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What exactly does it mean by "half a unit in the nth place"? I am reading numerical analysis and this term is popping out very frequent in the first chapter itself,my first encounter with this is while reading the rounding-off error rules.

From few solved examples I can understand (or probably guess) it mean $5$ but what I still don't understand why such a abstruse term ?


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up vote 5 down vote accepted

You need to think in terms of significant figures. The nth place is the nth significant figure. Consider, for example, the number 123.45 (base 10). A unit in the second place is 10, in the 3rd place is 1, in the 4th place is 0.1, etc. So half a unit in the second place is 5, in the 3rd place is 0.5, in the 4th place is 0.05, etc.

Now, in the context in which the book uses it I expect that n is the number of digits used to represent a number. So if all numbers are represented with 5 significant decimal digits, 123.45 really represents the range 123.445 to 123.455 (probably half-open). Then the maximum absolute error in that representation is half a unit in the nth place.

Does that make sense?

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