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List Of Logarithmic Identities
In mathematics, many logarithmic identities exist. The following is a compilation of the notable of these, many of which are used for computational purposes. Trivial identities ''Trivial'' mathematical identities are relatively simple (for an experienced mathematician), though not necessarily unimportant. The trivial logarithmic identities are as follows: Explanations By definition, we know that: \log_b(y) = x \iff b^x = y, where b \neq 0 and b \neq 1. Setting x = 0, we can see that: b^x = y \iff b^ = y \iff 1 = y \iff y = 1 So, substituting these values into the formula, we see that: \log_b (y) = x \iff \log_b (1) = 0, which gets us the first property. Setting x = 1, we can see that: b^x = y \iff b^ = y \iff b = y \iff y = b So, substituting these values into the formula, we see that: \log_b (y) = x \iff \log_b (b) = 1, which gets us the second property. Cancelling exponentials Logarithms and exponentials with the same base cancel each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Number Theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet ''L''-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem). Branches of analytic number theory Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. * Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. *Additive numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Telescoping Series
In mathematics, a telescoping series is a series whose general term t_n is of the form t_n=a_-a_n, i.e. the difference of two consecutive terms of a sequence (a_n). As a consequence the partial sums of the series only consists of two terms of (a_n) after cancellation. The cancellation technique, with part of each term cancelling with part of the next term, is known as the method of differences. An early statement of the formula for the sum or partial sums of a telescoping series can be found in a 1644 work by Evangelista Torricelli, ''De dimensione parabolae''. Definition Telescoping sums are finite sums in which pairs of consecutive terms partly cancel each other, leaving only parts of the initial and final terms. Let a_n be the elements of a sequence of numbers. Then \sum_^N \left(a_n - a_\right) = a_N - a_0. If a_n converges to a limit L, the telescoping series gives: \sum_^\infty \left(a_n - a_\right) = L-a_0. Every series is a telescoping series of its own parti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Series Test
In mathematical analysis, the alternating series test proves that an alternating series is convergent when its terms decrease monotonically in absolute value and approach zero in the limit. The test was devised by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. The test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test. For a generalization, see Dirichlet's test. History Leibniz discussed the criterion in his unpublished ''De quadratura arithmetica'' of 1676 and shared his result with Jakob Hermann in June 1705 and with Johann Bernoulli in October, 1713. It was only formally published in 1993. Formal statement Alternating series test A series of the form \sum_^\infty (-1)^ a_n = a_0-a_1 + a_2 - a_3 + \cdots where either all ''a''''n'' are positive or all ''a''''n'' are negative, is called an alternating series. The alternating series test guarantees that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Ideal
In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P generated by a single element x \in P, which is to say the set of all elements less than or equal to x in P. The remainder of this article addresses the ring-theoretic concept. Definitions * A ''left principal ideal'' of R is a subset of R given by Ra = \ for some element a. * A ''right principal ideal'' of R is a subset of R given by aR = \ for some element a. * A ''two-sided principal ideal'' of R is a subset of R given by RaR = \ for some element a, namely, the set of all finite sums of elements of the form ras. While the definition for two-sided principal ideal may seem more complicated than for the one-sided principal ideals, it is necessary to ensure that the ideal remains closed under ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Series (mathematics)
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: \sum_^\infty\frac = 1 + \frac + \frac + \frac + \frac + \cdots. The first n terms of the series sum to approximately \ln n + \gamma, where \ln is the natural logarithm and \gamma\approx0.577 is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it is a divergent series. Its divergence was proven in the 14th century by Nicole Oresme using a precursor to the Cauchy condensation test for the convergence of infinite series. It can also be proven to diverge by comparing the sum to an integral, according to the integral test for convergence. Applications of the harmonic series and its partial sums include Divergence of the sum of the reciprocals of the primes, Euler's proof that there are infinitely many prime numbers, the analysis of the coupon collector's problem on how many random trials are nee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Number Difference
In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st harmonic''; the other harmonics are known as ''higher harmonics''. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also periodic at that frequency. The set of harmonics forms a '' harmonic series''. The term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio technology, and other fields. For example, if the fundamental frequency is 50 Hz, a common AC power supply frequency, the frequencies of the first three higher harmonics are 100 Hz (2nd harmonic), 150 Hz (3rd harmonic), 200 Hz (4th harmonic) and any addition of waves with these frequencies is periodic at 50 Hz. In music, harmonics are used on string instruments and w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Sum
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is in numerical integration, i.e., approximating the area of functions or lines on a graph, where it is also known as the rectangle rule. It can also be applied for approximating the length of curves and other approximations. The sum is calculated by partitioning the region into shapes (rectangles, trapezoids, parabolas, or cubics—sometimes infinitesimally small) that together form a region that is similar to the region being measured, then calculating the area for each of these shapes, and finally adding all of these small areas together. This approach can be used to find a numerical approximation for a definite integral even if the fundamental theorem of calculus does not make it easy to find a closed-form solution. Because the region by the small shapes is usually not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Test For Convergence
In mathematics, the integral test for convergence is a method used to test infinite series of monotonic terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test. Statement of the test Consider an integer and a function defined on the unbounded interval , on which it is monotone decreasing. Then the infinite series :\sum_^\infty f(n) converges to a real number if and only if the improper integral :\int_N^\infty f(x)\,dx is finite. In particular, if the integral diverges, then the series diverges as well. Remark If the improper integral is finite, then the proof also gives the lower and upper bounds for the infinite series. Note that if the function f(x) is increasing, then the function -f(x) is decreasing and the above theorem applies. Many textbooks require the function f to be positive, but this condition is not really necessary, since when f is negative and decreasing both \sum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Constant
Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limit of a sequence, limiting difference between the harmonic series (mathematics), harmonic series and the natural logarithm, denoted here by : \begin \gamma &= \lim_\left(-\log n + \sum_^n \frac1\right)\\[5px] &=\int_1^\infty\left(-\frac1x+\frac1\right)\,\mathrm dx. \end Here, represents the floor and ceiling functions, floor function. The numerical value of Euler's constant, to 50 Decimal Places, decimal places, is: History The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled ''De Progressionibus harmonicis observationes'' (Eneström Index 43), where he described it as "worthy of serious consideration". Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Euler used the notations and for the cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Number
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \dots Harmonic numbers are related to the harmonic mean in that the -th harmonic number is also times the reciprocal of the harmonic mean of the first positive integers. Harmonic numbers have been studied since antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions. The harmonic numbers roughly approximate the natural logarithm function and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. His work was extended into the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
