Table_of_bases
A table of bases is a reference compilation listing properties of various numeral systems, such as digit sets, place values, and powers, to facilitate understanding and conversion between different bases in mathematics and computing.[1] In positional numeral systems, the base (or radix) is the number of distinct digits used to represent numbers, with each position indicating a power of the base. For example, the decimal system uses base-10 with digits 0–9, where the number 123 equals 1×10² + 2×10¹ + 3×10⁰. Other common bases include binary (base-2, digits 0–1, used in digital electronics) and hexadecimal (base-16, digits 0–9 and A–F, used in programming).[2][1] The concept of positional notation originated in ancient Mesopotamia with the Babylonians using base-60 around 2000 BCE, while the modern base-10 system with zero as a placeholder developed in India by the 5th century CE and spread globally via the Hindu–Arabic numeral system.[3]Fundamentals of Number Bases
Definition and Positional Notation
In a positional numeral system, the base, also known as the radix, is defined as the number of distinct digits, including zero, that are used to represent numbers. For instance, the decimal system employs base 10 with digits ranging from 0 to 9. This foundational concept allows for compact and systematic representation of integers and other numerical values across various mathematical and computational contexts.[4][5] Positional notation operates by assigning a weight to each digit based on its location relative to a radix point, which separates the integer and fractional parts (though the focus here is on integer representations). The rightmost digit corresponds to the zeroth power of the base (b^0 = [1](/page/1)), the next digit to the left to the first power (b^1), and subsequent digits to higher powers accordingly. This weighting enables the encoding of arbitrarily large numbers using a finite set of symbols, with the position determining the magnitude contribution of each digit. Bases greater than 1 are essential for this system, as base 1 relies on unary tally marks—repetitions of a single symbol without positional weighting—and thus lacks the structure of a true positional numeral system.[5][6] The numerical value of a positional representation d_n d_{n-1} \cdots d_1 d_0 in base b is calculated as the sum of each digit multiplied by the corresponding power of the base: \sum_{k=0}^{n} d_k \cdot b^k Here, each d_k must be an integer satisfying $0 \leq d_k < b, ensuring uniqueness and completeness in representation for non-negative integers. This formula underpins conversions between bases and arithmetic operations in positional systems.[5][7]Historical Origins
The earliest known use of a base-10 numeral system emerged around 3000 BCE in ancient Egyptian mathematics, where hieroglyphic symbols represented powers of 10, likely influenced by finger counting on human hands.[8] This decimal approach facilitated practical calculations in administration and construction during the Old Kingdom (c. 2700–2200 BCE). Similarly, Sumerian scribes in Mesopotamia around the same period developed a sexagesimal (base-60) system, reflecting early cultural adaptations to counting for trade and record-keeping. By c. 2000 BCE, the Babylonians refined a positional sexagesimal (base-60) system inherited from Sumerian traditions, using it extensively in astronomy for dividing circles and tracking celestial movements due to 60's high divisibility.[9] This innovation persisted in timekeeping and angular measurements. In parallel, binary-like patterns appeared in ancient China with the I Ching (c. 1000 BCE), where yin-yang lines formed hexagrams for divination, prefiguring dualistic representations without direct arithmetic application.[10] Around the 3rd century BCE, Indian mathematician Pingala developed binary sequences in his Chandah-sutra to classify Sanskrit poetic meters, using short and long syllables as 0 and 1 analogs to enumerate combinations recursively.[11] The adoption of Hindu-Arabic base-10 numerals in Europe accelerated in the 13th century through Leonardo of Pisa (Fibonacci)'s Liber Abaci (1202), which demonstrated their efficiency for commerce over Roman numerals, spreading via Italian merchants.[12] Medieval English measurements retained base-12 (dozenal) influences from ancient Roman and Germanic systems, evident in units like 12 inches per foot and 12 pennies per shilling for practical divisions.[13] During the Renaissance, Gottfried Wilhelm Leibniz formalized binary arithmetic in 1703, inspired by I Ching hexagrams, viewing it as a philosophical tool for computation.[14] In the 20th century, base-2 became central to digital electronics following John von Neumann's 1945 EDVAC report, which outlined a binary stored-program computer architecture influencing modern designs.[15] Bases 8 (octal) and 16 (hexadecimal) emerged post-1940s as compact representations for binary data in early computers, standardizing programming and hardware interfaces.Standard Integer Bases
