I believe that I have found a major error in Max Black's paper. Either he doesn't understand the mathematics he discusses (which I view as being unlikely) or he was simply careless in some respect; however, he made a mistake in the first instance of his math that is so glaringly obvious I'm surprized that his editors didn't catch it!

The "convergent geometrical series" that he names doesn't converge to 11.1 - it can't! Why? Because his first term is 100 - which is greater than 11.1! Instead, here's a calculation of the value that the series converges to:

S=a/(1-r)

S=100/(1-1/10)

S=111.11111...

What I find to be really disturbing is that he keeps on naming 11.1 as the value at which the series converges - while high school math shows this to be incorrect. Also, he names 11.1 as one value to which his series apparently converges and 11 and 1/9 as another, which differ by 0.0111.... or 0.1111...% While I admit that this deviation is pretty small, it still doesn't change the fact that its quite obvious that 11 and 1/9 is a different number than 11.1. So, not only does he quote the wrong number as the value at which the series converges, but he also contradicts himself in naming the same number!

In case you were wondering, he makes the mistake right after he lists out the terms of the series. He says:

"So the total number of yards he must travel in order to catch the tortoise is

100+10+1+1/10+...

This is a convergent series whose sum can be expressed in decimal notation as

The "convergent geometrical series" that he names doesn't converge to 11.1 - it can't! Why? Because his first term is 100 - which is greater than 11.1! Instead, here's a calculation of the value that the series converges to:

S=a/(1-r)

S=100/(1-1/10)

S=111.11111...

What I find to be really disturbing is that he keeps on naming 11.1 as the value at which the series converges - while high school math shows this to be incorrect. Also, he names 11.1 as one value to which his series apparently converges and 11 and 1/9 as another, which differ by 0.0111.... or 0.1111...% While I admit that this deviation is pretty small, it still doesn't change the fact that its quite obvious that 11 and 1/9 is a different number than 11.1. So, not only does he quote the wrong number as the value at which the series converges, but he also contradicts himself in naming the same number!

In case you were wondering, he makes the mistake right after he lists out the terms of the series. He says:

"So the total number of yards he must travel in order to catch the tortoise is

100+10+1+1/10+...

This is a convergent series whose sum can be expressed in decimal notation as

**11.1, that is to say exactly 11 1/9.**"
## 3 Comments:

Good eye! You're right that there is obviously a mistake. But it is clear that it is a typo (or two typos--and admittedly egregious ones at that). Just below the passage that you quote, he correctly points out that 11.1bar = 11 1/9. Assuming he is not so dumb as to miss that 11.1bar is distinct from 11.1, I think we should take the omission of the bar in the first occurrence as a typo.

Similar remarks apply to the omission of the first '1' in '111.1bar'. (!) When he checks the calculations in the paragraph at the bottom of the page, he correctly notes that the distance is 111 1/9, which he notes is "as before". Finally, he re-states the distance correctly in several places on the next page. So assuming he's not crazy enough to think that 111 1/9 = 11 1/9, we should take this to be a typo. Though admittedly a bad one.

I don't think Max Black has made an error in the numbers to which he refers. When Max Black mentions 11 1/9, he is speaking of the time necessary for Achilles to achieve the overall distance. The overall distance is 111 1/9 and the rate is 10 yards per second. Since rate*time is equal to distance, we get (10 yd/sec)*(11 1/9 sec)=111 1/9 yards. I don't believe this to be a typo. The only typo I find is that the first 11.1 does not have a bar. This, too, I don't believe to be a typo because he says "this is a convergent geometrical series whose sum can be expressed in decimal notation as 11.1, that is to say exactly 11 1/9." Here, I think he realizes that 11.1 is just an estimate, which is why he says immediately after "more exactly 11 1/9."

I am still inclined to think that there are two typos. Before the first occurrence of '11.1', he says 'This is a convergent geometrical series whose sum can be expressed in decimal notation as 11.1, that is to say exactly 11 1/9.' The occurrence of 'this' refers to the total number of yards Achilles must travel to catch the tortoise: 100 + 10 + 1 + 1/10 + . . . And he does not say 'more exactly'. Rather, he says that the sum can be expressed as 11.1, that is to say exactly 11 1/9. This seems to pretty clearly convey that he takes the first number to be equivalent to the second number rather than taking the first to be an approximation where the second is more precise.

Granted, all other occurrences of '11 1/9' are correctly used to describe the *time* it will take Achilles to catch the tortoise. Regardless of which interpretation of what is written in Black's article is correct, I'm sure we can all agree that the distance Achilles must travel to catch the tortoise, given the set-up, is 111.1bar and the time it will take is 11.1bar.

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