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In elementary arithmetic, a carry is a digit that is transferred from one column of digits to another column of more significant digits. It is part of the standard algorithm to add numbers together by starting with the rightmost digits and working to the left. For example, when 6 and 7 are added to make 13, the "3" is written to the same column ...
This number is renowned for the following rule: Take any four-digit number, using at least two different digits (leading zeros are allowed). Arrange the digits in descending and then in ascending order to get two four-digit numbers, adding leading zeros if necessary. Subtract the smaller number from the bigger number. Go back to step 2 and repeat.
For example, through the standard addition algorithm, the sum can be obtained by following three rules: a) line up the digits of each addend by place value, longer digit addends should go on top, b) each addend can be decomposed -- ones are added with ones, tens are added with tens, and so on, and c) if the sum of the digits of the current place value is ten or greater, then the number must be ...
The nines' complement of a number given in decimal representation is formed by replacing each digit with nine minus that digit. To subtract a decimal number y (the subtrahend) from another number x (the minuend) two methods may be used: In the first method, the nines' complement of x is added to y. Then the nines' complement of the result ...
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.
Dividing 272 and 8, starting with the hundreds digit, 2 is not divisible by 8. Add 20 and 7 to get 27. The largest number that the divisor of 8 can be multiplied by without exceeding 27 is 3, so it is written under the tens column. Subtracting 24 (the product of 3 and 8) from 27 gives 3 as the remainder.
The basic principle of Karatsuba's algorithm is divide-and-conquer, using a formula that allows one to compute the product of two large numbers and using three multiplications of smaller numbers, each with about half as many digits as or , plus some additions and digit shifts.
Given numbers and , the naive attempt to compute the mathematical function by the floating-point arithmetic ( ()) is subject to catastrophic cancellation when and are close in magnitude, because the subtraction can expose the rounding errors in the squaring.