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Golden Ratio Base
Golden ratio base is a non-integer positional numeral system that uses the golden ratio (the irrational number ≈ 1.61803399 symbolized by the Greek letter φ) as its base. It is sometimes referred to as base-φ, golden mean base, phi-base, or, colloquially, phinary. Any non-negative real number can be represented as a base-φ numeral using only the digits 0 and 1, and avoiding the digit sequence "11" – this is called a ''standard form''. A base-φ numeral that includes the digit sequence "11" can always be rewritten in standard form, using the algebraic properties of the base φ — most notably that φ (φ1) + 1 (φ0) = φ2. For instance, 11φ = 100φ. Despite using an irrational number base, when using standard form, all non-negative integers have a unique representation as a terminating (finite) base-φ expansion. The set of numbers which possess a finite base-φ representation is the ring Z .html" ;"title="/a>"> ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-integer Representation
A non-integer representation uses non-integer numbers as the radix, or base, of a positional notation, positional numeral system. For a non-integer radix ''β'' > 1, the value of :x = d_n \dots d_2d_1d_0.d_d_\dots d_ is :\begin x &= \beta^nd_n + \cdots + \beta^2d_2 + \beta d_1 + d_0 \\ &\qquad + \beta^d_ + \beta^d_ + \cdots + \beta^d_. \end The numbers ''d''''i'' are non-negative integers less than ''β''. This is also known as a ''β''-expansion, a notion introduced by and first studied in detail by . Every real number has at least one (possibly infinite) ''β''-expansion. The set (mathematics), set of all ''β''-expansions that have a finite representation is a subset of the ring (mathematics), ring Z[''β'', ''β''−1]. There are applications of ''β''-expansions in coding theory and models of quasicrystals (; ). Construction ''β''-expansions are a generalization of decimal expansions. While infinite decimal expansions are not unique (for example, 1.000... = 0.9 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negative Number
In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset. If a quantity, such as the charge on an electron, may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as ''positive'' and ''negative''. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic. For example, −(−3) = 3 because the opposite of an opposite is the original value. Negative numbers are usually written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of three, and is pronounced "min ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. The addition of two Natural number, whole numbers results in the total amount or ''summation, sum'' of those values combined. The example in the adjacent image shows a combination of three apples and two apples, making a total of five apples. This observation is equivalent to the Expression (mathematics), mathematical expression (that is, "3 ''plus'' 2 is Equality (mathematics), equal to 5"). Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root Of 2
The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. The fraction (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator. Sequence in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places: : History The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of in four sexagesimal figures, , which is accurate to about six ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E (mathematical Constant)
The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series e = \sum\limits_^ \frac = 1 + \frac + \frac + \frac + \cdots. It is also the unique positive number such that the graph of the function has a slope of 1 at . The (natural) exponential function is the unique function that equals its own derivative and satisfies the equation ; hence one can also define as . The natural logarithm, or logarithm to base , is the inverse function to the natural exponential function. The natural logarithm of a number can be defined directly as the area under the curve between and , in which case is the value of for which this area equals one (see image). There are various other characteriz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each successive term can be obtained by multiplying the previous term by 1/2. In general, a geometric series is written as a + ar + ar^2 + ar^3 + ..., where a is the coefficient of each term and r is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the Taylor series, the complex Fourier series, and the matrix exponential. The name geometric series indicates each term is the geometric mean of its two neighboring terms, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighboring terms. The sequence of geometric series term ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Long Division
In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit Hindu-Arabic numerals (Positional notation) that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. It enables computations involving arbitrarily large numbers to be performed by following a series of simple steps. The abbreviated form of long division is called short division, which is almost always used instead of long division when the divisor has only one digit. Chunking (also known as the partial quotients method or the hangman method) is a less mechanical form of long division prominent in the UK which contributes to a more holistic understanding of the division process. While related algorithms have existed since the 12th century, the specific algorithm in modern use was introduced by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root Of 5
The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as: :\sqrt. \, It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are: : . which can be rounded down to 2.236 to within 99.99% accuracy. The approximation (≈ 2.23611) for the square root of five can be used. Despite having a denominator of only 72, it differs from the correct value by less than (approx. ). As of January 2022, its numerical value in decimal has been computed to at least 2,250,000,000,000 digits. Rational approximations The square root of 5 can be expressed as the continued fraction : ; 4, 4, 4, 4, 4,\ldots= 2 + \cfrac 1 . The successive partial evaluations of the continue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each successive term can be obtained by multiplying the previous term by 1/2. In general, a geometric series is written as a + ar + ar^2 + ar^3 + ..., where a is the coefficient of each term and r is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the Taylor series, the complex Fourier series, and the matrix exponential. The name geometric series indicates each term is the geometric mean of its two neighboring terms, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighboring terms. The sequence of geometric series term ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square (algebra)
In mathematics, a square is the result of multiplication, multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as exponentiation, raising to the power 2 (number), 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations ''x''^2 (caret) or ''x''**2 may be used in place of ''x''2. The adjective which corresponds to squaring is ''wikt:quadratic, quadratic''. The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expression (mathematics), expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear function (calculus), linear polynomial is the quadratic polynomial . One of the imp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product of multiplying bases: b^n = \underbrace_. The exponent is usually shown as a superscript to the right of the base. In that case, is called "''b'' raised to the ''n''th power", "''b'' (raised) to the power of ''n''", "the ''n''th power of ''b''", "''b'' to the ''n''th power", or most briefly as "''b'' to the ''n''th". Starting from the basic fact stated above that, for any positive integer n, b^n is n occurrences of b all multiplied by each other, several other properties of exponentiation directly follow. In particular: \begin b^ & = \underbrace_ \\[1ex] & = \underbrace_ \times \underbrace_ \\[1ex] & = b^n \times b^m \end In other words, when multiplying a base raised to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
