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If necessary, simplify the long division problem by moving the decimals of the divisor and dividend by the same number of decimal places, to the right (or to the left), so that the decimal of the divisor is to the right of the last digit. When doing long division, keep the numbers lined up straight from top to bottom under the tableau.
Arbitrary precision is used in applications where the speed of arithmetic is not a limiting factor, or where precise results with very large numbers are required. It should not be confused with the symbolic computation provided by many computer algebra systems , which represent numbers by expressions such as π ·sin(2) , and can thus represent ...
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.
A standardized way of writing very large numbers allows them to be easily sorted in increasing order, and one can get a good idea of how much larger a number is than another one. To compare numbers in scientific notation, say 5×10 4 and 2×10 5 , compare the exponents first, in this case 5 > 4, so 2×10 5 > 5×10 4 .
Setting up for Division. Division in the Trachtenberg System is done much the same as in multiplication but with subtraction instead of addition. Splitting the dividend into smaller Partial Dividends, then dividing this Partial Dividend by only the left-most digit of the divisor will provide the answer one digit at a time.
In fact, most of the quotients are very small, so a fair number of steps of the Euclidean algorithm can be collected in a 2-by-2 matrix of single-word integers. When Lehmer's algorithm encounters a quotient that is too large, it must fall back to one iteration of Euclidean algorithm, with a Euclidean division of large numbers.
divide a 16-digit number by an 8-digit divisor. Addition or subtraction is performed in a single step, with a turn of the crank. Multiplication and division are performed digit by digit on the multiplier or divisor digits, in a procedure equivalent to the familiar long multiplication and long division procedures taught in school.
For single digit numbers simply duplicate the number into the tens digit, for example: 1 × 11 = 11, 2 × 11 = 22, up to 9 × 11 = 99. The product for any larger non-zero integer can be found by a series of additions to each of its digits from right to left, two at a time. First take the ones digit and copy that to the temporary result.