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Large numbers are numbers above one million that are usually represented either with the use of an exponent such as 109 or by terms such as billion or thousand millions that frequently differ from system to system. The American system of numeration for denominations above one million was modeled on.
- The Editors of Encyclopaedia Britannica
- Overview
- Properties of logarithms
- History of logarithms
logarithm, the exponent or power to which a base must be raised to yield a given number. Expressed mathematically, x is the logarithm of n to the base b if bx = n, in which case one writes x = logb n. For example, 23 = 8; therefore, 3 is the logarithm of 8 to base 2, or 3 = log2 8. In the same fashion, since 102 = 100, then 2 = log10 100. Logarithms of the latter sort (that is, logarithms with base 10) are called common, or Briggsian, logarithms and are written simply log n.
Invented in the 17th century to speed up calculations, logarithms vastly reduced the time required for multiplying numbers with many digits. They were basic in numerical work for more than 300 years, until the perfection of mechanical calculating machines in the late 19th century and computers in the 20th century rendered them obsolete for large-scale computations. The natural logarithm (with base e ≅ 2.71828 and written ln n), however, continues to be one of the most useful functions in mathematics, with applications to mathematical models throughout the physical and biological sciences.
Logarithms were quickly adopted by scientists because of various useful properties that simplified long, tedious calculations. In particular, scientists could find the product of two numbers m and n by looking up each number’s logarithm in a special table, adding the logarithms together, and then consulting the table again to find the number with that calculated logarithm (known as its antilogarithm). Expressed in terms of common logarithms, this relationship is given by log mn = log m + log n. For example, 100 × 1,000 can be calculated by looking up the logarithms of 100 (2) and 1,000 (3), adding the logarithms together (5), and then finding its antilogarithm (100,000) in the table. Similarly, division problems are converted into subtraction problems with logarithms: log m/n = log m − log n. This is not all; the calculation of powers and roots can be simplified with the use of logarithms. Logarithms can also be converted between any positive bases (except that 1 cannot be used as the base since all of its powers are equal to 1), as shown in the Click Here to see full-size tabletable of logarithmic laws.
Only logarithms for numbers between 0 and 10 were typically included in logarithm tables. To obtain the logarithm of some number outside of this range, the number was first written in scientific notation as the product of its significant digits and its exponential power—for example, 358 would be written as 3.58 × 102, and 0.0046 would be written as 4.6 × 10−3. Then the logarithm of the significant digits—a decimal fraction between 0 and 1, known as the mantissa—would be found in a table. For example, to find the logarithm of 358, one would look up log 3.58 ≅ 0.55388. Therefore, log 358 = log 3.58 + log 100 = 0.55388 + 2 = 2.55388. In the example of a number with a negative exponent, such as 0.0046, one would look up log 4.6 ≅ 0.66276. Therefore, log 0.0046 = log 4.6 + log 0.001 = 0.66276 − 3 = −2.33724.
The invention of logarithms was foreshadowed by the comparison of arithmetic and geometric sequences. In a geometric sequence each term forms a constant ratio with its successor; for example, …1/1,000, 1/100, 1/10, 1, 10, 100, 1,000… has a common ratio of 10. In an arithmetic sequence each successive term differs by a constant, known as the common difference; for example, …−3, −2, −1, 0, 1, 2, 3… has a common difference of 1. Note that a geometric sequence can be written in terms of its common ratio; for the example geometric sequence given above: …10−3, 10−2, 10−1, 100, 101, 102, 103…. Multiplying two numbers in the geometric sequence, say 1/10 and 100, is equal to adding the corresponding exponents of the common ratio, −1 and 2, to obtain 101 = 10. Thus, multiplication is transformed into addition. The original comparison between the two series, however, was not based on any explicit use of the exponential notation; this was a later development. In 1620 the first table based on the concept of relating geometric and arithmetic sequences was published in Prague by the Swiss mathematician Joost Bürgi.
The Scottish mathematician John Napier published his discovery of logarithms in 1614. His purpose was to assist in the multiplication of quantities that were then called sines. The whole sine was the value of the side of a right-angled triangle with a large hypotenuse. (Napier’s original hypotenuse was 107.) His definition was given in terms of relative rates.
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The logarithme, therefore, of any sine is a number very neerely expressing the line which increased equally in the meene time whiles the line of the whole sine decreased proportionally into that sine, both motions being equal timed and the beginning equally shift.
In cooperation with the English mathematician Henry Briggs, Napier adjusted his logarithm into its modern form. For the Naperian logarithm the comparison would be between points moving on a graduated straight line, the L point (for the logarithm) moving uniformly from minus infinity to plus infinity, the X point (for the sine) moving from zero to infinity at a speed proportional to its distance from zero. Furthermore, L is zero when X is one and their speed is equal at this point. The essence of Napier’s discovery is that this constitutes a generalization of the relation between the arithmetic and geometric series; i.e., multiplication and raising to a power of the values of the X point correspond to addition and multiplication of the values of the L point, respectively. In practice it is convenient to limit the L and X motion by the requirement that L = 1 at X = 10 in addition to the condition that X = 1 at L = 0. This change produced the Briggsian, or common, logarithm.
E, mathematical constant that is the base of the natural logarithm function f (x) = ln x and of its related inverse, the exponential function y = ex. To five decimal places, the value used for the constant is 2.71828. The number e is an irrational number; that is, it cannot be expressed as the ratio.
2024年6月27日 · computer science, the study of computers and computing, including their theoretical and algorithmic foundations, hardware and software, and their uses for processing information. The discipline of computer science includes the study of algorithms and data structures, computer and network design, modeling data and information ...
2024年6月17日 · trigonometry, the branch of mathematics concerned with specific functions of angles and their application to calculations. There are six functions of an angle commonly used in trigonometry. Their names and abbreviations are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).
Periodic table, in chemistry, the organized array of all the chemical elements in order of increasing atomic number. When the elements are thus arranged, there is a recurring pattern called the ‘periodic law’ in their properties, in which elements in the same column (group) have similar properties.
momentum, product of the mass of a particle and its velocity. Momentum is a vector quantity; i.e., it has both magnitude and direction. Isaac Newton ’s second law of motion states that the time rate of change of momentum is equal to the force acting on the particle. See Newton’s laws of motion.
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