A binary operation in mathematics, the inverse operation
of multiplication. Its operands are its
dividend and its divisor; its result is its quotient.
A word of warning:
A long time ago, the author, quite an enthusiast of E2, sought
a way to make a contribution. A great
comfort and pleasure in
his life was Mathematics; for example he spent many hours reading
about the Extraction of Roots in the Encyclopedia Britannica, and
the occasional mathematical problem within his means was all the
sustenance he needed.
Having seen many attempts at the
Rigorization of Mathematics,
directed at the Youths in High Schools today, the Foundations seemed
important. Certainly they are poorly understood, if the
many writeups here on, for example, division by zero
are any indication. So the task became the telling of what he knew of
division - particularly its DEFINITION, which is not
commonly known.
Various concepts of division exist
- it is how we check multiplication;
it is an idealization of
physical partitioning into equal groups;
it creates formal symbols for which multiplication
by the divisor returns the dividend - in this way we can divide by numbers
which are not factors of the dividend...
There is a mess of algorithms and
notations and definitions, but in the end it seems that division
is not very important or interesting. The author is at a loss.
(If there is any interest, he will tell about formal division
— how mathematicians narrowly escape dividing by zero —,
division viewed as a continuous function, or division into
factors as a method for understanding an algebraic
structure. Let him know.)
Operation
Division is breaking a dividend into parts each equal in magnitude
to the divisor and counting them. Equivalently, one can break
a dividend into a number of equal parts represented by the
divisor and measure each part's size. But perhaps the most precise and
simple definition is that division is the inverse operation of multiplication.
(a ÷ b = c)
if and only if (c × b = a).
Dividing by a number in a group (such as a nonzero real number), then,
is equivalent to multiplying by its multiplicative inverse.
(a × b-1)
× b = a × (b-1 × b) = a × 1 = a;
applying the definition of division, a/b = ab-1 if the latter is defined.
Addition distributes on the left over division
because it distributes over multiplication.
When a/c + b/c = q, (a/c + b/c)c = qc,
and since multiplication is distributive, a + b = qc;
so (a + b)/ c = q.
It is not always possible, however, to
distribute (a + b)/c = q; for example, as integers (1 + 3)/2 = 2,
but 1 and 3 are not divisible by two.
Notation
Division is usually written horizontally with the operator in the
middle. There are at least three such ways to write a quotient:
-
As a fraction, a divided by b is written
with a above and b below a horizontal
bar. By analogy, fractions are often
written diagonally or horizontally, as a/b.
-
As a ratio, a divided by b is written
horizontally, as a : b.
-
The generic symbol for division is the division sign ÷,
a colon superimposed on a horizontal bar, which reads as
"divided by" instead of "over" or "to".
In long division, the long division symbol or the colon is used to separate the divisor from the dividend.
Algorithms
The division algorithm (see abiessu's writeup there)
allows splitting division problems into smaller problems.
People usually divide small two digit numbers by one digit
numbers in their heads using shortcuts and multiplication tables.
If all else fails, you can guess and check.
Multiplication by the multiplicative inverse
To divide fractions, you may multiply the dividend by the reciprocal
of the divisor. To divide integers modulo n, multiply
the dividend by the multiplicative inverse of the divisor.
In other cases, another method is probably simpler.
See Also