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In more generality: For all p ≥ 1 and odd h, f p − 1 (2 p h − 1) = 2 × 3 p − 1 h − 1. (Here f p − 1 is function iteration notation.) For all odd h, f(2h − 1) ≤ 3h − 1 / 2; The Collatz conjecture is equivalent to the statement that, for all k in I, there exists an integer n ≥ 1 such that f n (k) = 1. Undecidable generalizations
Unit fractions can also be expressed using negative exponents, as in 2 −1, which represents 1/2, and 2 −2, which represents 1/(2 2) or 1/4. A dyadic fraction is a common fraction in which the denominator is a power of two, e.g. 1 / 8 = 1 / 2 3. In Unicode, precomposed fraction characters are in the Number Forms block. Proper and improper ...
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 ...
One half is a rational number that lies midway between nil and unity (which are the elementary additive and multiplicative identities) as the quotient of the first two non-zero integers, . It has two different decimal representations in base ten, the familiar and the recurring , with a similar pair of expansions in any even base; while in odd ...
The 1/3–2/3 conjecture was formulated by Sergey Kislitsyn in 1968, [5] and later made independently by Michael Fredman [6] and by Nati Linial in 1984. [3] It was listed as a featured unsolved problem at the founding of the journal Order, and remains unsolved; being called "one of the most intriguing problems in the combinatorial theory of ...
Mathematical fallacy. In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known ...
Other symbols for division include the slash or solidus /, the colon:, and the fraction bar (the horizontal bar in a vertical fraction). [3] [4] The ISO 80000-2 standard for mathematical notation recommends only the solidus / or "fraction bar" for division, or the "colon" : for ratios ; it says that the ÷ sign "should not be used" for division.
For power-of-2 integer division, a simple binary counter can be used, clocked by the input signal. The least-significant output bit alternates at 1/2 the rate of the input clock, the next bit at 1/4 the rate, the third bit at 1/8 the rate, etc. An arrangement of flipflops is a classic method for integer-n division. Such division is frequency ...