How To Use This Converter Type a number in any of the Number-Base Radix input fields and press Enter or Return on your keyboard or keypad. The other Number-Base Radix fields will show the number you entered, converted to the other radixes. TIP: Entry Errors If the data you type into any Base field contains disallowed symbols for that field, the converter will notify you. In that case, simply re-type valid data in the field of your choice and press Enter or Return. TIP: Copy and Paste You can Copy and Paste any of the fields by selecting the text in the field, then pressing Control-C to copy, and Control-V to paste.

Radix-2

Binary or Radix-2 Number System Binary is a Radix-2 positional notation number system. Radix-2 or Base-2 means that each Binary "digit" (called a Bit), represents a multiple of increasing powers of 2, from right to left, such as: ... 2³, 2², 2¹, 2⁰ The valid Binary symbols are: 0, 1 For example, the Binary number 101 represents the number 5 in the Decimal number system, like so: (1 x 2²) + (0 x 2¹) + (1 x 2⁰) History of the Binary Number System The modern Binary number system was introduced by Gottfried Leibniz in his 1703 work entitled: "Explication de l'Arithmétique Binaire" In 1854, George Boole, a British mathematician, developed what would later become modern Binary algebra (known today as Boolean algebra). In 1937, Claude Shannon, an American electrical engineer and mathematician, in his Master's thesis at MIT, demonstrated a machine that used electrical Binary relays to implement Boolean algebra. Shannon's work showed that an electrical computer based on Boolean algebra could construct and solve any logical, numerical relationship. Shannon is credited as the father of Information Theory, the mathematical basis of modern digital communications and data transmission.

Radix-3

Ternary or Radix-3 Number System Ternary is a Radix-3 positional notation number system. Radix-3 or Base-3 means that each Ternary "digit" represents a multiple of increasing powers of 3, from right to left, such as: ... 3³, 3², 3¹, 3⁰ The valid Ternary symbols are: 0, 1, 2 For example, the Ternary number 210 represents the number 21 in the Decimal number system, like so: (2 x 3²) + (1 x 3¹) + (0 x 3⁰) Applications of the Ternary Number System In modern use, ternary can improve the performance of elliptic curve cryptology algorithms. During the late 1950s, Nikolay Brusentsov, at Moscow State University in the former Soviet Union, built a ternary computer, which he named "Setun". In 1970, Brusentsov built an enhanced version, which he named "Setun-70". Possible future applications of ternary may combine an optical computer based on "balanced ternary" logic, where no-light represents 0 (zero), and two orthogonal polarizations of light represent 1 and -1. A base-3 system is used in Islam to count Tasbih (short devotional prayers), to 99 or to 100 on a single hand, as an alternative to using a string of prayer beads called a Misbaha.

Radix-4

Quaternary or Radix-4 Number System Quaternary is a Radix-4 positional notation number system. Radix-4 or Base-4 means that a Quaternary "digit" (called a Crumb), represents a multiple of increasing powers of 4, from right to left, such as: ... 4³, 4², 4¹, 4⁰ The valid Quaternary symbols are: 0, 1, 2, 3 For example, the Quaternary number 302 represents the number 50 in the Decimal number system, like so: (3 x 4²) + (0 x 4¹) + (2 x 4⁰) History of the Quaternary Number System Many or all of the now extinct Chumashan languages originally used a base-4 counting system, in which the names for numbers were structured according to multiples of 4 and 16, not 10. Chumashan was a family of languages that were spoken on the southern California coast, from San Luis Obispo to Malibu, in neighboring Coastal and Transverse range valleys bordering the San Joaquin Valley, and on the adjacent Channel islands of San Miguel, Santa Rosa, and Santa Cruz. Parallels can be drawn between Quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in alphabetical order, abbreviated A, C, G and T, can be represented by the Quaternary digits 0, 1, 2, and 3.

