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Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
The most significant digit (10) is "dropped": 10 1 0 11 <- Digits of 0xA10B ----- 10 Then we multiply the bottom number from the source base (16), the product is placed under the next digit of the source value, and then add: 10 1 0 11 160 ----- 10 161 Repeat until the final addition is performed: 10 1 0 11 160 2576 41216 ----- 10 161 2576 41227 ...
Long division of integers can easily be extended to include non-integer dividends, as long as they are rational. This is because every rational number has a recurring decimal expansion. The procedure can also be extended to include divisors which have a finite or terminating decimal expansion (i.e. decimal fractions).
For instance, Fibonacci represents the fraction 8 / 11 by splitting the numerator into a sum of two numbers, each of which divides one plus the denominator: 8 / 11 = 6 / 11 + 2 / 11 . Fibonacci applies the algebraic identity above to each these two parts, producing the expansion 8 / 11 = 1 / 2 ...
The Archimedean property: any point x before the finish line lies between two of the points P n (inclusive).. It is possible to prove the equation 0.999... = 1 using just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series and limits.
Dirac delta function: everywhere zero except for x = 0; total integral is 1. Not a function but a distribution, but sometimes informally referred to as a function, particularly by physicists and engineers. Dirichlet function: is an indicator function that matches 1 to rational numbers and 0 to irrationals. It is nowhere continuous.
In the following example of conversion from non-standard to standard form, the notation 1 is used to represent the signed digit −1.. 211.01 φ is not a standard base-φ numeral, since it contains a "11" and additionally a "2" and a "1" = −1, which are not "0" or "1".
Declarative solutions are easier to understand than imperative solutions, [12] and there has been a long-term trend from imperative to declarative methods. [13] [14] Formula calculators are part of this trend. Many software tools for the general user, such as spreadsheets, are declarative. Formula calculators are examples of such tools.