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The two types of calculus in quantum calculus are q-calculus and h-calculus. The goal of both types is to find "analogs" of mathematical objects, where, after taking a certain limit, the original object is returned. In q-calculus, the limit as q tends to 1 is taken of the q-analog.
For example, the equation z 2 + 1 = 0, has infinitely many quaternion solutions, which are the quaternions z = b i + c j + d k such that b 2 + c 2 + d 2 = 1. Thus these "roots of –1" form a unit sphere in the three-dimensional space of vector quaternions.
The distance between the base of the ladder and the wall, x, and the height of the ladder on the wall, y, represent the sides of a right triangle with the ladder as the hypotenuse, h. The objective is to find dy/dt, the rate of change of y with respect to time, t, when h, x and dx/dt, the rate of change of x, are known. Step 1: =
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. [1] In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
An example of calculus of relations arises in erotetics, the theory of questions. In the universe of utterances there are statements S and questions Q. There are two relations π and α from Q to S: q α a holds when a is a direct answer to question q. The other relation, q π p holds when p is a presupposition of question q.
Examples of proper fractions are 2/3, –3/4, and 4/9; examples of improper fractions are 9/4, –4/3, and 3/3. improper integral In mathematical analysis , an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number , ∞ {\displaystyle \infty } , − ∞ ...
In mathematics, in the area of combinatorics and quantum calculus, the q-derivative, or Jackson derivative, is a q-analog of the ordinary derivative, introduced by Frank Hilton Jackson. It is the inverse of Jackson's q-integration. For other forms of q-derivative, see Chung et al. (1994).
For example, the solutions to the quadratic Diophantine equation x 2 + y 2 = z 2 are given by the Pythagorean triples , originally solved by the Babylonians ( c. 1800 BC ). [ 2 ] Solutions to linear Diophantine equations, such as 26 x + 65 y = 13, may be found using the Euclidean algorithm (c. 5th century BC).
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