TITLE:
Elaboration of Division—A Systematic Search for True Digits
AUTHORS:
Milosav M. Marjanović
KEYWORDS:
Canonical Division, Long Division, Rounding up Divisor, Rounding down Dividend, Algorithm to Find True Digits
JOURNAL NAME:
Open Access Library Journal,
Vol.4 No.2,
February
21,
2017
ABSTRACT: In this paper, a new procedure of
division is proposed and the corresponding didactical elaboration is sketched.
A division task A:B is called canonical when A is less than tenfold B. As it is well known, each procedure
of division splits into a number of canonical ones. The proposed elaboration is
carried out in two main steps: the cases of canonical division by a two-digit divisor and the cases of long
division. In order to make division easier, in the first step, a divisor is
rounded up, increasing its first digit by 1 and replacing the second one by 0.
In the same time, the dividend is rounded down replacing its last digit by 0.
In this way the calculation is reduced to divisions by a one-digit divisor. This step is technically
important and should precede the case of long divisions. Let A:B be a case of canonical long division. The divisor B is rounded up, increasing its second digit by 1 and replacing all
those that follow by 0’s. In the same time, the dividend A is rounded down, replacing by 0’s the same number of its final
digits as in the case of B. Thus, this division task is reduced to a “short”
canonical division whose divisor is a two-digit number. According to a fact proved by this author in his
paper Division—A Systematic Search for True Digits, II, The Teaching of
Mathematics, XVIII, 2, (2015), 84-92, the quotient of the “short” division is equal or just 1 less than the
number representing the true digit. This fact is the basis for the algorithm of
producing true digits that we propose which is a contrast to the traditional
“trial and error” method.