Division By Repeated Subtraction
May 12, 2017
I traded several emails this week with a programming student who was having trouble with this assignment:
Write a function that divides two numbers and returns their quotient. Use recursive subtraction to find the quotient.
The student went on to explain that he thought he needed to use a counter variable that he incremented each time the function recurred, but he didn’t know how to do that because the function could only take two arguments; it had to be of the form (define (divide a b) ...).
Your task is to write the function, and explain to the student how it works. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.
Programming Praxis 
Nice story, especially how division by zero helped :)
Division itself is a bit of a distraction when it’s not specified that the numbers are non-negative integers, so I spent some time worrying about that, but what bothered me most was not being able to use three arguments when implementing a function that takes two. One must get over that. So I wrote the following. I make no attempt to explain recursion.
;; There are many ways to define division when there is a remainder. ;; While truncation towards zero is not necessarily the best, it may ;; be the one that people are most familiar with. Also, let's just ;; implement one kind and worry about its analysis later. (But this ;; one does turn out to truncate towards zero, so, whatever.) (define (divide a b) ;; Let an auxiliary procedure, aux, deal with two positive numbers ;; only (not zero, not negative, but "strictly" greater than zero). ;; It's wishful thinking at this point, but there will be such an ;; auxiliary. See below. (cond ((zero? b) (raise "result is not a finite number")) ((zero? a) a) ((and (positive? a) (positive? b)) (aux a b 0)) ((and (negative? a) (positive? b)) (- (divide (- a) b))) ((and (positive? a) (negative? b)) (- (divide a (- b)))) ((and (negative? a) (negative? b)) (divide (- a) (- b))) (else (raise "arguments are a new kind of number")))) (define (aux a b q) ;; You need a three-argument procedure to implement a two-argument ;; procedure? You define the auxiliary that you need. (It can be ;; internal to the main procedure. This one maybe should be.) ;; ;; Got that? You need a function that you don't have, you get to ;; define that function. Then you have that function. This may be ;; the single most important idea in programming. ;; ;; (This auxiliary has a lousy name. That's a relatively minor ;; issue. Naming things well is actually not so easy.) (if (< a b) q (aux (- a b) b (+ q 1)))) (let ((test (lambda (a b) ;; The above division truncates towards zero, so test it ;; against Scheme's standard integer division, which ;; also truncates towards zero. It's usually a good idea ;; to have a simple test set available while developing ;; software - it's not a proof of correctness, but it ;; can easily be a proof that something went funny. (display `((divide ,a ,b) => ,(divide a b) = ,(quotient a b))) (newline)))) (test 5 2) (test 4 2) (test 3 2) (test 3 3) (test 3 4) (test -3 2) (test 3 -2) (test -3 -2) (test 0 3)) ;; TL;DR; Define the auxiliary procedure that you need to use.@Jussi: Division by zero was the clincher. When the student realized that the computation failed because recursion never reached the base case, he finally understood the true meaning of the base case, and how it makes everything work.
@Praxis: Yes, I understood that. Note that I followed your rules and wrote my program and its comments before reading your solution.
Negative numbers might have a similar effect, possibly even more so: there would be cases where the computation “progresses” but in the wrong direction, away from the base case.
(* assuming positive integers repeatedly subtracting b from a will result in 0 1. 20 / 4 = 5 2. 20 - 4 - 4 - 4 - 4 - 4 = 0 Since we know the result of performing the repeated subtraction in (2) is 0, any computation that we make alongside the subtraction (counting) will be the result of the function *) fun divide a b = if a < b then 0 else 1 + (divide (a - b) b); (* divide 10 2 1 + divide 8 2 1 + 1 + divide 6 2 1 + 1 + 1 + divide 4 2 1 + 1 + 1 + 1 + divide 2 2 1 + 1 + 1 + 1 + 1 + divide 0 2 1 + 1 + 1 + 1 + 1 + 0 ------------------------------ 5 *)C11 solution:
#include <stdio.h>
#include <stdlib.h>
int
divide (int a, int b);
int
main (int argc, char **argv)
{
const int a = 15;
const int b = 4;
printf("%d / %d = %d\n", a, b, divide(a, b));
exit(0);
}
int
divide (int a, int b)
{
if (a < b)
return 0;
else
return 1 + divide(a - b, b);
}
We build
divideby asking what the base case is: in our case, whena < b. If this is so, then division will yield0. If this is not the case, then we recursively call the method by noticing thata - bhas one lessb‘s worth of magnitude thana, meaning thatdivide(a, b) - divide(a - b, b) = 1. We restructure the formula to get the recursive invocationdivide(a - b, b) + 1.Gforth:
[sourcecode]
: / over over < IF 0 ELSE swap over – swap recurse 1 + THEN ;
[/sourecode]
Which language is that? It seems very foreign to me.
