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Narcissistic Number
In number theory, a narcissistic number 1 F_ : \mathbb \rightarrow \mathbb to be the following: : F_(n) = \sum_^ d_i^k. where k = \lfloor \log_ \rfloor + 1 is the number of digits in the number in base b, and : d_i = \frac is the value of each digit of the number. A natural number n is a narcissistic number if it is a fixed point for F_, which occurs if F_(n) = n. The natural numbers 0 \leq n 0: digit_count = digit_count + 1 y = y // b total = 0 while x > 0: total = total + pow(x % b, digit_count) x = x // b return total def ppdif_cycle(x, b): seen = [] while x not in seen: seen.append(x) x = ppdif(x, b) cycle = [] while x not in cycle: cycle.append(x) x = ppdif(x, b) return cycle The following Python program determines whether the integer entered is a Narcissistic / Armstrong number or not. def no_of_digits(num): i = 0 while num > 0: num //= 10 i+=1 retu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-7
There are many different numeral systems, that is, writing systems for expressing numbers. By culture / time period By type of notation Numeral systems are classified here as to whether they use positional notation (also known as place-value notation), and further categorized by radix or base. Standard positional numeral systems The common names are derived somewhat arbitrarily from a mix of Latin and Greek, in some cases including roots from both languages within a single name. There have been some proposals for standardisation. Non-standard positional numeral systems Bijective numeration Signed-digit representation Negative bases The common names of the negative base numeral systems are formed using the prefix ''nega-'', giving names such as: Complex bases Non-integer bases ''n''-adic number Mixed radix * Factorial number system * Even double factorial number system * Odd double factorial number system * Primorial number system * Fibono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer (computer Science)
In computer science, an integer is a datum of integral data type, a data type that represents some range of mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain negative values. Integers are commonly represented in a computer as a group of binary digits (bits). The size of the grouping varies so the set of integer sizes available varies between different types of computers. Computer hardware nearly always provides a way to represent a processor register or memory address as an integer. Value and representation The ''value'' of an item with an integral type is the mathematical integer that it corresponds to. Integral types may be ''unsigned'' (capable of representing only non-negative integers) or ''signed'' (capable of representing negative integers as well). An integer value is typically specified in the source code of a program as a sequence of digits optionally prefixed with + or −. Some programming languages allow oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Python (programming Language)
Python is a high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation. Python is dynamically-typed and garbage-collected. It supports multiple programming paradigms, including structured (particularly procedural), object-oriented and functional programming. It is often described as a "batteries included" language due to its comprehensive standard library. Guido van Rossum began working on Python in the late 1980s as a successor to the ABC programming language and first released it in 1991 as Python 0.9.0. Python 2.0 was released in 2000 and introduced new features such as list comprehensions, cycle-detecting garbage collection, reference counting, and Unicode support. Python 3.0, released in 2008, was a major revision that is not completely backward-compatible with earlier versions. Python 2 was discontinued with version 2.7.18 in 2020. Python consistently ranks as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle Detection
In computer science, cycle detection or cycle finding is the algorithmic problem of finding a cycle in a sequence of iterated function values. For any function that maps a finite set to itself, and any initial value in , the sequence of iterated function values : x_0,\ x_1=f(x_0),\ x_2=f(x_1),\ \dots,\ x_i=f(x_),\ \dots must eventually use the same value twice: there must be some pair of distinct indices and such that . Once this happens, the sequence must continue periodically, by repeating the same sequence of values from to . Cycle detection is the problem of finding and , given and . Several algorithms for finding cycles quickly and with little memory are known. Robert W. Floyd's tortoise and hare algorithm moves two pointers at different speeds through the sequence of values until they both point to equal values. Alternatively, Brent's algorithm is based on the idea of exponential search. Both Floyd's and Brent's algorithms use only a constant number of memory ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signed-digit Representation
In mathematical notation for numbers, a signed-digit representation is a positional numeral system with a set of signed digits used to encode the integers. Signed-digit representation can be used to accomplish fast addition of integers because it can eliminate chains of dependent carries. In the binary numeral system, a special case signed-digit representation is the '' non-adjacent form'', which can offer speed benefits with minimal space overhead. History Challenges in calculation stimulated early authors Colson (1726) and Cauchy (1840) to use signed-digit representation. The further step of replacing negated digits with new ones was suggested by Selling (1887) and Cajori (1928). In 1928, Florian Cajori noted the recurring theme of signed digits, starting with Colson (1726) and Cauchy (1840). In his book ''History of Mathematical Notations'', Cajori titled the section "Negative numerals". For completeness, Colson uses examples and describes addition (pp. 163–4), mult ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-16