Radix-5

Quinary or Radix-5 Number System Quinary is a Radix-5 positional notation number system. Radix-5 or Base-5 means that each Quinary "digit" represents a multiple of increasing powers of 5, from right to left, such as: ... 5³, 5², 5¹, 5⁰ The valid Quinary symbols are: 0, 1, 2, 3, 4 For example, the Quinary number 402 represents the number 102 in the Decimal system, like so: (4 x 5²) + (0 x 5¹) + (2 x 5⁰) History of the Quinary Number System Many languages use Quinary, including the three Aboriginal Australian languages: Nunggubuyu, Gumatj, and Kuurn Kopan Noot, as well as Saraveca, the extinct Arawakan language once spoken in Bolivia by the Sarave people. Other languages use Biquinary, where 5 is the sub-base for a Decimal counting system, as for example, Wolof, a language spoken in Senegal, The Gambia, and Mauritania, the native language of the ethnic group known as the Wolof people, and also for example in Khmer or Cambodian, the language of the Khmer people, the official language of Cambodia. Additionally, other languages use 5 as a sub-base for a base-20 counting system, as found in Nahuatl and the Maya numerals. Nahuatl is a group of related dialects of the Nahuan or "Aztecan" language family native to Central Mexico. The Maya numerals were invented and used by the Maya, an advanced pre-Columbian civilization native to Central America. Roman numerals employ a Biquinary number system, as do the Chinese and Japanese abaci.

Radix-8

Octal or Radix-8 Number System Octal is a Radix-8 positional notation number system. Radix-8 or Base-8 means that each Octal "digit" represents a multiple of increasing powers of 8, from right to left, such as: ... 8³, 8², 8¹, 8⁰ The valid Octal symbols are: 0, 1, 2, 3, 4, 5, 6, 7 For example, the Octal number 657 represents the number 431 in the Decimal number system, like so: (6 x 8²) + (5 x 8¹) + (7 x 8⁰) History of the Octal Number System The Yuki language in California and the Pamean languages in central Mexico use Octal because the speakers count using the spaces between their fingers rather than the fingers themselves. Octal is occassionally used in computing, for example to represent the disk file permissions in the Unix operating system and its variants. Octal has the advantage of not requiring additional symbols beyond the digits 0 through 7. One Octal digit perfectly represents 3-bits of Binary, and for this reason Octal was used on computers whose word-length was a multiple of 3, such as the 24-bit and 36-bit word-length IBM computers that were popular in the 1960's. Those word-lengths could be efficiently represented by numerical displays consisting of 8 and 12 Octal digits. Modern computers, however, use 16-bit, 32-bit, or 64-bit words, which are not perfectly represented by Octal. Instead, modern computers use Hexadecimal (or Hex) to represent their words for display to humans, because a Hex digit can perfectly represent 4-bits. Consequently, modern computer words are represented perfectly with 4, 8, or 16 Hex digits.

Radix-10

Decimal or Radix-10 Number System Decimal is a Radix-10 positional notation number system. Radix-10 or Base-10 means that each Decimal "digit" represents a multiple of increasing powers of 10, from right to left, such as: ... 10³, 10², 10¹, 10⁰ The valid Decimal symbols are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 For example, the Decimal number 927 represents the following numerical expression: (9 x 10²) + (2 x 10¹) + (7 x 10⁰) History of the Decimal Number System The modern Decimal number system, known as the Hindu-Arabic number system, originated in India around the 9th century CE. The system spread to the western world during the Middle Ages (1000 to 1300 CE) as a result of trade. Some scholars attribute the first documented use of a Decimal system to China in the 1st century BCE.

Radix-12

Duodecimal or Radix-12 Number System Duodecimal is a Radix-12 positional notation number system. Radix-12 or Base-12 means that each Duodecimal "digit" represents a multiple of increasing powers of 12, from right to left, such as: ... 12³, 12², 12¹, 12⁰ The valid Duodecimal symbols are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B Where: A=10, B=11 For example, the Duodecimal number BA9 represents the number 1713 in the Decimal system, like so: (11 x 12²) + (10 x 12¹) + (9 x 12⁰) History of the Duodecimal Number System Languages using Duodecimal number systems are rare. Languages in the Nigerian Middle Belt such as Janji, Gbiri-Niragu (Kahugu), the Nimbia dialect of Gwandara, the Chepang language of Nepal, and the Mahl language of Minicoy Island in India are known to use Duodecimal numerals. J. R. R. Tolkien's Elvish languages use a Duodecimal number system. Units of time in many civilizations are a Duodecimal system. There are 12 signs of the zodiac, 12 months in a year, and 12 hours in a day or night. Traditional Chinese calendars, clocks, and compasses are based on the 12 Earthly Branches.

