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In mathematics, Minkowski's question-mark function, denoted ?(x), is a function with unusual fractal properties, defined by Hermann Minkowski in 1904. [1] It maps quadratic irrational numbers to rational numbers on the unit interval , via an expression relating the continued fraction expansions of the quadratics to the binary expansions of the ...
The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7. In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9. Hence, every triangular number is either divisible by three or has a ...
[4] [3] However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135): [11] Variations of two earlier meters [is the variation] ... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens.
Sieve of Eratosthenes: algorithm steps for primes below 121 (including optimization of starting from prime's square). In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit.
Display questions are more directive than authentic questions, and they promote greater ability in thinking by spurring students to have to back up their contribution. Utilising display questions that build on previous statements made by the students in a rephrased or simplified form facilitates the production of a more elaborate dialogue. [10]
A grade 8 is also equivalent to an A*, however the grade 9 is the top end of the A*. The former C grade is set at grade 4 (known as a 'standard pass') and grade 5 (considered a 'strong pass') under the numerical scheme. Although fewer qualifications have tiered examinations than before, the tiering system still exists.