Continuing with the abacus theme two and three days ago, we move on the division today. (You can compare my traditional approach to MarkCC's approach.)
Division
Abacus division is the inverse of abacus multiplication in a lot of ways: in multiplication, the second factor is placed flush left. in division the divisor is placed flush left. In multiplication, the first factor is placed near the right end of the abacus leaving a space just wide enough to hold the second factor, and when we are done, the result is flush right. In division, the dividend is placed flush right, and when we are done, the result's unit digit floats to the left. If the dividend is an integral multiple of the divisor, there would have been a space to the right of the result that's as wide as the divisor. In other words, with multiplication, the decimal point floats to the right; and with division, it floats to the left.
As with pencil and paper division, you start the division my dividing the first digit of the divisor into the first digit of the dividend and obtain a tentative first digit of the quotient. You then multiply the divisor by the tentative quotient digit and subtract the result from the dividend. When this is done, you move to the next digit of the dividend and repeat the process.
Here are the division rhymes:
see one advance one
see two advance two
see three advance three
see four advance four
see five advance five
see six advance six
see seven advance seven
see eight advance eight
see nine advance nine
two one turn to five
see two advance one
see four advance two
see six advance three
see eight advance four
three one three leave one
three two six leave two
see three advance one
see six advance two
see nine advance three
four one two leave two
four two turn to five
four three seven leave two
see four advance one
see eight advance two
five one double as two
five two double as four
five three double as six
five four double as eight
see five advance one
six one next add four
six two three leave two
six three turn to five
six four six leave four
six five seven leave two
see six advance one
seven one next add three
seven two next add six
seven three four leave two
seven four five leave five
seven five seven leave one
seven six eight leave four
see seven advance one
eight one next add two
eight two next add four
eight three next add six
eight four turn to five
eight five six leave two
eight six seven leave four
eight seven eight leave six
see eight advance one
nine one next add one
nine two next add two
nine three next add three
nine four next add four
nine five next add five
nine six next add six
nine seven next add seven
nine eight next add eight
see nine advance one
That's a lot of lines to memorize. And they pretty much are useless unless you are using an abacus. But when you are six or seven or eight years old, it's really not that big a deal. (I was able to recite 10000-word essays out of memory, too, when I was six.)
Here's how they work:
Each groups of lines handle one one digit divisor. The first group of lines are used when dividing a number by two, etc.
The lines of the form "see k*d advance k" is the easiest to carry out. Take the "see six advance three" line from the "divide by two" group as an example. It means if you see a six, you should take away that six and add three to the digit on the left. The result? 6 ÷ 2 = 3. What about 7 ÷ 2? Well you can still use "see six advance three". You take away six in the current digit and put a three on the digit to the left. There will be a one left on the current digit. And that's when the "two one turn to five" line gets applied. It literally means that you should turn that one into a five. This gives you 7 ÷ 2 = 3.5. (Remember the decimal floats to the left by one digit when you divide by two, a one digit number.)
In general, the "2k k turn to five" lines: two one turn to five, four two turn to five, six three turn to five, and eight four turn to five, corresponds to 1 ÷ 2 = 0.5, 2 ÷ 4 = 0.5, 3 ÷ 6 = 0.5, and 4 ÷ 8 = 0.5.
The set of lines of the form "d n m leave r" all satisfy the equation 10n = d * m + r. It corresponds to the action, when dividing by d, that if you see n, you should change it into an m and add r to the digit on the right. Thus "three one three leave one" is used when dividing 3 into 1. You literally turn that 1 into a 3 and add 1 to the next digit. If you extend the abacus to have infinite length, then you can keep on applying "three one three leave one" to the 1 that you have just added, turning it into a 3 and add a 1 to the right. This way you get the repeat decimal result 1 ÷ 3 = 0.333333...
The lines "d n next add r" are degenerate forms of "d n n leave r." Since the would be quotient is the same as the dividend digit, you don't have to do any thing to that digit. You can just add the remainder to the next digit.
OK. Let's apply what we learned so far to the division 1000000 ÷ 7:
7 1000000|
7 1300000| seven one next add three
7 1420000| seven three four leave two
7 1426000| seven two next add six
7 1428400| seven six eight leave four
7 1428550| seven four five leave five
7 1428571| seven five seven leave one
Therefore 1000000 ÷ 7 = 142857 with a remainder of 1. And if you paid attention, you realize that the pattern will repeat indefinitely if we had an infinite abacus. BTW, 142857 is one of those numbers that show up in math quizzes because its multiples are also its decimal circular shifts:
The only remaining lines are the divide-by-five ones: five one double as two, five two double as four, five three double as six, five four double as eight. When dividing by 5, you simply double all digits that are less than five. For digits greater then five, one application of "see five advance one" will reduce it to less than five.
Fun exercises for division include the 123456789 ÷ n, where n is 2, 3, 4, 5, 6, 7, 8, 9.
So far, all of our divisors are single digit. Things gets a little bit complicated when we have multiple digit divisors. Unlike single digit division, where the tentative quotient is always the true quotient, in multiple digit division, there is a chance that the tentative quotient is too big. For example, when divide 40 by 19, the tentative quotient is 4 ÷ 1 = 4, which is too big (the real quotient is 2.) A mechanism exists to return one unit of the tentative quotient to the dividend. The rhymes are:
one withdraw one return one
two withdraw one return two
three withdraw one return three
four withdraw one return four
five withdraw one return five
six withdraw one return six
seven withdraw one return seven
eight withdraw one return eight
nine withdraw one return nine
Note that the division rhymes already take care of subtracting the product of the quotient and the first digit of the divisor from the dividend, we only need to subtract the product of the rest of the digits and the quotient from the dividend. Here's a worked example of 87 ÷ 27 = 3 with a remainder of 6:
27 87|
27 407| see eight advance four, but 4 * 7 = 28 > 07
27 327| two withdraw one return two
27 306| 3 * 7 = 21, 27 - 21 = 06
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