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For example, 10 3 = 1000 and 10 −4 = 0.0001. Exponentiation with base 10 is used in scientific notation to denote large or small numbers. For instance, 299 792 458 m/s (the speed of light in vacuum, in metres per second ) can be written as 2.997 924 58 × 10 8 m/s and then approximated as 2.998 × 10 8 m/s .
If the sum is of the form = ()where ƒ is a smooth function, we could use the Euler–Maclaurin formula to convert the series into an integral, plus some corrections involving derivatives of S(x), then for large values of a you could use "stationary phase" method to calculate the integral and give an approximate evaluation of the sum.
Euler's sum of powers conjecture § k = 3, relating to cubes that can be written as a sum of three positive cubes; Plato's number, an ancient text possibly discussing the equation 3 3 + 4 3 + 5 3 = 6 3; Taxicab number, the smallest integer that can be expressed as a sum of two positive integer cubes in n distinct ways
In these cases, iterated exponential notation is used to express them in base 10. The values containing a decimal point are approximate. Usually, the limit that can be calculated in a numerical calculation program such as Wolfram Alpha is 3↑↑4, and the number of digits up to 3↑↑5 can be expressed.
In mathematics and statistics, sums of powers occur in a number of contexts: . Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities.
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
Any real number can be written in the form m × 10 ^ n in many ways: for example, 350 can be written as 3.5 × 10 2 or 35 × 10 1 or 350 × 10 0. In normalized scientific notation (called "standard form" in the United Kingdom), the exponent n is chosen so that the absolute value of m remains at least one but less than ten ( 1 ≤ | m | < 10 ).
For polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree (sometimes called the total degree) of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial x 2 y 2 + 3x 3 + 4y has degree 4, the same degree as the term x ...