Here is a Haskell version. The Natural number type represents unbounded, non-negative integers.
import Numeric.Natural -- Let's *derive* a solution to our problem! If a and b are natural numbers, -- then define "n = a divide b" to be the largest natural number, n, such that -- n*b ≤ a. (For now, let's assume b ≠ 0. We'll deal with 0 later.) If a < b -- then n must be 0 for the inequality to hold. Otherwise, a ≥ b and the -- statement n*b ≤ a is equivalent to saying (n-1)*b + b ≤ a, or (n-1)*b ≤ a-b. -- -- We can rewrite this last inequality as m*b ≤ c, where m = n-1 and c = a-b. -- But, aside from using different variable names, this is saying "m = c divide -- b", which is just a restatement of our original problem! (But, with c being -- smaller than a.) -- -- Since m = n - 1, then n = 1 + m -- = 1 + c divide b -- = 1 + (a-b) divide b -- -- Translating our two cases to code gives us: divide :: Natural -> Natural -> Natural a `divide` b | a < b = 0 | otherwise = 1 + (a-b) `divide` b -- Also, still assuming b ≠ 0, we know that this function terminates because if -- a < b it gives 0 immediately, otherwise it calls itself recursively with a -- strictly smaller first argument. Since the argument is strictly decreasing -- it must become less than b after a finite number of calls, at which point the -- first branch will apply. -- -- Okay, but what about 0? Since we said that n is the *largest* number such -- that n*b ≤ a, then b = 0 means that we can satisfy the inequality while -- increasing n without bound. And, in fact, divide will try to do exactly -- that! So, to ensure that we always get a result we'll say that "a divide 0" -- is undefined for any a. We can represent this in code as follows: divideZ :: Natural -> Natural -> Maybe Natural _ `divideZ` 0 = Nothing a `divideZ` b = Just (a `divide` b)Sample output from the REPL:
buoyantair: The language you asked about is Forth.
Firstly, we need to remember that if
is 
then 
is undefined, and so we need to specify that the function will return not only the quotient and the remainder of 
but also whether 
is undefined:
type t_WholeNumber = 0 .. maxint; { As defined by the College Algebra (MA008) (<http://Udacity.com/course/ma008>) course offered by Udacity. } t_DivModResult = record m_quotient : integer; m_remainder : t_WholeNumber; m_undefined : boolean end;function f_DivMod( a { The dividend. }, b { The divisor. } : integer ) : t_DivModResult;Secondly, we also need to remember that
may be positive or negative (or zero) and that 
may be positive or negative, and so to function correctly regardless of the signs of 
and 
whilst remaining simple the function will first determine if the quotient will be negative (if either 
or 
– but not both – is negative) and will use the absolute values of 
and 
thereafter…
function f_IsNegative( p : integer ) : boolean; begin f_IsNegative := p < 0 end;…setting the quotient to be negative (if necessary) just before it is returned:
if quotient_is_negative then result.m_quotient := result.m_quotient * -1; f_DivMod := result end;Let’s specify that the base cases of the function – viz., the cases wherein the function will not invoke itself – are the cases where either
is greater than 
(and so the quotient and remainder of 
are 
and 
, respectively) or where 
is 
(and so 
is undefined):
if b = 0 then result.m_undefined := true else if b > a then begin result.m_undefined := false; result.m_quotient := 0; result.m_remainder := a endThe remaining non-base case of the function – viz., the case wherein the function will invoke itself – is the case where
is greater than- or equal to- 
; we know that:
in that case the quotient is at least
;
in that case the function will invoke itself and will pass to itself as an argument a dividend of not
but of 
;
and that the function invoked in 2. will return a data structure containing the quotient and remainder of
and whether 
is undefined.
And so, given 1. and 3., the function will increment the quotient in the data structure returned to it after 2.:
else { a >= b } begin result := f_DivMod( a - b, b ); result.m_quotient := result.m_quotient + 1 end;And the following is the complete function:
type t_WholeNumber = 0 .. maxint; { As defined by the College Algebra (MA008) (<http://Udacity.com/course/ma008>) course offered by Udacity. } t_DivModResult = record m_quotient : integer; m_remainder : t_WholeNumber; m_undefined : boolean end; function f_DivMod( a { The dividend. }, b { The divisor. } : integer ) : t_DivModResult; var quotient_is_negative : boolean; result : t_DivModResult; function f_IsNegative( p : integer ) : boolean; begin f_IsNegative := p < 0 end; begin quotient_is_negative := ( f_IsNegative( a ) and not f_IsNegative( b )) or ( not f_IsNegative( a ) and f_IsNegative( b )); a := abs( a ); b := abs( b ); if b = 0 then result.m_undefined := true else if b > a then begin result.m_undefined := false; result.m_quotient := 0; result.m_remainder := a end else { a >= b } begin result := f_DivMod( a - b, b ); result.m_quotient := result.m_quotient + 1 end; if quotient_is_negative then result.m_quotient := result.m_quotient * -1; f_DivMod := result end;(I hope that the above is comprehensible; although I’m interested in teaching – and more specifically in improving the teaching and assessment of GCSEs and AS & A levels in the STEM subjects and in making the teaching and assessment of those qualifications available to more students (more specifically to students who are studying at secondary schools or sixth form colleges which don’t offer those qualifications or who are, like myself, not in education) – I’m not, and have never been, a teacher or anything approaching that :/ )
Here’s a solution in x86 assembly that takes unsigned integers.
The handling of division-by-zero is non-portable, as it uses Linux system calls. For handling divide-by-zero, I wanted to trigger the same behavior as calling the div instruction with a zero divisor (without calling the div instruction directly). I tried to do this using interrupts, but was unsuccessful (I have limited experience using assembly code). I asked about this on Stack overflow. Any help would be appreciated.
https://stackoverflow.com/questions/46756369/how-can-i-use-interrupts-to-trigger-a-divide-by-zero-error-exception-in-x86-asse
Here’s a C program that calls the function.
/* main.c */ #include <stdio.h> unsigned int divide(unsigned int, unsigned int); int main(int argc, char* argv[]) { printf("%d / %d = %d\n", 10, 3, divide(10, 3)); printf("%d / %d = %d\n", 3, 10, divide(3, 10)); }Example Usage:
Output: