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For example, 20 apples divide into five groups of four apples, meaning that "twenty divided by five is equal to four". This is denoted as 20 / 5 = 4, or 20 / 5 = 4. [2] In the example, 20 is the dividend, 5 is the divisor, and 4 is the quotient.
Find the shortest sequence of digits starting from the left end of the dividend, 500, that the divisor 4 goes into at least once. In this case, this is simply the first digit, 5. The largest number that the divisor 4 can be multiplied by without exceeding 5 is 1, so the digit 1 is put above the 5 to start constructing the quotient.
(For example, the quotient digit pairs (0, +2) and (1, −2) are equivalent, since 0×4+2 = 1×4−2.) This tolerance allows quotient digits to be selected using only a few most-significant bits of the dividend and divisor, rather than requiring a full-width subtraction. This simplification in turn allows a radix higher than 2 to be used.
The first number to be divided by the divisor (4) is the partial dividend (9). One writes the integer part of the result (2) above the division bar over the leftmost digit of the dividend, and one writes the remainder (1) as a small digit above and to the right of the partial dividend (9).
For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually be mentioned. 1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself. Integers divisible by 2 are called even, and integers not divisible by 2 are called odd.
If the sum is a two-digit number then the leftmost digit, called the "carry", is added to the next pair of digits to the left. This process is repeated until all digits have been added. [ 65 ] Other methods used for integer additions are the number line method, the partial sum method, and the compensation method. [ 66 ]
The following exposition assumes that the numbers are broken into two-digit pieces, separated by commas: e.g. 3456 becomes 34,56. In general x,y denotes x⋅100 + y and x,y,z denotes x⋅10000 + y⋅100 + z, etc. Suppose that we wish to divide c by a, to obtain the result b. (So a × b = c.)
(omitted) Dividing 5884 by 594 yields 9 which is written as the new digit of the quotient. 58 − 5×9 = 13 so cross out the 5 and 8 and above them write 1 and 3. Cross out the 5 of the divisor. The resulting dividend is now 1384. 138 − 9×9 = 57. Cross out 1,3, and 8 of the dividend and write 5 and 7 above. Cross out the 9 of the divisor ...