Radix-16

Hexadecimal or Radix-16 Number System Hexadecimal is a Radix-16 positional notation number system. Radix-16 or Base-16 means that each Hexadecimal "digit" represents a multiple of increasing powers of 16, from right to left, such as: ... 16³, 16², 16¹, 16⁰ The valid Hexadecimal symbols are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Where: A=10, B=11, C=12, D=13, E=14, F=15 For example, the Hexadecimal number FE8 represents the number 4072 in the Decimal system, like so: (15 x 16²) + (14 x 16¹) + (8 x 16⁰) Applications of Hexadecimal Number System A Hexadecimal digit perfectly represents 4 Binary digits (bits), often called a "Nibble". The principal use of Hexadecimal notation is in displaying to humans the Binary values used in computing and digital electronics. For example, byte values (8-bits) can range from 0 to 255 Decimal, but are more efficiently represented as two Hexadecimal digits in the range 00 through FF. Hexadecimal is commonly used to represent computer memory addresses and memory contents, CPU register values, computer colors, computer character encodings, and many other values that are routinely used in modern computers and digital electronics.

Radix-20

Vigesimal or Radix-20 Number System Vigesimal is a Radix-20 positional notation number system. Radix-20 or Base-20 means that each Vigesimal "digit" represents a multiple of increasing powers of 20, from right to left, such as: ... 20³, 20², 20¹, 20⁰ The valid Vigesimal symbols are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J Where: A=10, B=11, C=12, D=13, E=14, F=15, G=16, H=17, I=18, J=19 For example, the Vigesimal number JHG represents the number 7956 in the Decimal system, like so: (19 x 20²) + (17 x 20¹) + (16 x 20⁰) History of the Vigesimal Number System In many languages, especially in Europe, 20 is a base with respect to the linguistic structure of the names used in those languages for certain numbers. In the pre-Columbian civilizations of central America, 20 was an important linguistic and counting base. Twenty was a base in Maya number systems. The Maya had specific names for many of the powers of twenty, as did the Nahuatl (Aztecs) of central Mexico. French has many words to represent various larger numbers, which are based on the number 20. This is also true of many other European languages, including Danish, Breton, Welch, Irish and Scottish Gaelic, Albanian, Georgian, and Basque. In old traditional English counting by the "score" (i.e. by 20's), was common, but is no longer practiced.

Radix-24

Quadrovigesimal or Radix-24 Number System Quadrovigesimal is a Radix-24 positional notation number system. Radix-24 or Base-24 means that each Quadrovigesimal "digit" represents a multiple of increasing powers of 24, from right to left, such as: ... 24³, 24², 24¹, 24⁰ The valid Quadrovigesimal symbols are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J, K, L, M, N Where: A=10, B=11, C=12, D=13, E=14, F=15, G=16, H=17, I=18, J=19, K=20, L=21, M=22, N=23 For example, the Quadrovigesimal number M8N represents the number 12887 in the Decimal number system, like so: (22 x 24²) + (8 x 24¹) + (23 x 24⁰) History of Quadrovigesimal Systems There is little linguistic or practical use of base-24. However, Umbu-Ungu, a language of Papua New Guinea, is reported to make some use of 24 as a linguistic base for certain larger numbers.

Radix-26

Hexavigesimal or Radix-26 Number System Hexavigesimal is a Radix-26 positional notation number system. Radix-26 or Base-26 means that each Hexavigesimal "digit" represents a multiple of increasing powers of 26, from right to left, such as: ... 26³, 26², 26¹, 26⁰ The valid Hexavigesimal symbols are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P Where: A=10, B=11, C=12, D=13, E=14, F=15, G=16, H=17, I=18, J=19, K=20, L=21, M=22, N=23, O=24, P=25 For example, the Hexavigesimal number PO9 represents the number 17533 in the Decimal number system, like so: (25 x 26²) + (24 x 26¹) + (9 x 26⁰) Applications of Hexavigesimal Systems Microsoft Excel uses a variant of Radix-26 to identify columns in its spreadsheets. The column labels use the following alphabetic series: A B C .. Y Z AA AB AC .. AY AZ BA BB BC .. BZ .. This Radix-26 variant is called "Bijective Base-26 or 26-adic". Its symbols are A, B, C, ... X, Y, Z, which represent decimal 1 to 26. The symbols 0 to 9 are not used. Mathematician John Nash worked extensively with Radix-26, while at Princeton University in the 1970s.

