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Namely, an attacker observing the sequence of squarings and multiplications can (partially) recover the exponent involved in the computation. This is a problem if the exponent should remain secret, as with many public-key cryptosystems. A technique called "Montgomery's ladder" [2] addresses this concern.
Plain text, programming languages, and calculators also use a single asterisk to represent the multiplication symbol, [6] and it must be explicitly used; for example, 3x is written as 3 * x. Rather than using the ambiguous division sign (÷), [ a ] division is usually represented with a vinculum , a horizontal line, as in 3 / x + 1 .
It addition to standard features such as trigonometric functions, exponents, logarithm, and intelligent order of operations found in TI-30 and TI-34 series of calculators, it also include base (decimal, hexadecimal, octal, binary) calculations, complex values, statistics. Conversions include polar-rectangular coordinates (P←→R), angles.
For example, on a simple calculator, typing 1 + 2 × 3 = yields 9, while a more sophisticated calculator will use a more standard priority, so typing 1 + 2 × 3 = yields 7. Calculators may associate exponents to the left or to the right.
This pen-and-paper method uses the same algorithm as polynomial long division, but mental calculation is used to determine remainders. This requires less writing, and can therefore be a faster method once mastered. The division is at first written in a similar way as long multiplication with the dividend at the top, and the divisor below it.
A simple arithmetic calculator was first included with Windows 1.0. [5]In Windows 3.0, a scientific mode was added, which included exponents and roots, logarithms, factorial-based functions, trigonometry (supports radian, degree and gradians angles), base conversions (2, 8, 10, 16), logic operations, statistical functions such as single variable statistics and linear regression.
In abstract algebra, given a magma with binary operation ∗ (which could nominally be termed multiplication), left division of b by a (written a \ b) is typically defined as the solution x to the equation a ∗ x = b, if this exists and is unique. Similarly, right division of b by a (written b / a) is the solution y to the equation y ∗ a = b ...
An orange that has been sliced into two halves. In mathematics, division by two or halving has also been called mediation or dimidiation. [1] The treatment of this as a different operation from multiplication and division by other numbers goes back to the ancient Egyptians, whose multiplication algorithm used division by two as one of its fundamental steps. [2]