Binary, Octal, and Hexadecimal
The binary number system, base-2, employs only the digits 0 and 1 to represent values through positional notation, where each position corresponds to a power of 2.[16] It serves as the foundational representation in digital logic circuits and Boolean algebra, enabling the manipulation of electrical signals as true (1) or false (0) states in computing hardware.[17] For instance, the binary number $1011_2 equals $1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 11_{10}.[18] In practical applications, binary data is organized into bits (individual binary digits), with a byte comprising 8 bits that can represent 256 unique values, from 0 to 255, forming the basic unit for data storage and processing in most computer systems.[19] The octal system, base-8, utilizes digits from 0 to 7 and is closely tied to binary because 8 equals $2^3, allowing each octal digit to directly correspond to a 3-bit binary group.[20] It has been employed in legacy computing environments and remains prominent in Unix-like systems for representing file permissions, where each set of three bits denotes read, write, or execute access for user, group, and others, often expressed as a three-digit octal value like 755.[21] As an example, $17_8 converts to $1 \times 8^1 + 7 \times 8^0 = 15_{10}.[18] Hexadecimal, or base-16, extends this efficiency further with digits 0-9 and A-F (where A=10, B=11, up to F=15), leveraging the fact that 16 is $2^4 to map each hex digit to exactly 4 binary bits.[22] It is a standard in programming for denoting memory addresses, which are typically displayed in hex to simplify reading large binary values, and for color codes in web development, such as #FF0000 for pure red.[23][24] For example, $1A_{16} equals $1 \times 16^1 + 10 \times 16^0 = 26_{10}.[18] Interconversions between these bases exploit their power-of-2 relationships: to convert from binary to octal, group the binary digits into sets of 3 starting from the right (padding with leading zeros if needed) and replace each group with its octal equivalent; for binary to hexadecimal, use groups of 4 bits.[18] These methods facilitate efficient handling of binary data without full decimal intermediation. Octal and hexadecimal offer advantages over pure binary representations by providing more compact notations—hexadecimal, in particular, condenses 4-bit sequences into single characters, reducing the length of expressions for memory addresses or machine code while remaining straightforward to translate back to binary.[25] This brevity enhances readability and error detection in computing tasks involving large datasets.[26]Decimal and Dozenal
The decimal system, also known as base-10, employs the digits 0 through 9 to represent numbers and stands as the universal standard for numerical notation in global commerce, scientific communication, and everyday transactions.[27] This widespread adoption stems from its alignment with human physiology, particularly the ten fingers historically used for counting and tallying.[28] In positional notation, a number like 123 in decimal expands to $1 \times 10^2 + 2 \times 10^1 + 3 \times 10^0 = 100 + 20 + 3 = 123./01:_Number_Representation_in_Different_Bases_and_Cryptography/1.03:_Converting_to_Different_Base_Systems) Despite its practicality for human-scale counting, the decimal system encounters challenges with divisibility, as fractions like one-third result in a non-terminating repeating expansion: \frac{1}{3} \approx 0.333\ldots./03:_Logic_and_Proofs/3.05:_Divisibility_and_Prime_Factors) In contrast, the dozenal system, or base-12, builds on digits 0-9 and introduces additional symbols for the values ten (commonly denoted as X or ↊) and eleven (E or ↋), enabling representation of all integers up to 11.[29] A key advantage of base-12 lies in its high divisibility, as 12 factors evenly into 1, 2, 3, 4, 6, and 12, which facilitates cleaner arithmetic operations and fraction handling compared to base-10's factors of 1, 2, 5, and 10.[29] This property underpins its practical applications in timekeeping, with 12 hours marking half a day on analog clocks, and in imperial measurement systems, such as 12 inches per foot or 12 ounces per pound (troy).