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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
|
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For math purists, here’s the real reason:
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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
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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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