A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
has a base of twenty-six.
Base-26 may be represented by using conventional numerals for the digits 0 to 9, and then the letters A to P for the tenth to twenty-fifth digits. "10" would represent 26, "11" = 27, "AB" = 271 and "NP" = 623.
Alternatively, base-26 may be represented using only letters of the Latin alphabet
Latin alphabet
The Latin alphabet, also called the Roman alphabet, is the most recognized alphabet used in the world today. It evolved from a western variety of the Greek alphabet called the Cumaean alphabet, which was adopted and modified by the Etruscans who ruled early Rome...
English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...
, base-26 is also the highest base in which this is possible and hence utilizes every letter. 0 is represented by A, 1 = B, 2 = C ... 24 = Y, 25 = Z. Some examples: 26 = BA, 678 = BAC.
These systems are of limited practical value, although letters used in nominal
Nominal number
Nominal numbers are numerals used for identification only. The numerical value is irrelevant, and they do not indicate quantity, rank, or any other measurement.-Definition:...
26 is the natural number following 25 and preceding 27.- In mathematics :26 is the only positive integer that is one greater than a square and one less than a cube .A rhombicuboctahedron has twenty-six sides....
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
25 is the natural number following 24 and preceding 26.-In mathematics:It is a square number, being 5² = 5 × 5. It is the smallest square that is also a sum of two squares: 25 = 3² + 4²...
27 is the natural number following 26 and preceding 28.- In mathematics :Twenty-seven is a perfect cube, being 33 = 3 × 3 × 3. 27 is also 23 . There are exactly 27 straight lines on a smooth cubic surface, which give a basis of the fundamental representation of the E6 Lie algebra...
The fractions B/G, B/I, B/J, B/K, B/M, B/N, B/P, B/Q are also simple.
Conversion algorithm (alphabet-only system)
Any number may be converted to base-26 by repeatedly dividing the number by 26. The remainder
Remainder
In arithmetic, the remainder is the amount "left over" after the division of two integers which cannot be expressed with an integer quotient....
s of each division will be the base-26 digits from right to left (least-significant to most-significant place). For example, to convert 678 to "BAC", the first division yields 26 remainder 2, so 2 (C) is the last digit. The quotient
Quotient
In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...
26 is divided again, yielding 1 remainder 0, so 0 (A) is the second-last digit. The next quotient 1 is then divided to give 0 remainder 1, so the final digit is 1 (B). This is extensible to fractions.
Java is a programming language originally developed by James Gosling at Sun Microsystems and released in 1995 as a core component of Sun Microsystems' Java platform. The language derives much of its syntax from C and C++ but has a simpler object model and fewer low-level facilities...
to convert a non-negative integer to a base-26 character string as follows:
public static String toBase26(int number){
number = Math.abs(number);
String converted = "";
// Repeatedly divide the number by 26 and convert the
// remainder into the appropriate letter.
do
{
int remainder = number % 26;
converted = (char)(remainder + 'A') + converted;
number = (number - remainder) / 26;
} while (number > 0);
return converted;
}
The reverse conversion is achieved by processing each base-26 digit from left to right. The value of the first (leftmost) digit is multiplied by 26 and then added to the subsequent digit. If digits remain, then the cumulative sum is multiplied by 26 before adding the next digit, and so on. Note that this works for any base as long as one has the tools to perform multiplication by 26 and addition in that base. For example, to convert "BAC" to 678, B (1) is multiplied to give 26 and added to A (0) to yield 26. This is multiplied to give 676 and added to C (2) to yield 678.
The reverse conversion algorithm may be represented in Java
Java (programming language)
Java is a programming language originally developed by James Gosling at Sun Microsystems and released in 1995 as a core component of Sun Microsystems' Java platform. The language derives much of its syntax from C and C++ but has a simpler object model and fewer low-level facilities...
to convert a base-26 character string to an integer as follows:
public static int fromBase26(String number) {
int s = 0;
if (number != null && number.length > 0) {
s = (number.charAt(0) - 'A');
for (int i = 1; i < number.length; i++) {
s *= 26;
s += (number.charAt(i) - 'A');
}
}
return s;
}
John Nash
While at Princeton University in the 1970s, mathematician John Nash worked extensively in base-26 as part of his obsession with numerology
Numerology
Numerology is any study of the purported mystical relationship between a count or measurement and life. It has many systems and traditions and beliefs...
and the uncovering of "hidden" messages. Nash composed letters and other messages in which the actual English text also represented a base-26 equation, the answer to which would itself be both a base-26 number and a word or phrase in English. Nash is believed to have devised algorithms with which he was able to adapt primitive electronic calculators to assist him in performing hexavigesimal calculations.
Bijective base 26
Using bijective numeration, it is possible to operate in base 26 without a zero; it uses digits "A" to "Z" to represent one to twenty-six
26 (number)
26 is the natural number following 25 and preceding 27.- In mathematics :26 is the only positive integer that is one greater than a square and one less than a cube .A rhombicuboctahedron has twenty-six sides....
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
. Many spreadsheets including Microsoft Excel use the 26-adic counting system with the "digits" A-Z to label the columns of a spreadsheet, starting A, B, C... Z, AA, AB... AZ, BA... ZZ, AAA, etc. A variant of this system is used to name variable stars
Variable star designation
Variable stars are named using a variation on the Bayer designation format of an identifying label combined with the Latin genitive of the name of the constellation in which the star lies...
.
The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.