[29] For instance, the dozenal number 1X expands to $1 \times 12^1 + \text{X} \times 12^0 = 12 + 10 = 22 in decimal.[30] Dozenal excels in fractional efficiency, where common divisions yield terminating expansions; notably, one-third equals exactly 0.4 in base-12, avoiding the infinite repetition seen in decimal's 0.333....[29] Such precision reduces approximation errors in measurements and calculations involving thirds, quarters, or sixths, highlighting base-12's conceptual superiority for anthropocentric tasks like dividing resources or angular measurements.[29] Advocacy for dozenal persists through organizations like the Dozenal Society of America, established in 1944 as a nonprofit dedicated to research and education on base-12's benefits for mathematics instruction and societal applications.[31] To convert a decimal number to dozenal, apply repeated division by 12, recording each remainder as a digit from the least significant place; the quotients guide subsequent steps until reaching zero.[30] This remainder method mirrors general base conversion techniques and underscores the shared positional principles between the systems.Specialized Integer Bases
Ternary and Quinary
Ternary, or base-3, numeral systems employ digits 0, 1, and 2, known as trits, to represent values in positional notation where each position corresponds to a power of 3.[32] This structure provides higher information density compared to binary, with each trit encoding approximately \log_2 3 \approx 1.58 bits of information, making it more efficient for storage in certain computational contexts.[33] A notable historical application is the Setun computer, developed in the late 1950s at Moscow State University in the USSR, which utilized ternary logic for its arithmetic operations and demonstrated practical viability in early computing.[33] For instance, the ternary number $12_3 equals $1 \times 3^1 + 2 \times 3^0 = 5_{10}.[34] Balanced ternary represents a variant of base-3 using digits -1, $0, and $1 (often denoted as \bar{1}, $0, $1), which eliminates the need for a separate sign bit by inherently supporting positive and negative values through the negative digit. This system simplifies arithmetic operations relative to binary, as addition and subtraction can be performed without distinguishing between signed and unsigned numbers, reducing hardware complexity in implementations like the Setun. An example is \bar{1}1_3 = (-1) \times 3^1 + 1 \times 3^0 = -3 + 1 = -2_{10}, or more complexly, $1\bar{1}1_3 = 1 \times 3^2 + (-1) \times 3^1 + 1 \times 3^0 = 9 - 3 + 1 = 7_{10}.[34] The inherent redundancy in ternary representations also enables effective error detection, as deviations from valid trit values can be identified using techniques like cyclic redundancy checks adapted for base-3.[35] Quinary, or base-5, systems use digits 0 through 4 to denote values, with each position weighted by powers of 5, offering a compact representation for certain cultural and computational purposes.[36] Balanced quinary extends this by employing digits -2, -1, $0, $1, $2. A simple example is 14_5 = 1 \times 5^1 + 4 \times 5^0 = 9_{10}$.Senary, Septenary, and Higher Bases
Senary, or base-6, numeral systems utilize digits from 0 to 5 and represent a midpoint between quinary and decimal systems. As half of base 12, senary shares divisibility properties with duodecimal, facilitating subgrouping in counting tasks. Certain Central African cultures incorporated base-6 elements in traditional counting, expressing numbers beyond 6 additively, such as 7 as 6 + 1 and 8 as 6 + 2.[37] For example, the numeral $10_6 equals $6_{10}. Septenary, or base-7, systems employ digits 0 through 6 and remain uncommon in historical numeral applications. While rare, base-7 concepts appear in modern dice games like Sevens, where players score by avoiding combinations summing to 7 with standard six-sided dice.[38] The primality of 7 enhances base-7's utility in modular arithmetic, as the ring of integers modulo 7 forms a field, enabling efficient operations in cryptography and number theory without zero divisors.[39] Higher bases from 11 to 36 offer compact representations for large numbers but require extended digit sets, typically 0-9 followed by letters A-Z for values 10-35. Base 36 finds practical use in computing for alphanumeric encoding of identifiers in URLs, reducing length while remaining human-readable; for instance, the digits 0-9 and A-Z allow efficient mapping of numeric IDs to short strings.[40] An example is $11_{36} = 1 \times 36 + 1 = 37_{10}. These higher bases, while advantageous for brevity in digital contexts, prove impractical for everyday human use due to the cognitive load of memorizing numerous symbols. Vestiges of even larger bases persist in specialized domains, such as the Babylonian base-60 system, which endures in divisions of time (hours, minutes, seconds) and angles (degrees, arcminutes, arcseconds).[41]Tables of Base Properties
