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Turtle graphics are often associated with the Logo programming language. [2] Seymour Papert added support for turtle graphics to Logo in the late 1960s to support his version of the turtle robot, a simple robot controlled from the user's workstation that is designed to carry out the drawing functions assigned to it using a small retractable pen set into or attached to the robot's body.
In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, [a] which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size.
Since 7 October 2024, Python 3.13 is the latest stable release, and it and, for few more months, 3.12 are the only releases with active support including for bug fixes (as opposed to just for security) and Python 3.9, [55] is the oldest supported version of Python (albeit in the 'security support' phase), due to Python 3.8 reaching end-of-life.
Python: The standard library includes a Fraction class in the module fractions. [6] Ruby: native support using special syntax. Smalltalk represents rational numbers using a Fraction class in the form p/q where p and q are arbitrary size integers. Applying the arithmetic operations *, +, -, /, to fractions returns a reduced fraction. With ...
Python: the built-in int (3.x) / long (2.x) integer type is of arbitrary precision. The Decimal class in the standard library module decimal has user definable precision and limited mathematical operations (exponentiation, square root, etc. but no trigonometric functions). The Fraction class in the module fractions implements rational numbers ...
A fractional program in which f is nonnegative and concave, g is positive and convex, and S is a convex set is called a concave fractional program.If g is affine, f does not have to be restricted in sign.
The first working Logo turtle robot was created in 1969. A display turtle preceded the physical floor turtle. Modern Logo has not changed very much from the basic concepts predating the first turtle. The first turtle was a tethered floor roamer, not radio-controlled or wireless. At BBN Paul Wexelblat developed a turtle named Irving that had ...
In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis.The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain ...