1.1: The Arrow Paradox

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In his famous arrow paradox, Zeno contends that an arrow cannot move since at every instant of time it is at rest. There are at least two logical problems hidden in this claim.

Zero Divided by Zero

In one interpretation, Zeno seems to be saying that, since at every instant of time the arrow has a definite position, and hence does not travel any distance during that instant of time, the velocity of the arrow is \(0 .\) The question is, if an object travels a distance 0 in time of duration \(0,\) is the velocity of the object \(0 ?\)

That is, is

\[\frac{0}{0}=0 ?\]

To answer this question, we need to examine the meaning of dividing one number by another. If \(a\) and \(b\) are real numbers, with \(b \neq 0\), then

\[\frac{a}{b}=c\]

means that

\[ a=b \times c .\]

In particular, for any real number \(b \neq 0\),

\[ \frac{0}{b}=0\]

since \(b \times 0=0 .\) Note that if \(a \neq 0,\) then

\[ \frac{a}{0}\]

is undefined since there does not exist a real number \(c\) for which \(0 \times c\) is equal to a. We say that division of a non-zero number by zero is meaningless. On the other hand,

\[ \frac{0}{0}\]

is undefined because \(0 \times c=0\) for all real numbers \(c .\) For this reason, we say that division of zero by zero is indeterminate.

The first logical problem exposed by Zeno's arrow paradox is the problem of giving determinate meaning to ratios of quantities with zero magnitude. We shall see that infinitesimals give us one way of giving definite meanings to ratios of quantities with zero magnitudes, and these ratios will provide the basis for what we call the differential calculus.

Adding Up Zeros

Another possible interpretation of the arrow paradox is that if at every instant of time the arrow moves no distance, then the total distance traveled by the arrow is equal to 0 added to itself a large, or even infinite, number of times. Now if \(n\) is any positive integer, then, of course,

\[ n \times 0=0 .\]

That is, zero added to itself a finite number of times is zero. However, if an interval of time is composed of an infinite number of instants, then we are asking for the product of infinity and zero, that is,

\[ \infty \times 0 .\]

One might at first think this result should also be zero; however, more careful reasoning is needed.

Note that an interval of time, say the interval \([0,1]\), is composed of an infinity of instants of no duration. Hence, in this case, the product of infinity and 0 must be \(1,\) the length of the interval. However, the same reasoning applied to the interval \([0,2]\) would lead us to think that infinity times 0 is \(2 .\) Indeed, as with the problem of zero divided by \(0,\) infinity times 0 is indeterminate.

Thus the second logical problem exposed by Zeno's arrow paradox is the problem of giving determinate meaning to infinite sums of zero magnitudes, or, in the simplest cases, to products of infinitesimal and infinite numbers.

Since division is the inverse operation of multiplication we should expect a close connection between these questions. This is, in fact, the case, as we shall see when we discuss the fundamental theorem of calculus.