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 bs together such that the number of bs 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 doesnt 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:
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:
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:
(+)(+) = (+)
()() = (+)
()(+) = ()
(+)() = ()
For math purists, heres the real reason:
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.
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
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 Έ 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: