You are correct. We started with the problem
16 × (1)/(8). We know, by ordinary arithmetic, that
16 × (1)/(8) = 2. Now, if we use powers of 2 to make the calculation
16
= 24,
(1)/(8)
= (1)/(23) = 2 − 3,
and the multiplication rule
(bm)(bn) = b(m + n),
we get
16 × (1)/(8) = (24)(2 − 3) = 2(4 − 3) = 21 = 2
which is the answer we expect, of course. So our use of negative exponents works out properly in multiplication.
In this lesson you have seen that negative numbers may represent real quantities and can be used for very practical purposes. In fact, the whole idea of negative numbers probably originated in a malodorous marketplace some centuries ago when grain brought in by farmers for sale was weighed and found wanting, and the bookkeeper of the day, needing some easy sign to represent the shortage, invented the minus sign.
The Hindus brought the idea into use as a mathematical concept. For each positive number, they introduced an opposite, or negative, value. Then they found a practical use for these negative numbers in representing debts.
In dealing with negative quantities, it may help to think of some scale on which zero is a midpoint, such as a thermometer or a bank statement. Then the three basic rules for handling negative numbers can be visualised:
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To add two numbers of like sign, add their absolute values as in any arithmetic addition, and then give the sum the sign common to the two numbers. 5 + 3 = 8. Similarly ( − 3) + ( − 5) = − 8. So if you have £5 in your bank account and deposit another £3, you have £8. On the other hand, if you already owe £5 and increase your indebtedness by another £3, you then owe a total of £8.
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To add numbers of unlike signs, find the difference of their absolute values and attach the sign of the larger number to the answer. If the temperature is -10 degrees and rises 5 degrees, the reading is -5 degrees, because − 10 + ( + 5) = − 5.
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To subtract signed numbers, change the sign of the number you are subtracting - the subtrahend - and add it instead. 9 − ( − 6) = 15.
Yesterday the temperature was 14 degrees. Today it’s -3 degrees. What is the difference between yesterday’s temperature and today’s? Change the sign of the subtrahend and add: 14 − ( − 3) = 14 + 3 = 17. The difference between yesterday’s temperature and today’s is 17 degrees.
Now we are ready to use this information about the arithmetic of negative numbers. When a whole number has a negative exponent, we know it is a fraction. You may remember from arithmetic that a proper fraction is a number in which the denominator (the value below the line) is larger than the numerator (the part above the line).
To divide two numbers, each expressed as powers of the same base, one into the other, we have learned to subtract the exponent of the denominator from the exponent of the numerator. A negative answer (exponent) would tell us that the denominator is larger than the numerator and that the number is a fraction. But let’s look at an example to make sure that is really true:
(bm)/(bn) = b(m − n).
To remember the meaning of a negative exponent, b − n, try thinking about it this way: A negative exponent means that the value of the number expressed by the term is smaller than the base if the base is larger than 1 - and the larger the negative exponent, the smaller the value of the number, since the denominator becomes larger in proportion to the numerator. If b is 2, b − 1 = (1)/(2), b − 2 = (1)/(4), b − 3 = (1)/(8), and so on. In short, larger positive exponents lead to larger values, larger negative exponents lead to smaller values.
If
m = 2, and
n = 4,
(b2)/(b4) = (b × b)/(b × b × b × b) = (1)/(b2)
or
b(2 − 4) = b − 2
Therefore,
(1)/(b2) = b − 2
or, in general
b − n = (1)/(bn).
Dealing with negative exponents requires a knowledge of how to deal with negative numbers. We multiply numbers with negative exponents by simply adding the exponents in the same way that we add signed numbers. It is no surprise to you that
(4 − 2)(43) = 4( − 2 + 3) = 41.
Or, if you should be called upon to divide one power of a given number by another power of the same number (or base) where one of them has a negative exponent, the arithmetic would go like this:
(32)/(3 − 3) = 3(2 − ( − 3)) = 3(2 + 3) = 35.
To be sure you understand these operations with negative exponents, answer the Self-Test Questions for Lesson 3 in Appendix.
