|
We
think of a multiplication
statement like “2 ´ 3”
as meaning “Add two threes
together”, or
3 + 3
and
“4 ´ 9” as “add 4 nines together”, or
9 + 9 + 9 + 9.
In
general, a ´ b means to add b’s together such that
the number of b’s is
equal to a:
a ´ b = b + b + b + . . . + b
(a times)
We can apply this same rule
to make sense out of what
we mean by a positive number
times a negative number. For
example,
3 ´ (–4)
just means to take 3 of the
number “negative four” and
add them together:
3 ´ (–4) = (–4) + (–4) + (–4) = –12
Unfortunately, this scheme
breaks down when we try to
multiply a negative number
times a number. It doesn’t
make sense to try to write
down a number a negative number
of times. There are two ways
to look at this problem.
One way is to use the fact
that multiplication obeys
the commutative law,
which means that the order
of multiplication does not
matter:
a ´ b = b ´ a.
This lets us write a negative
times a positive as a positive
times a negative and proceed
as before:
(–3) ´ 4 = 4 ´ (–3) = (–3) + (–3) + (–3) + (–3) = –12
However, we are still in
trouble when it comes to multiplying
a negative times a negative.
A better way to look at this
problem is to demand that
multiplication obey a consistent
pattern. If we look at a multiplication
table for positive numbers
and then extend it to include
negative numbers, the results
in the table should continue
to change in the same pattern.
For example, consider the
following multiplication table:
| a
|
b
|
a
´ b |
|
3
|
2
|
6
|
|
2
|
2
|
4
|
|
1
|
2
|
2
|
|
0
|
2
|
0
|
The numbers in the last column
are decreasing by 2 each time,
so if we let the values for
a continue into the
negative numbers we should
keep decreasing the product
by 2:
| a
|
b
|
a
´ b |
|
3
|
2
|
6
|
|
2
|
2
|
4
|
|
1
|
2
|
2
|
|
0
|
2
|
0
|
|
–1
|
2
|
–2
|
|
–2
|
2
|
–4
|
|
–3
|
2
|
–6
|
We can make a bigger multiplication
table that shows many different
possibilities. By keeping
the step sizes the same in
each row and column, even
as we extend into the negative
numbers, we see that the following
sign rules hold for multiplication:
(+)(+) = (+)
(–)(–) = (+)
(–)(+) = (–)
(+)(–) = (–)
Multiplication
Table
Notice how the step
size in each row or
column remains consistent,
regardless of whether
we are multiplying positive
or negative numbers.
| |
-5 |
-4 |
-3 |
-2 |
-1 |
0 |
1 |
2 |
3 |
4 |
5 |
| -5 |
25
|
20
|
15
|
10
|
5
|
0
|
-5
|
-10
|
-15
|
-20
|
-25
|
| -4 |
20
|
16
|
12
|
8
|
4
|
0
|
-4
|
-8
|
-12
|
-16
|
-20
|
| -3 |
15
|
12
|
9
|
6
|
3
|
0
|
-3
|
-6
|
-9
|
-12
|
-15
|
| -2 |
10
|
8
|
6
|
4
|
2
|
0
|
-2
|
-4
|
-6
|
-8
|
-10
|
| -1 |
5
|
4
|
3
|
2
|
1
|
0
|
-1
|
-2
|
-3
|
-4
|
-5
|
| 0 |
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
| 1 |
-5
|
-4
|
-3
|
-2
|
-1
|
0
|
1
|
2
|
3
|
4
|
5
|
| 2 |
-10
|
-8
|
-6
|
-4
|
-2
|
0
|
2
|
4
|
6
|
8
|
10
|
| 3 |
-15
|
-12
|
-9
|
-6
|
-3
|
0
|
3
|
6
|
9
|
12
|
15
|
| 4 |
-20
|
-16
|
-12
|
-8
|
-4
|
0
|
4
|
8
|
12
|
16
|
20
|
| 5 |
-25
|
-20
|
-15
|
-10
|
-5
|
0
|
5
|
10
|
15
|
20
|
25
|
|
For math purists, here’s
the real reason:
|
It
should be obvious that
the presentation of
the rules of arithmetic
given here is just a
collection of motivational
arguments, not a formal
development. The formal
development of the real
number system starts
with the field axioms.
The field axioms are
postulated, and then
all the other properties
follow from them. The
field axioms are
- The
associative and
commutative laws
for addition and
multiplication
- The
existence of the
additive and multiplicative
identities (0 and
1)
- The
existence of the
additive inverse
(opposites, or negatives)
and the multiplicative
inverse (the reciprocal)
- The
distributive law
All
of these are essential,
but the distributive
law is particularly
important because it
is what distinguishes
the behavior of multiplication
from addition. Namely,
multiplication distributes
over addition but not
vice-versa.
The
rules of arithmetic
like “a negative times
a negative gives a positive”
are what they are because
that is the only way
the field axioms would
still hold. For example,
the distributive law
requires that
–2(3 – 2) = (–2)(3) + (–2)(–2)
We
can evaluate the left
side of this equation
by following the order
of operations, which
says to do what is in
parentheses first, so
–2(3 – 2) = –2(1) = –2.
Now
for the distributive
law to be true, the
right side must also
be equal to
-2, so
(–2)(3) + (–2)(–2) = –2
If
we use our sign rules
for multiplication then
it works out the way
it should:
(–2)(3) + (–2)(–2) = –6 + 4 = –2
|
We are used to using the
symbol “´”
to represent multiplication
in arithmetic, but in algebra
we prefer to avoid that symbol
because we like to use the
letter “x” to represent
a variable, and the two symbols
can be easily confused. So
instead, we adopt the following
notation for multiplication:
1.
Multiplication is implied
if two quantities are written
side-by-side with no other
symbol between them.
Example: ab means a ´ b.
2. If
a symbol is needed to prevent
confusion, we use a dot.
Example:
If
we need to show 3 times 5,
we cannot just write them
next to each other or it would
look like the number thirty-five,
so we write 3 · 5.
- We
can also use parentheses
to separate factors. 3 times
5 could be written as 3(5)
or (3)5 or (3)(5).
There are two ways to think
of division: as implying a
related multiplication, or
as multiplying by the reciprocal.
The statement “12 ¸
3 = 4” is true only
because 3 ´ 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 ´ 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.
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 the reciprocal:
a ¸ b = a ´ (1/b)
This is usually the most
convenient way to think of
division when you are doing
algebra.
Instead
of using the symbol “ ¸ ” to represent division, we prefer to write it using the fraction
notation:
Because
division can always be written
as a multiplication by the
reciprocal, it obeys the same
sign rules as multiplication.
If
a positive is divided by a
negative, or a negative divided
by a positive, the result
is negative:
but
if both numbers are the same
sign, the result is positive:
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