In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, hexadecimal uses 16 distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9, and "A"–"F" (or alternatively "a"–"f") to represent values from 10 to 15. Software developers and system designers widely use hexadecimal numbers because they provide a human-friendly representation of binary-coded values. Each hexadecimal digit represents four bits (binary digits), also known as a nibble (or nybble). For example, an 8-bit byte can have values ranging from 00000000 to 11111111 in binary form, which can be conveniently represented as 00 to FF in hexadecimal. In mathematics, a subscript is typically used to specify the base. For example, the decimal value would be expressed in hexadecimal as . In programming, a number of no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-15
There are many different numeral systems, that is, writing systems for expressing numbers. By culture / time period By type of notation Numeral systems are classified here as to whether they use positional notation (also known as place-value notation), and further categorized by radix or base. Standard positional numeral systems The common names are derived somewhat arbitrarily from a mix of Latin and Greek, in some cases including roots from both languages within a single name. There have been some proposals for standardisation. Non-standard positional numeral systems Bijective numeration Signed-digit representation Negative bases The common names of the negative base numeral systems are formed using the prefix ''nega-'', giving names such as: Complex bases Non-integer bases ''n''-adic number Mixed radix * Factorial number system * Even double factorial number system * Odd double factorial number system * Primorial number system * Fibono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-14
There are many different numeral systems, that is, writing systems for expressing numbers. By culture / time period By type of notation Numeral systems are classified here as to whether they use positional notation (also known as place-value notation), and further categorized by radix or base. Standard positional numeral systems The common names are derived somewhat arbitrarily from a mix of Latin and Greek, in some cases including roots from both languages within a single name. There have been some proposals for standardisation. Non-standard positional numeral systems Bijective numeration Signed-digit representation Negative bases The common names of the negative base numeral systems are formed using the prefix ''nega-'', giving names such as: Complex bases Non-integer bases ''n''-adic number Mixed radix * Factorial number system * Even double factorial number system * Odd double factorial number system * Primorial number system * Fibono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-13
There are many different numeral systems, that is, writing systems for expressing numbers. By culture / time period By type of notation Numeral systems are classified here as to whether they use positional notation (also known as place-value notation), and further categorized by radix or base. Standard positional numeral systems The common names are derived somewhat arbitrarily from a mix of Latin and Greek, in some cases including roots from both languages within a single name. There have been some proposals for standardisation. Non-standard positional numeral systems Bijective numeration Signed-digit representation Negative bases The common names of the negative base numeral systems are formed using the prefix ''nega-'', giving names such as: Complex bases Non-integer bases ''n''-adic number Mixed radix * Factorial number system * Even double factorial number system * Odd double factorial number system * Primorial number system * Fibono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-12
The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead written as "10" in duodecimal (meaning "1 dozen and 0 units", instead of "1 ten and 0 units"), whereas the digit string "12" means "1 dozen and 2 units" (decimal 14). Similarly, in duodecimal, "100" means "1 gross", "1000" means "1 great gross", and "0.1" means "1 twelfth" (instead of their decimal meanings "1 hundred", "1 thousand", and "1 tenth", respectively). Various symbols have been used to stand for ten and eleven in duodecimal notation; this page uses and , as in hexadecimal, which make a duodecimal count from zero to twelve read 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, , , 10. The Dozenal Societies of America and Great Britain (organisations promoting the use of duodecimal) use turned digits in their published mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-11
The undecimal numeral system (also known as the base-11 numeral system) is a positional numeral system that uses eleven as its base. While no known society counts by elevens, two are purported to have done so: the Māori, one of the two Polynesian peoples of New Zealand, and the Pañgwa (Pangwa), a Bantu-speaking people of Tanzania. The idea of counting by elevens remains of interest for its relation to a traditional method of tally-counting practiced in Polynesia. During the French Revolution, base-11 was briefly mentioned as a possible basis for the reformed system of measurement. Base-11 numerals also appear in the International Standard Book Number system. Alleged use of base-11 in cultural number systems Use by the Māori Conant and Williams For about a century, the idea that Māori counted by elevens was best known from its mention in the writing of the American mathematician Levi Leonard Conant. He identified it as a "mistake" originating with a 19th-century dicti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
