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For example, 2 3 = 8 ≠ 3 2 = 9. Also unlike addition and multiplication, exponentiation is not associative. For example, (2 3) 2 = 8 2 = 64, whereas 2 (3 2) = 2 9 = 512. Without parentheses, the conventional order of operations for serial exponentiation in superscript notation is top-down (or right-associative), not bottom-up [22] [23] [24 ...
In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix. Some variants are commonly referred to as square-and-multiply algorithms or binary exponentiation.
Modular exponentiation is the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus); that is, c = be mod m. From the definition of division, it follows that 0 ≤ c < m. For example, given b = 5, e = 3 and m = 13, dividing 53 = 125 by 13 leaves a remainder of c = 8.
Addition-chain exponentiation. In mathematics and computer science, optimal addition-chain exponentiation is a method of exponentiation by a positive integer power that requires a minimal number of multiplications. Using the form of the shortest addition chain, with multiplication instead of addition, computes the desired exponent (instead of ...
Horner's method evaluates a polynomial using repeated bracketing: + + + + + = + (+ (+ (+ + (+)))). This method reduces the number of multiplications and additions to just Horner's method is so common that a computer instruction "multiply–accumulate operation" has been added to many computer processors, which allow doing the addition and multiplication operations in one combined step.
Bernoulli's inequality. An illustration of Bernoulli's inequality, with the graphs of and shown in red and blue respectively. Here, In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of . It is often employed in real analysis. It has several useful variants: [1]
When the right factor β = 0, ordinary multiplication gives α · 0 = 0 for any α. For β > 0, the value of α · β is the smallest ordinal greater than or equal to (α · δ) + α for all δ < β. Writing the successor and limit ordinals cases separately: α · 0 = 0. α · S(β) = (α · β) + α, for a successor ordinal S(β).
In 2022, Python 3.10.4 and 3.9.12 were expedited [59] and 3.8.13, because of many security issues. [60] When Python 3.9.13 was released in May 2022, it was announced that the 3.9 series (joining the older series 3.8 and 3.7) would only receive security fixes in the future. [61]