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There are two ways to think of division: as implying a
related multiplication, or as multiplying by the reciprocal.
Division as Related Multiplication
The statement "12 ?3 = 4" is true only because 3 x 4 = 12.
A division problem is really asking the question "What number
can I multiply the divisor by to get the dividend?"; and so
every division equation implies an equivalent multiplication
equation. In general:
a ?b = c if and only if a = b x c
This also shows why you cannot divide by zero. If we asked
"What is six divided by zero?" we would mean "What number
times zero is equal to six?", but any number times zero
gives zero, so there is no answer to this question.
Multiplicative Inverse (The Reciprocal)
For every real number a (except zero) there exists a real
number denoted by 1/a such that
a(1/a) = 1
* The number 1/a is called the reciprocal or multiplicative
inverse of a.
* Note that the reciprocal of 1/a is a. The reciprocal of the
reciprocal gives you back what you started with.
This allows us to define division as multiplication by
a ?b = a x (1/b)
This is usually the most convenient way to think of
division when you are doing algebra.
Notation for Division
Instead of using the symbol " ?" to represent division,
we prefer to write it using the fraction notation:
a ?b = ---
Sign Rules for Division
Because division can always be written as a multiplication
by the reciprocal, it obeys the same sign rules as
If a positive is divided by a negative, or a negative divided
by a positive, the result is negative:
a -a a
- --- = ---- = ----
b b -b
but if both numbers are the same sign, the result is positive:
---- = ----