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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 ...
Problems 1–6 compute divisions of a certain number of loaves of bread by 10 men and record the outcome in unit fractions. Problems 7–20 show how to multiply the expressions 1 + 1/2 + 1/4 = 7/4, and 1 + 2/3 + 1/3 = 2 by different fractions. Problems 21–23 are problems in completion, which in modern notation are simply subtraction problems.
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 ...
v. t. e. In celestial mechanics, the Lagrange points ( / ləˈɡrɑːndʒ /; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the gravitational influence of two massive orbiting bodies. Mathematically, this involves the solution of the restricted three-body problem. [ 1]
The analogy of the Sun (or simile of the Sun or metaphor of the Sun) is found in the sixth book of The Republic (507b–509c), written by the Greek philosopher Plato as a dialogue between his brother Glaucon and Socrates, and narrated by the latter. Upon being urged by Glaucon to define goodness, a cautious Socrates professes himself incapable ...
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.