How to formally define decimal truncation?fn(r)=10-nsign(r)10n|r|

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Landyn Whitney · Accepted Answer

Step 1fn(r)=10-nsign(r)10n|r|Where&amp;nbsp;⌊x⌋&amp;nbsp;is the greatest integer&amp;nbsp;≤xThis can be written as:1)&amp;nbsp;fn(r)=10−nf0(10nr)The use of absolute values and sign reflects that it is somewhat arbitrary to “truncate” towards zero.For example, if r is positive and n is a positive integer, then&amp;nbsp;f0(r+n)=f0(r)+n,, but this doesn’t generalize to all real r and integers n.You could substituteg0(r)=⌊r+12r⌋and define&amp;nbsp;gn&amp;nbsp;using the equivalent of (1) to define rounding to n digits.Rounding is more natural because&amp;nbsp;g0(r+n)=g0(r)+n&amp;nbsp;for all real r and integer n. The only time you will get&amp;nbsp;f0(−r)≠−f0(r)&amp;nbsp;is when r is exactly a half-integer.

How to formally define decimal truncation? f_n(r)=10^{-n}\operatorname{sign}(r)\left\lfloor 10^n|r|\right\rfloor

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