Arithmetic of Computers

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Lesson 3

Negative Numbers

Real values of less than nothing

Page 115

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Your answer :

(b²)/(b³) = b^( − 1).

You are correct. Using our rule for division

(b^m)/(bⁿ) = b^{(m − n)}

in the case of (b²)/(b³) we get

(b²)/(b³) = b^{(2 − 3)} = b^( − 1).

Incidentally, we used the parentheses, () in b^( − 1) for clarity, but ordinarily the symbol is written simply as b^− 1, and we will follow this usage from now on.

b^− 1 means b^( − 1).

So here we have reached something new, a negative exponent. Now let’s see if we can find out what this negative exponent means by the same method we used to investigate an exponent of 0. We have just shown , by our division rule, that

b^− 1 = (b²)/(b³)

and we can write b² = b × b, and b³ = b × b × b, so we have

b^− 1 = (b²)/(b³) = (b × b)/(b × b × b) = (b × b)/(b × b × b) = ?

So now how shall we define b^− 1?

Answer :

b^− 1 = (0)/(b).