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A common error in reasoning is the off by one error which I found that has a more "formal" name as "Fence post error".
The error is that if we do e.g. $11 - 8$ while subtraction gives $3$ in the context of where we try to count items including $8$ the correct answer is actually $4$ i.e. $N - i + 1$.
E.g. clean floors $8$ to $11$ that is $4$ floors and not $3$.

I have been bitten by this error occasionally and I am always curious how these logical falacies arise in the thought process. So looking online I found this post which I first thought seemed not to make sense to me: How to avoid fence post error
Basically it states:

Subtraction is a span between numbers, not a count. The equation 11 - 8 = 3 means there are 3 "spans" between 8 and 11, but four numbers in that range! A span is a distance measure, like time from 8am and 11am (3 hours) or the distance between 8 and 11 inches (3 inches).

To me this whole explanation seemed weird at first reading and wrong. Because the core of the problem is when we say $11 - 8$ we mean that $8$ is removed too exactly like you have $11$ apples and remove $8$ to end up with $3$ which you can count (apple $9th$, $10th$ and $11th$).

But then I thought may be there is something about the basic operation of subtraction that escapes me and this post might be alluding to something I am not getting.

Are there multiple perspectives of what is subtraction fundamentally and if it is related to the context of counting or something different changes?

Bill Dubuque
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Jim
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  • I don't find that explanation to be very clear. When I think of $11 -8$ I think of: 11 apples take away 8 apples; the number that when added to 8 gives a sum of 11; the distance from 8 to 11 on the number line. – littleO Feb 16 '21 at 17:49
  • @littleO: If you check the post I link to it also has a diagram to try to get the point home. I.e. that subtraction is a span. Paragraph "Number line" https://betterexplained.com/articles/learning-how-to-count-avoiding-the-fencepost-problem/ – Jim Feb 16 '21 at 18:16
  • Somehow the post equates the gap in the number line with the distinct items produced by division – Jim Feb 16 '21 at 18:19
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    Now that I read that explanation, I remember one thing that helped me avoid off-by-one errors was to start with the simple observation that the list of numbers $1, 2, \ldots, n$ contains $n$ numbers (as the article mentions). Then if I need to know for example how many numbers are in the list $37, 38, \ldots, 72$, I will subtract $36$ from each number in the list so that the list starts with $1$. The new list is $1, 2, \ldots, 72 - 36$. So the number of numbers in the list is $72 - 36$, rather than $72 - 37$ (which would be a common error). – littleO Feb 16 '21 at 19:02
  • Does this answer your question? Why do $n-1$ lines connect $n$ dots, when the lines share dots? –  May 27 '23 at 21:57

2 Answers2

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Interesting view as a span. This works basically for natural numbers.

Another view is to consider subtraction as a derived operation from addition if you consider the set of integers, i.e.,

$x + y = x + (-y)$

such has $11 - 8 = 11 + (-8)$, where the right-hand side is the addition of $x$ and the additive inverse of $y$.

Wuestenfux
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    I think the scope is about natural numbers since the fence post error is only relevant there right? But to me viewing it as span seems wrong. In that case for instance in the elementary school example: From $11$ apples remove $8$ how many do you have the answer based on the view as span is what? $3$ spans? – Jim Feb 16 '21 at 17:12
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When I am confronted by problems like this. I try to think of a very simple example, figure that out, and then try to generalize.

So suppose we have three circles lined up on the ground that are each exactly one step apart from the previous circle.

If we want to measure the distance from the first circle to the third,

  • We stand on the first circle. Step count = $0$.
  • We step to the second circle. Step count = $1$.
  • We step to the third circle. Step count = $2$.

So the distance between the $3$ circles is $2$ steps.

The important concept is that you start counting at $0$.

If we want to plant a fence post in each circle.

  • We stand on the first circle and plant a fence post. Fence post count = $1$.
  • We step to the second circle and plant a fence post. Fence post count = $2$.
  • We step to the third circle and plant a fence post. Fence post count = $3$.

So the number of fence posts planted is $3$.

The important concept is that you start counting at $1$.

This isn't about subtraction. It's about counting. Are you going to start at $0$ or are you going to start at $1$? Figure that out and choose your number line accordingly.

Steven Alexis Gregory
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