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Arrange the digits 1 to 9 in order so that the first two digits form a multiple of 2, the first three digits form a multiple of 3, the first four digits form a multiple of 4 etc. and finally the entire number is a multiple of 9.
Furthermore, it is clear that even-digits with greater than or equal to 8, [10] and with 9 digits, [11] or odd-digits with greater than or equal to 15 digits [12] have multiple solutions. Although 11-digit and 13-digit numbers have only one solution, it forms a loop of five numbers and a loop of two numbers, respectively. [ 13 ]
There are no self-descriptive numbers in bases 2, 3 or 6. In bases 7 and greater, there is exactly one self-descriptive number: () + + +, which has b−4 instances of the digit 0, two instances of the digit 1, one instance of the digit 2, one instance of digit b – 4, and no instances of any other digits.
By their nature, all numbers expressed in floating-point format are rational numbers with a terminating expansion in the relevant base (for example, a terminating decimal expansion in base-10, or a terminating binary expansion in base-2). Irrational numbers, such as π or √2, or non-terminating rational numbers, must be approximated. The ...
All other four-digit numbers eventually reach 6174 if leading zeros are used to keep the number of digits at 4. For numbers with three identical digits and a fourth digit that is one higher or lower (such as 2111), it is essential to treat 3-digit numbers with a leading zero; for example: 2111 – 1112 = 0999; 9990 – 999 = 8991; 9981 – 1899 ...
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
For example, 6 // 9 == 2 // 3 && typeof (-4 // 9) == Rational {Int64}. [2] Haskell provides a Rational type, which is really an alias for Ratio Integer (Ratio being a polymorphic type implementing rational numbers for any Integral type of numerators and denominators). The fraction is constructed using the % operator. [3] OCaml's Num library ...
All integers are rational, but there are rational numbers that are not integers, such as −2/9. Real numbers (): Numbers that correspond to points along a line. They can be positive, negative, or zero. All rational numbers are real, but the converse is not true. Irrational numbers (): Real numbers that are not rational.