0.999… = 1 and Zeno’s Paradox

Overview

One surprisingly difficult concept for many students of mathematics is understanding that 0.999… (more properly depicted as \(0.9\overline{9}\)), that is, a decimal with an infinite number of 9s, is equal to 1. There are various proofs of it, and various arguments against it.

Below, I’m going to present a discussion of this problem in terms of Zeno’s Paradox, but first I’ll show a fairly simple proof:

\[ \begin{align} x &= 0.99\overline{9} \\ 10x &= 9.99\overline{9} \\ 10x – x &= 9.00\overline{0} \\ 9x &= 9 \\ x &= 1 \\ 0.99\overline{9} &= 1 \end{align} \]

This proof really just moves the goalposts, though. If people have trouble understanding an infinite number of 9s, they’re also likely to have trouble understanding why \(9.99\overline{9} – 0.99\overline{9} = 9.00\overline{0}\). What about the last 9?

Another proof is likewise problematic:

\[\begin{align} x &= 1 \\ \frac{x}{3} &= \frac{1}{3} \\ \frac{1}{3} &= 0.33\overline{3} \\ 3\left( 0.33\overline{3} \right) &= 0.99\overline{9} \\ 3\left( \frac{x}{3}\right) &= x \\ x &= 0.99\overline{9} \\ 1 &= 0.99\overline{9} \end{align} \]

This relies on accepting that \( \frac{1}{3} = 0.33\overline{3} \) exactly, as opposed to merely approaching it closer and closer. People who have trouble understanding that \( 0.99\overline{9} \) eventually reaches 1 are likewise likely to struggle with this (although PurpleMath, among other sites, points out that students don’t overtly balk at it nearly as often).

There are other proofs, all of which are related in some way to trying to explain the notion of limits towards infinity (see PurpleMath).

So let’s talk about Zeno.

Zeno’s Paradox

In its common form, Zeno’s Paradox presents a race between Achilles and a turtle. Because Achilles is known to run exactly twice as fast as the turtle, the turtle is placed at the halfway point of the race, to be fair. The race starts. By the time Achilles gets to where the turtle started, the turtle is now halfway to the finish line. By the time Achilles gets to that new spot, the turtle has again gotten to the halfway point. Achilles cannot catch up to the turtle, but we know that the race should end in a tie. Hence, Zeno argues that the race is never finished, because a tie would imply that Achilles caught up, which he can’t.

Zeno’s Paradox is a theoretical one. In the real world, of course, we go places. Perhaps Achilles is having a good day and runs fast; perhaps the turtle gets distracted. As Achilles gets closer to the turtle, it takes him less time to cover the remaining distance; it eventually becomes impossible in the real world to measure “half” the distance, as well as the time taken to cover the distance.

The key element that Zeno appears to leave out is that of time. If it takes 10 seconds for Achilles to catch up to where the turtle started the race, it will take 5 seconds for him to cover the next distance, then 2.5, then 1.25, and so on. As the distance gets smaller, so does the time, until eventually they’re both approaching 0. In other words, both the time and the distance are contracting as they approach that single point in time and space where Achilles catches the turtle.

The Magic Cookie Cutter

To discuss thirds, I’m going to modify Zeno’s Paradox significantly, to the point that it might not be recognizable.

Let us say we have a device capable of cutting any object into ten perfectly equal pieces. Such a device could not exist in the real world, of course, but this is a theoretical device for a theoretical case of repeating a number to infinity.

Let us say we have three children, and we are giving each of them equal parts of a cookie. We use our device to create 10 pieces, and distribute 3 to each of them. We have one piece of cookie left over. We use our device to create 10 pieces from that one, and distribute 3 to each of the children. We have one (much smaller) piece of cookie left over. We will always be able to distribute nine of the pieces, with one left over for our device. We will never be able to distribute that final piece, but we will also always be able to cut it into smaller pieces. In the theoretical world, we will never reach the end of our project.

In order to say that \(3(0.33\overline{3})\), that is, \(0.99\overline{9}\) is not equal to 1, we would be saying that we’re able to reach the point where we can no longer cut the cookie. We have two intertwined infinite processes in our hypothetical scenario: Cutting the cookie, and having a remainder.

All this said, a fair response is that saying that \(0.99\overline{9}=1\) is saying that there’s some point where there’s no remainder. True, but the reality is that Zeno’s Paradox, and this variation of it, is indeed a paradox. The natural extension of Zeno’s Paradox is that it’s impossible to go anywhere because we always have to go to someplace between where we are and where we’re going first. Just as we have to pass through C to get from A to B (going in a straight line), we also have to pass through D to get from A to C, and through E to get to D, and so on. However, we clearly do go places.

When we acknowledge that we go places, we’re acknowledging that at some point the distance between A and B and the time taken to travel between them get sufficiently small as to balance each other.

When we state that \(0.99\overline{9}=1\), we’re not disregarding the remainder. We’re saying that that the ability to cut the remainder of the cookie and the size of the resulting wedges become sufficiently small so as to balance each other.

The general (although not universal) result of two infinite processes acting in the same direction on their variables is to balance to 1.

Other bases

In our theoretical universe, we’re technologically capable of creating a device that can cut things into exactly ten even pieces. Why can’t we create a device that can cut things into some other number of pieces? Wouldn’t that just get rid of this whole mess?

Actually, we can. We can also build a device that can cut things into exactly nine even pieces. We take our cookie, and with one cut we make nine pieces of our cookie. Each child gets three pieces, and there is no remainder. Hooray!

However, the next day, there’s a friend visiting. Now we have four children. When we apply our 9-cutter, we get nine pieces. Two go to each child, and there’s one left over. When we apply it again, two go to each child, and there’s one left over. Uh-oh.

We could apply our 10-cutter instead, when we have four children. We get ten pieces; two go to each child, and we have two left over. We apply our 10-cutter to each left over piece, and we get 20 pieces, five for each child. Each child gets two larger wedges and five smaller wedges, with none left over.

This is how number bases work with fractions. In base 9, 1/3 is a nice, clean number: 0.3. \(0.3 + 0.3 + 0.3 = 1\). However, 1/4 is repeating: \(0.22\overline{2}\). In base 10, the situation is the opposite: 1/4 is 0.25, while 1/3 is \(0.33\overline{3}\).

We could make a device capable of cutting things into exactly six pieces. With three children, everyone gets two. With four children, everyone gets one the first round, then three the second round. With five children, everyone gets one the first round, with one left over; the second round, they get one each, with one left over. Uh-oh.

In base 6, in other words, 1/3 = 0.2, 1/4 = 0.13, and 1/5 = 0.111….

There is no base in which all ratios can be expressed with a non-repeating “decimal” (“decimal” technically refers to base 10). Which denominators cause repeating series depends on the base.

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