), and dividing it by the implied denominator in the target radix. Approximation may be needed due to a possibility of non-terminating digits if the reduced fraction's denominator has a prime factor other than any of the base's prime factor(s) to convert to. For example, 0.1 in decimal (1/10) is 0b1/0b1010 in binary, by dividing this in that radix, the result is 0b0.0 0011 (because one of the prime factors of 10 is 5). For more general fractions and bases see the algorithm for positive bases. Alternatively, Horner's method can be used for base conversion using repeated multiplications, with the same computational complexity as repeated divisions. A number in positional notation can be thought of as a polynomial, where each digit is a coefficient. Coefficients can be larger than one digit, so an efficient way to convert bases is to convert each digit, then evaluate the polynomial via Horner's method within the target base. Converting each digit is a simple lookup table, removing the need for expensive division or modulus operations; and multiplication by x becomes right-shifting. However, other polynomial evaluation algorithms would work as well, like repeated squaring for single or sparse digits. Example: Convert 0xA10B to 41227 A10B = (10*16^3) + (1*16^2) + (0*16^1) + (11*16^0) Lookup table: 0x0 = 0 0x1 = 1 ... 0x9 = 9 0xA = 10 0xB = 11 0xC = 12 0xD = 13 0xE = 14 0xF = 15 Therefore 0xA10B's decimal digits are 10, 1, 0, and 11. Lay out the digits out like this. The most significant digit (10) is "dropped": 10 1 0 11 \frac{\N_0}{b^{\N_0}} := \left\{mb^{-\nu}\mid m\in \N_0 \wedge \nu\in \N_0 \right\} . More explicitly, if p_1^{\nu_1} \cdot \ldots \cdot p_n^{\nu_n} := b is a factorization of b into the primes p_1, \ldots ,p_n \in \mathbb P with exponents then with the non-empty set of denominators S := \{ p_1, \ldots, p_n \} we have \Z_S := \left\{x \in \Q \left | \, \exists \mu_i \in \Z : x \prod_{i=1}^n {p_i}^{\mu_i} \in \Z \right . \right\} = b^{\Z} \, \Z = {\langle S\rangle}^{-1}\Z where \langle S\rangle is the group generated by the p\in S and {\langle S\rangle}^{-1}\Z is the so-called localization of \Z with respect to The denominator of an element of \Z_S contains if reduced to lowest terms only prime factors out of S . This ring of all terminating fractions to base b is dense in the field of rational numbers \Q . Its completion for the usual (Archimedean) metric is the same as for \Q , namely the real numbers \R . So, if S = \{ p\} then \Z_{\{ p\}} has not to be confused with \Z_{(p)} , the discrete valuation ring for the prime p , which is equal to \Z_{T} with T = \mathbb P \setminus \{ p\} . If b divides c , we have b^{\Z} \, \Z \subseteq c^{\Z} \, \Z. 2.42\overline{314}_5 = 2.42314314314314314\dots_5 This is the repeating decimal notation (to which there does not exist a single universally accepted notation or phrasing). For base 10 it is called a repeating decimal or recurring decimal. An irrational number has an infinite non-repeating representation in all integer bases. Whether a rational number has a finite representation or requires an infinite repeating representation depends on the base. For example, one third can be represented by: 0.1_3 0.\overline3_{10} = 0.3333333\dots_{10} or, with the base implied: 0.\overline3 = 0.3333333\dots (see also 0.999...) 0.\overline{01}_2 = 0.010101\dots_2 0.2_6 For integers p and q with gcd (p, q) = 1, the fraction p/q has a finite representation in base b if and only if each prime factor of q is also a prime factor of b. For a given base, any number that can be represented by a finite number of digits (without using the bar notation) will have multiple representations, including one or two infinite representations: A finite or infinite number of zeroes can be appended: 3.46_7 = 3.460_7 = 3.460000_7 = 3.46\overline0_7 The last non-zero digit can be reduced by one and an infinite string of digits, each corresponding to one less than the base, are appended (or replace any following zero digits): 3.46_7 = 3.45\overline6_7 1_{10} = 0.\overline9_{10}\qquad (see also 0.999...) 220_5 = 214.\overline4_5 y = \sqrt[n]{x} with y^n = x and , numbers which are called algebraic, or numbers like \pi,e which are transcendental. The number of transcendentals is uncountable and the sole way to write them down with a finite number of symbols is to give them a symbol or a finite sequence of symbols. Each position does not need to be positional itself. Babylonian sexagesimal numerals were positional, but in each position were groups of two kinds of wedges representing ones and tens (a narrow vertical wedge | for the one and an open left pointing wedge ⟨ for the ten) — up to 5+9=14 symbols per position (i.e. 5 tens ⟨⟨⟨⟨⟨ and 9 ones ||||||||| grouped into one or two near squares containing up to three tiers of symbols, or a place holder (⑊) for the lack of a position). Hellenistic astronomers used one or two alphabetic Greek numerals for each position (one chosen from 5 letters representing 10–50 and/or one chosen from 9 letters representing 1–9, or a zero symbol). The Development of Hindu Arabic and Traditional Chinese Arithmetics Implementation of Base Conversion at cut-the-knot Learn to count other bases on your fingers Online Arbitrary Precision Base Converter

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