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The Division Algorithm
Theorem [Division Algorithm]. Given any strictly
positive integer d and any integer a, there
exist unique integers q and r such that
a = qd + r,
0≤ r < d.
Before discussing the proof, I want to make some general remarks about what this theorem really
says, why it says it in what seems at first such a perversely obscure way, and why it's worth proving
something at all which, as we shall see, actually seems quite obvious.
The Language in Which Mathematics Is Done
Even when talking about day to day life, it sometimes happens that one has a hard time finding
the words to express one's thoughts. Furthermore, even when one believes that one has been totally
clear, it sometimes happens that there is a vagueness in one's words that results in the listener
understanding something completely different from what the speaker intended.
Even the simplest mathematical reasoning is usually more complicated than the most complicated
things one attempts to discuss in everyday life. For this reason, mathematics has developed an
extremely specialized (and often stylized) language. To see the value of this , one only has to think of
how difficult it would be to solve even the simple quadratic equation x2 − 5x = 14 if one's calculation
had to be done without symbols, using only ordinary language.
"Suppose that a quantity has the property that when five times that quantity is
subtracted from the product of the quantity times itself then the result is 14.
Determine the possible values of this quantity."
During the middle ages, algebraic calculations were in fact done in this fashion | except that the
language was Latin, not English. But progress beyond the level of simple algebra only became possible
with the introduction of modern algebraic notation | which the West learned from the Arabs.
By combining symbolic notation with very precisely defined concepts and a very formalized
language, any mathematical proof can be expressed in a way that can be understood by any
"mathematically mature" reader. One is not dependent on the reader's ability to "see what is meant."
In principle, at least, every step is described in terms so precise that no possible misunderstanding can
occur, and anyone can follow the reasoning from one step to the next, provided only that he is skilled
in making the most basic sort of algebraic, set theoretic, and logical manipulations. In particular, the
language is so explicit that one never needs reference to illustrative examples in order to explain the
meaning of what is said. (In practice, it has to be admitted that sometimes even the experienced
mathematician gets stopped at certain points in the proof and has to rack his brains for a minute or
two |or even longer| before he says, "Oh, yeah, that's why." Students, of course, have this
difficulty much more frequently.)
It's important to realize, however, that useful as the language of mathematics is, it doesn't
represent the way humans normally gain understanding. So even something quite familiar can
sometimes seem strange when described in a precise , mathematical way.
One might think of the language in which mathematics is done as analogous to programming
languages. A computer program describes how to perform a given task without omitting even a single
step, no matter how small, and without any conceivable ambiguity. One might think that for this
reason, a computer program would be the ideal way to explain a procedure so that no one could
possibly fail to understand it. But in fact, human beings find computer programs extremely difficult
The language of mathematical proofs is a little more human than that of computer programs.
Nonetheless, learning to read this kind of language is a formidible challenge for students. Learning to
write it is even more formidible. The endeavor can be quite worthwhile, however, since the reward for
success is access to the vast realm of modern mathematics.
With this in mind, take another look at the theorem we started with.
Theorem [Division Algorithm]. Given any strictly positive integer d and any integer a, there
exist unique integers q and r such that
a = qd + r,
0 ≤r < d.
One might be excused for racking one's brains for quite some time to figure out what this theorem
"really means." And yet it simply describes the process of dividing one integer a (the "dividend") by
another d (the 'divisor") to get a quotient (q ) and a remainder (r ). For instance,
In this example, we have a = 1052 and b = 29. The theorem
above states that there exist unique
numbers q and r such that
1052 = 29q + r and 0 ≤r < 29.
The calculation shows that in fact, q = 36 and r = 8.
The dividend a for the Division Algorithm is allowed to be negative . In this case, the quotient will
also be negative (or possibly 0), but the remainder is still required to be positive.
This calculuation represents the equation −121 = 29(−5) +
Operational, Procedural, and Notational Definitions
The statement of the division algorithm as given in the theorem describes very explicitly and
formally what long division is. To borrow a word from physics, the description of long division by the
two conditions a = qd + r and 0 ≤r < d is operational. Given two numbers, for instance, 1052
and 29, the conditions give a very explicit way of testing whether or not 36 is the quotient and 8 the
remainder when the first number is divided by the second.
On the other hand, these two conditions are not procedural: they do not provide a recipe for
actually finding the quotient and remainder. In fact, until one recognizes what the theorem is
describing as simply long division, it is not obvious that the theorem is even true. Good definitions in
mathematics must always be operational (at least in some idealized sense), but they are often not
Analogously, consider the definition of the absolute value function give in many college algebra
This definition is both operational and procedural , since
it describes exactly how to compute lxl .
And yet most students find it completely bewildering. (Students who do make some attempt to make
sense out of it often think it says, "If x is positive, leave it alone, and if it's negative, make it
negative." They forget that if x is negative, then −x if positive: for instance, −(−13) = 13. )
Unless I consider a student fairly bright, when he asks me what the above definition of the
absolute value function really means, I usually say, "Just make x positive by throwing away the minus
sign if it has one." One can say that this second definition ("throw away the minus sign") is
operational and procedural, since it certainly enables one to easily compute the absolute value of a
number. So what's the advantage of the first one, which seems so bewildering?
