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"What is the value of 0.02 cm rounded to the nearest centimeter?"

Is it logical to approximate a real value (however small) to zero? I know that following a simple 'rounding' or approximation algorithm, the answer is zero cm. This offends my sense of logic, given that we are talking about a real value.

What do others think?

cormac
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2 Answers2

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Pythagoras would say no as zero is not a number, but then he would also say no to rounding to one, for the same reason. However the rest of the modern world would say yes (with the reservation that it does depend on what you are going to do with the result). But if this is homework of an exercise in rounding then very definitely it rounds to zero if you are asked to round to the nearest centimetre.

Summary: it is not a matter of logic but of what you are going to do with the approximation.

Conrad Turner
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  • Thasnks CT! I think that pretty well covers it. I notice lots of the other comments about approximation mention the purpose as the determinant. It sets my teeth on edge a bit as I recoil from the idea of 'pragmatic' rather than stone cold logical maths. BTW, I didn't realise that Pythagoras had a problem with 1! – cormac Feb 20 '15 at 06:59
  • @cormac Half remembered factoid: number represents multiplicity, one is the generator of the numbers, but does not represent a multiplicity so is not a number itself. The trouble with half remembered factoids is that they do not always correspond to facts. – Conrad Turner Feb 20 '15 at 07:21
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You could be taking the difference in two person's heights, rounded to the nearest centimeter. If the measured difference is only $0.02$ cm then it may make sense to say the two persons are the same height, that is, the height difference is zero.

If you are trying to measure the gap between two electrodes in an electronic device then it makes no sense to round $0.02$ cm to zero, but it makes no sense to round to the nearest centimeter then anyway. On the other hand it hardly makes sense to round the distance from Paris to Moscow to the nearest centimeter. You have to assume you're working at a scale where whole centimeters represent the values we care about.

David K
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  • Yes indeed, David. The question comes from my daughters' homework sheet and the phrasing of the question set me thinking, as it mentioned a widget of thickness of 0.02cm and then asked to approximate to the nearest cm. The seemingly illogical step is that after discounting the widget-width as 0cm, the question asks the student to calculate a value for the thickness of 400 widgets. – cormac Feb 20 '15 at 12:30
  • For me, the problem is one of logic and reality: if itis possible to assign a 0 value to a real (but inconsequential) measurement, are we not undermining the reality of the object itself? Your example is a good one, and illustrates very well the principle of setting the approximation in the context of the wider aim. Thanks for taking the time to reply. – cormac Feb 20 '15 at 12:34
  • I like to think that objects exist even when we do not assign them any number at all. But knowing the context of this measurement, I agree that the homework question is incoherent. Knowing that they were about to stack $400$ objects, it is absurd to ask about the thickness of a single object rounded off to a precision that can barely give even one significant digit for the thickness of the entire stack. – David K Feb 20 '15 at 20:20