一個都不能遺漏：柵欄錯誤 Not One Less: Fencepost Errors You have f rom 1 p.m. to 4 p.m. to wor k on homework assignments for subjects #1 to #4. If you complete one assignment per hour you should get them all done on time. Yes or no? Now how about if you have from May 1 to May 4 to complete them? Can you get away with completing one assignment per day? These kinds of problems lead to more questions [1]. Why are hours and days counted differently? Where should we start counting from anyway? Applying what we learn in school is never as simple as it seems – even with something as simple as counting. To set things straight, let’s go back to preschool. We learned to count starting from one, two, three … and we also learned that this counting process lets us know how many things there are – pencils, houses, or days. To save time we can just let the labels do the counting for us: The days of the month in June are labelled 1 to 30, so there are 30 days [1]. When we get to subtraction, the teacher holds up six pencils and takes four of them away one by one to demonstrate that 6 – 4 = 2. Subtraction is an arithmetic operation, meaning an action (“operation”) is applied to change the number of objects: In this case, the act of taking a pencil away. Now we are considering a s l ight ly di f ferent concept, the span between two numbers. This isn’t the same as the numbers themselves! Once we are introduced to the number line, we get to represent the subtraction 6 – 4 = 2 as four arrows bumping down from 6 to 2: It’s clear that this refers to the four “spans” between 2 and 6. But the operation actual ly “touches” five numbers: 2, 3, 4, 5, 6. The discrepancy between four and f ive is the fencepost error. Suppose on the way to preschool you pass by an eight-meter fence with posts every two meters [2]. How many fenceposts are there? You might do the mindless division operation and think this means there are four fenceposts in the fence. But in an ordinary fence, with fenceposts at each end, there are actually five. (Is your fence really ordinary? We’ll return to this later.) What went wrong? There are two possible things you can count in this problem: the number of segments of fence between posts, or the number of posts. When By Peace Foo 胡適之

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