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The tortoise, with a 10-meter advantage, Zeno argued, would win. Achilles would have to move 10 meters to catch up to the tortoise, but the tortoise would already have moved another five meters by then. Achilles would then have to move 5 meters, where the tortoise would move 2.5 meters, and so on.
Today, a more standard phrasing of Archimedes' proposition is that the partial sums of the series 1 + 1 / 4 + 1 / 16 + ⋯ are: + + + + = +. This form can be proved by multiplying both sides by 1 − 1 / 4 and observing that all but the first and the last of the terms on the left-hand side of the equation cancel in pairs.
Demonstration of 2 / 3 via a zero-value game. A slight rearrangement of the series reads + + =. The series has the form of a positive integer plus a series containing every negative power of two with either a positive or negative sign, so it can be translated into the infinite blue-red Hackenbush string that represents the surreal number 1 / 3 :
The function () = (), defined for all real except for −1 and 1, is unbounded. As x {\displaystyle x} approaches −1 or 1, the values of this function get larger in magnitude. This function can be made bounded if one restricts its domain to be, for example, [ 2 , ∞ ) {\displaystyle [2,\infty )} or ( − ∞ , − 2 ] {\displaystyle (-\infty ...
An unmanned M-51 intercontinental cruise missile variant was developed, which would have delivered multiple warheads on targets in the contiguous United States. [ 4 ] Like most of the early 1960s supersonic strategic bomber projects, the M-50/52 program was terminated due to the development of intercontinental ballistic missiles and the ...
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
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where C is the circumference of a circle, d is the diameter, and r is the radius.More generally, = where L and w are, respectively, the perimeter and the width of any curve of constant width.