Multiplication and Division
When you combine positive and negative numbers through division, results vary according to the signs of the numbers. There are two general ways to view the results of multiplication and division operations involving numbers with different signs. First, consider multiplication and division problems involving only two numbers. Then consider multiplication and division problems involving more than two numbers.
Multiplication and Division with Two Numbers
When you carry out multiplication and division operations that involve two numbers, you deal with a few basic possibilities:
Both numbers are positive. If both numbers are positive, then the result of the multiplication or division is positive. Here are a few examples:
Both numbers are negative. If the two numbers are negative, then the result is positive. Here are a few examples:
In the last example, when you multiply the numerators, −4 and −4, you arrive at a positive value of 16. Along the same lines, multiplying the denominators, −2 and −2, results in a positive number. Alternatively, you can first carry out the divisions. In both cases, −4 ÷ −2, the quotient is positive 2.
One number is negative and the other is positive. If you multiply a negative number by a positive number, then the result is negative.
In the last example, when you carry out the multiplication of the denominator, you arrive at a positive number. When you multiply the numerators, you end up with a negative number. Alternatively, you can first carry out the divisions. In the first case, you divide a positive number by a negative number, resulting in a negative number. In the second case, you divide a negative number by a negative number resulting in a positive number. The final multiplication then is a negative number multiplied by a positive number, resulting in a negative number.
Multiplication and Division with More Than Two Numbers
When you deal with a sequence of divisions or multiplications, the outcome differs according to the last operation you carry out. Consider the following operations:
In the first example, three negative numbers multiplied together result in a number that is negative (−64). One way to examine this activity involves considering that the first two numbers when multiplied result in a positive number (16). When you multiply 16 by −4, however, you end up with a negative number.
To trace how a negative number results from multiplication of an odd sequence of negative numbers, consider that when you multiply the first two negative numbers, the result is positive. However, when you then multiply this number by a negative number, the result is negative. When you multiply by yet another negative number, then the result becomes positive. Consider a set of multiplications that proceed in this way:
| (−2)×(−2) = 4 | Two numbers are even. |
| (−2)×(−2)×(−2) = −8 | Three numbers are odd. |
| (−2)×(−2)×(−2)×(−2) = 16 | Four numbers are even. |
| (−2)×(−2)×(−2)×(−2)×(−2) = −32 | Five numbers are odd. |
With respect to division, the same relationship applies:
| (−32)÷(−2) = 16 | Two numbers are even. |
| (−32)÷(−2)÷(−2) = −8 | Three numbers are odd. |
| (−32)÷(−2)÷(−2)÷(−2) = 4 | Four numbers are even. |
| (−32)÷(−2)÷(−2)÷(−2)÷(−2) = −2 | Five numbers are odd. |
If you are dealing with a sequence of divisions, and if the sequence contains an odd number of negative numbers, the result is negative. If it contains an even number of negative numbers, then the result is positive.
Exercise Set 2.4Determine whether the result for each problem is negative or positive and then solve the problem.
|
