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If one rounds digits one by one starting from the end, then is the rounding same as when "cut-offing" around required the precision?

That is does (for $1/10^3$):

$0.84562...4356 \rightarrow 0.8456 \rightarrow 0.846$

produce the same as

$0.84562...4356 \rightarrow 0.84562...436$
$ \rightarrow 0.84562...44 \rightarrow 0.84562...4$
up until: $\rightarrow 0.846$

mavavilj
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1 Answers1

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No, you cant't use these methods equivalently.

For example by the sequence of rounding we have: $$.846 \approx .85 \approx .9$$ while the single rounding (which is the proper method) gives us: $$.846 \approx .8$$

Jaroslaw Matlak
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  • So how did they figure out the cutoff method as the valid one? Particularly, what's the property that makes it valid and the other invalid? – mavavilj Aug 06 '18 at 09:14
  • See the approximation error. In "cut off" method it is smaller. In sequentional rounding the approximation error accumulates and then it can be higher (like in the example above). – Jaroslaw Matlak Aug 06 '18 at 09:23
  • In the other words - in each step of the step-by-step rounding, you approximate different number. In most cases it would give you the same result, but when the number is close to the half, the results might differ. – Jaroslaw Matlak Aug 06 '18 at 09:26