Well, for one thing, the first definition of the absolute value function is stated in terms of intrinsic
properties of numbers, whereas the second definition is notational: it is stated in terms of the way
we write numbers down. The second definition works fine when we want to computer the absolute
value of a concrete number written down specifically, but it's not so useful when we want to talk
about numbers in generality, or we have a number that's not described in concrete form. (For
instance, if we say "Let x be the smallest solution of the equation x2 − x = 5," there's no minus sign
in the way we've named x, even though one can show that it is negative.) Most important, the first
definition is useful for proving theorems, whereas the second is not.
Nonetheless, I think that giving the first definition in college algebra books is approaching
mathematics in an overly formalistic way that drives many students away.
By the time a student gets to Number Theory, however, it's time he should begin to learn the kind
of language that is used in contemporary mathematics.
In general, there are often two approaches to defining a mathematical entity. The procedural
approach gives a method for actually calculating or constructing that entity. Alternatively, an entity
can be defined by specifying a set of rules which characterize its "behavior" (as it were). The
mathematics of the Twentieth Century tends to put emphasis on the rules by which
entities behave, rather on the way they are constructed.
As an example of this modern approach, one might describe the natural logarithm as a function L
with the properties that L(xy) = L(x) +L(y) for all x and y, and L'(1) = 1. This definition says
nothing about how to actually compute the natural logarithm. In fact, until one has proved the
appropriate theorem, it is not even clear that this natural logarithm function even exists. However
once the necessary existence theorem has been proved, this modern definition enables one to derive all
the important formulas for the logarithm function (including the formulas for its derivative and
integral) in a very clean manner.
The description of long division given in elementary school is notational. From a mathematician's
point of view, it doesn't describe what long division really is in an intrinsic way, as the theorem does.
Furthermore, the elementary school definition would not make any sense to someone who used Roman
numerals or some other non-standard way of writing numbers. (Although the use of Roman or
Chinese systems for representing numbers is not of great practical concern, there is non-decimal
representation of numbers which is of quite great importance| namely the representation of numbers
in computers. If one were required to write a computer program to do long division, one would not
want to start from the sixth-grade procedure, although that procedure-suitably modified-might
provide some useful insight.)
The description of the division algorithm by the conditions a = qd + r and 0 ≤r < d is complete
without reference to any illustrative example. It's also important to realize, though, that for us human
beings, simple examples, such as the example of long division given above, are an important aid in
understanding mathematics. As soon as one gives a simple numerical example for the Division
Algorithm, the student or the mathematician will almost instantly say, "Aha!" The example appeals
to a different sort of understanding, which is more natural for human beings. Unfortunately, in many
advanced mathematical texts the reader has to construct such examples for himself. (One can be sure,
however, that the author has constructed such examples. It's just that it's considered a little
amateurish | perhaps insulting to the reader's intelligence | to include them in an advanced text.)
The Importance of the Division Algorthm
The Division Algorithm talks about the form of division one first learns about in elementary
school, where one gets both a quotient and a remainder. Later on, after one learns about fractions , it
may seem like a rather simple-minded way of thinking about division. I know it seemed like that to
me. In particular, I was always annoyed by getting that damned remainder term. Somehow having a
remainder made the division process imperfect, and that was inconsistent with my idea of
mathematics as a subject where things are always exact.
In Number Theory, however, this is the way of looking at division which is most useful. And in
fact that apparently annoying remainder turns out to be often much more important than the
quotient. In computer science, and certain parts of mathematics, ones writes
a mod d
to denote the remainder when a is divided by d. For instance, 1052 mod 29 = 8. Most books on
number theory do not use this notation, but instead write 1052 ≡8 (mod 29).
As a simple example of why this concept might be useful, consider the question, "If June 21, 1997
is a Saturday, then what day of the week will July 4 come on in the year 2000?" To answer this, first
calculate, taking into account that the year 2000 will be a leap year, that there are
13 + 3 *365 + 1 = 1109 days between June 21, 1997 and July 4, 2001. Then divide 1109 by 7 and see
that the remainder is 3, so that 1109 mod 7 = 3. This tells us that there are a certain number of full
weeks (158, in fact, since 158 is the quotient when 1109 is divided by 7) plus 3 more days from
June 21, 1997 until July 4, 2001. The full weeks don't matter, so July 4, 2000 will be three days later
than Saturday, namely Tuesday.
This example may seem of rather limited value, but similar situations arise in mathematical
applications quite frequently, since many mathematical entities are cyclical in the same way as the
days of the week are. For instance, if one were to ask what the 500th digit in the decimal expansion of
1/13 is, is would be foolish to compute all 500 digits. Since the decimal expansion of 1/13,
viz .076923076923..., has a cyclic pattern of length 6, one only need note that 500 mod 6 = 2, so that
the 500th digit in this expansion will be the same as the second digit, namely 7.