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Number theory was once called arithmetic, but nowadays this term is mostly used for numerical calculations. [15] Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. [ 16 ]
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
The quotient division is generally used instead of the division method; at the same time, in order to make the multiplication and division digits consistently use the division multiplication. Later, Japan had a 3:5 abacus called 天三算盤, which is now in the Ize Rongji collection of Shansi Village in Yamagata City. Japan also used a 2:5 ...
Division and square roots took 13(d+1) cycles, where d is the number of digits in the result (quotient or square root). So a division or square root took up to 143 cycles, or 28,600 microseconds—a rate of 35 per second. (Wilkes 1956:20 [21] states that a division with a 10-digit quotient required 6 milliseconds.) If the result had fewer than ...
These are objects which can be treated like real numbers but which are, in some sense, "infinitely small". For example, an infinitesimal number could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, ... and thus less than any positive real number. From this point of view, calculus is a collection of techniques for ...
For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3 rd power: 1000 = 10 3 = 10 × 10 × 10. More generally, if x = b y, then y is the logarithm of x to base b, written log b x, so log 10 1000 = 3. As a single-variable function, the logarithm to base b is the inverse of exponentiation with base b.
While virtually all real numbers k will eventually have infinitely many convergents m / n whose distance from k is significantly smaller than this limit, the convergents for φ (i.e., the numbers 5 / 3 , 8 / 5 , 13 / 8 , 21 / 13 , etc.) consistently "toe the boundary", keeping a distance of almost exactly ...
A solubility equilibrium exists when a chemical compound in the solid state is in chemical equilibrium with a solution containing the compound. This type of equilibrium is an example of dynamic equilibrium in that some individual molecules migrate between the solid and solution phases such that the rates of dissolution and precipitation are equal to one another.