Digit Sets and Symbols
In positional numeral systems, the digit set for a given base b consists of b distinct symbols, each representing integer values from 0 to b-1. For bases 2 through 10, these symbols are the standard Arabic numerals 0 through 9, truncated to the required range (e.g., base 2 uses only 0 and 1). For bases greater than 10, additional symbols are introduced to represent values beyond 9, ensuring each digit uniquely corresponds to its value without ambiguity.[42] The conventional choice for these additional symbols in bases 11 through 36 is the uppercase Latin letters A through Z, where A denotes 10, B denotes 11, and so on up to Z for 35. This extension is widely adopted in computing and mathematics due to its simplicity and compatibility with alphanumeric character sets. Uppercase letters are preferred over lowercase to distinguish them clearly from numerals in mixed contexts, though lowercase variants (a-z) appear in some encoding schemes like certain URL shorteners.[43][44] Special cases arise for certain bases. In base 2 (binary), the digit set is limited to 0 and 1, reflecting its minimalistic use in digital logic. For base 60 (sexagesimal), ancient Babylonian notation employed only two wedge-shaped symbols repeated to form values up to 59, but modern representations—such as in timekeeping (hours:minutes)—use pairs of decimal digits from 00 to 59 rather than 60 unique symbols, avoiding the need for an expanded set.[9][45] These symbol choices matter because they enable unambiguous representation of numbers in any base, preventing confusion between digit values and facilitating computation or display. The ASCII character set practically limits standard alphanumeric notation to base 36, as it provides exactly 10 digits (0-9) plus 26 letters (A-Z), totaling 36 symbols; higher bases require custom or composite representations.[46] The following table summarizes the standard symbols for digit values 0 through 35, applicable to bases up to 36 (lower bases use the initial subset of these symbols):| Digit Value | Standard Symbol |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 7 |
| 8 | 8 |
| 9 | 9 |
| 10 | A |
| 11 | B |
| 12 | C |
| 13 | D |
| 14 | E |
| 15 | F |
| 16 | G |
| 17 | H |
| 18 | I |
| 19 | J |
| 20 | K |
| 21 | L |
| 22 | M |
| 23 | N |
| 24 | O |
| 25 | P |
| 26 | Q |
| 27 | R |
| 28 | S |
| 29 | T |
| 30 | U |
| 31 | V |
| 32 | W |
| 33 | X |
| 34 | Y |
| 35 | Z |
Place Values and Powers
In positional numeral systems with integer base b > 1, the place values represent successive powers of the base, where the rightmost digit (units place) corresponds to b^0 = 1, the next to b^1 = b, and the n-th place to the left to b^n. This hierarchical structure enables the efficient encoding of numerical values, with each additional place exponentially increasing the representable range by a factor of b.[48] The exponential growth inherent in these powers, defined as b^n for nonnegative integer n, underscores the scalability of different bases; higher bases yield steeper growth curves, allowing larger numbers to be expressed with fewer digits compared to lower bases. The table below enumerates these place values for commonly used integer bases (2, 3, 5, 8, 10, 12, and 16) across exponents 0 through 10, illustrating the rapid escalation in magnitude.| Exponent | Base 2 | Base 3 | Base 5 | Base 8 | Base 10 | Base 12 | Base 16 |
|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 2 | 3 | 5 | 8 | 10 | 12 | 16 |
| 2 | 4 | 9 | 25 | 64 | 100 | 144 | 256 |
| 3 | 8 | 27 | 125 | 512 | 1,000 | 1,728 | 4,096 |
| 4 | 16 | 81 | 625 | 4,096 | 10,000 | 20,736 | 65,536 |
| 5 | 32 | 243 | 3,125 | 32,768 | 100,000 | 248,832 | 1,048,576 |
| 6 | 64 | 729 | 15,625 | 262,144 | 1,000,000 | 2,985,984 | 16,777,216 |
| 7 | 128 | 2,187 | 78,125 | 2,097,152 | 10,000,000 | 35,831,808 | 268,435,456 |
| 8 | 256 | 6,561 | 390,625 | 16,777,216 | 100,000,000 | 429,981,696 | 4,294,967,296 |
| 9 | 512 | 19,683 | 1,953,125 | 134,217,728 | 1,000,000,000 | 5,159,780,352 | 68,719,476,736 |
| 10 | 1,024 | 59,049 | 9,765,625 | 1,073,741,824 | 10,000,000,000 | 61,917,364,224 | 1,099,511,627,776 |