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For instance, the first counterexample must be odd because f(2n) = n, smaller than 2n; and it must be 3 mod 4 because f 2 (4n + 1) = 3n + 1, smaller than 4n + 1. For each starting value a which is not a counterexample to the Collatz conjecture, there is a k for which such an inequality holds, so checking the Collatz conjecture for one starting ...
In The Republic (509d–510a), Socrates describes the divided line to Glaucon this way: . Now take a line which has been cut into two unequal parts, and divide each of them again in the same proportion, [1] and suppose the two main divisions to answer, one to the visible and the other to the intelligible, and then compare the subdivisions in respect of their clearness and want of clearness ...
Division is one of the four basic operations of arithmetic. The other operations are addition, subtraction, and multiplication. What is being divided is called the dividend, which is divided by the divisor, and the result is called the quotient. At an elementary level the division of two natural numbers is, among other possible interpretations ...
Various scholars also debate the possibility of a connection between the work in the allegory and the cave and the work done by Plato considering the analogy of the divided line and the analogy of the Sun. The divided line is a theory presented to us in Plato's work the Republic. This is displayed through a dialogue given between Socrates and ...
If the car is behind door 2 – with the player having picked door 1 – the host must open door 3, such the probability that the car is behind door 2 and the host opens door 3 is 1 / 3 × 1 = 1 / 3 . These are the only cases where the host opens door 3, so if the player has picked door 1 and the host opens door 3, the car is ...
A variant of the story has been told with 11 camels, to be divided into 1 ⁄ 2, 1 ⁄ 4, and 1 ⁄ 6. [22] [23] Another variant of the puzzle appears in the book The Man Who Counted, a mathematical puzzle book originally published in Portuguese by Júlio César de Mello e Souza in 1938. This version starts with 35 camels, to be divided in the ...
A simple fraction (also known as a common fraction or vulgar fraction, where vulgar is Latin for "common") is a rational number written as a / b or , where a and b are both integers. [ 9] As with other fractions, the denominator ( b) cannot be zero. Examples include 1 2 , − 8 5 , −8 5 , and 8 −5 .
Line clipping. In computer graphics, line clipping is the process of removing ( clipping) lines or portions of lines outside an area of interest (a viewport or view volume ). Typically, any part of a line which is outside of the viewing area is removed. There are two common algorithms for line clipping: Cohen–Sutherland and Liang–Barsky.