Radix-27

Septemvigesimal or Radix-27 Number System Septemvigesimal is a Radix-27 positional notation number system. Radix-27 or Base-27 means that each Septemvigesimal "digit" represents a multiple of increasing powers of 27, from right to left, such as: ... 27³, 27², 27¹, 27⁰ The valid Septemvigesimal symbols are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q Where: A=10, B=11, C=12, D=13, E=14, F=15, G=16, H=17, I=18, J=19, K=20, L=21, M=22, N=23, O=24, P=25, Q=26 For example, the Septemvigesimal number QPN represents the number 19652 in the Decimal number system, like so: (26 x 27²) + (25 x 27¹) + (23 x 27⁰) Applications of Septemvigesimal Systems A base-27 number system is used in the Telefol language and in the Oksapmin language of Papua New Guinea. Septemvigesimal notation can be used as a precise and efficient representation of Ternary data, where each Septemvigesimal digit perfectly represents three Ternary digits. This is similar to using Octal to represent Binary data, although the Septemvigesimal symbol set is closer in size to the Hexadecimal symbol set.

Radix-30

Trigesimal or Radix-30 Number System Trigesimal is a Radix-30 positional notation number system. Radix-30 or Base-30 means that each Trigesimal "digit" represents a multiple of increasing powers of 30, from right to left, such as: ... 30³, 30², 30¹, 30⁰ The valid Trigesimal symbols are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T Where: A=10, B=11, C=12, D=13, E=14, F=15, G=16, H=17, I=18, J=19, K=20, L=21, M=22, N=23, O=24, P=25, Q=26, R=27, S=28, T=29 For example, the Trigesimal number TSR represents the number 26967 in the Decimal system, like so: (29 x 30²) + (28 x 30¹) + (27 x 30⁰) Applications of Trigesimal Systems From the perspective of mathematics, 30 is a convenient base for a number system because 30 is divisible by the three smallest prime numbers (2, 3, and 5), and by their multiples 6, 10, and 15. The result of this characteristic is that many integer fractions (vulgar fractions) are concisely expressed in base-30. There are no known linguistic applications of base-30 number systems.

Radix-32

Duotrigesimal or Radix-32 Number System Duotrigesimal is a Radix-32 positional notation number system. Radix-32 or Base-32 means that each Duotrigesimal "digit" represents a multiple of increasing powers of 32, from right to left, such as: ... 32³, 32², 32¹, 32⁰ The valid Duotrigesimal symbols are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V Where: A=10, B=11, C=12, D=13, E=14, F=15, G=16, H=17, I=18, J=19, K=20, L=21, M=22, N=23, O=24, P=25, Q=26, R=27, S=28, T=29, U=30, V=31 For example, the Duotrigesimal number VUT represents the number 32733 in the Decimal number system, like so: (31 x 32²) + (30 x 32¹) + (29 x 32⁰) Applications of Duotrigesimal Systems The Ngiti language is reported to have a base-32 numeral system with base-4 cycles. The Ngiti is an ethnolinguistic group located in the Ituri Province of the Democratic Republic of the Congo. "Base32" is a transfer encoding scheme that implements a base-32 number system by using the twenty-six letters A-Z and the six digits 2-7. "Base32" is used in modern computer networks for tranferring data through the network in a compact and very lightly encrypted form.

Radix-36

Hexatridecimal or Radix-36 Number System Hexatridecimal is a Radix-36 positional notation number system. Radix-36 or Base-36 means that each Hexatridecimal "digit" represents a multiple of increasing powers of 36, from right to left, such as: ... 36³, 36², 36¹, 36⁰ The valid Hexatridecimal symbols are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z Where: A=10, B=11, C=12, D=13, E=14, F=15, G=16, H=17, I=18, J=19, K=20, L=21, M=22, N=23, O=24, P=25, Q=26, R=27, S=28, T=29, U=30, V=31, W=32, X=33, Y=34, Z=35 For example, the Hexatridecimal number ZYX represents the number 46617 in the Decimal number system, like so: (35 x 36²) + (34 x 36¹) + (33 x 36⁰) Applications of Hexatridecimal Systems From the perspective of mathematics, 36 is a convenient base for a number system because 36 is divisible by the two smallest prime numbers (2 and 3), and by their multiples 4, 6, 9, 12 and 18. There are many modern computer applications of base-36 systems, generally to encode Decimal numeric information in a more compact form, for later transmission over a network or for storage or both.

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