Editing Rounding (section)

===Directed rounding to an integer===
These four methods are called '''directed rounding to an integer''', as the displacements from the original number {{mvar|x}} to the rounded value {{mvar|y}} are all directed toward or away from the same limiting value (0, [[extended real number line|+∞]], or −∞). Directed rounding is used in [[interval arithmetic]] and is often required in financial calculations.

If {{mvar|x}} is positive, round-down is the same as round-toward-zero, and round-up is the same as round-away-from-zero. If {{mvar|x}} is negative, round-down is the same as round-away-from-zero, and round-up is the same as round-toward-zero. In any case, if {{mvar|x}} is an integer, {{mvar|y}} is just {{mvar|x}}.

Where many calculations are done in sequence, the choice of rounding method can have a very significant effect on the result. A famous instance involved a new [[stock index|index]] set up by the [[Vancouver Stock Exchange]] in 1982. It was initially set at 1000.000 (three decimal places of accuracy), and after 22 months had fallen to about 520, although the market appeared to be rising. The problem was caused by the index being recalculated thousands of times daily, and always being truncated (rounded down) to 3 decimal places, in such a way that the rounding errors accumulated. Recalculating the index for the same period using rounding to the nearest thousandth rather than truncation corrected the index value from 524.811 up to 1098.892.<ref>
{{cite book |title=Accuracy and stability of numerical algorithms  |author-first=Nicholas John |author-last=Higham |author-link=Nicholas Higham |page=54 |isbn=978-0-89871-521-7 |date=2002| url=https://books.google.com/books?id=7J52J4GrsJkC&dq=vancouver+stock+exchange+rounding&pg=PA54 |doi=10.1137/1.9780898718027.ch2 |edition=2nd |url-access=limited}} {{pb}} {{cite journal |last=Nievergelt |first=Yves |year=2000 |title=Rounding Errors to Knock Your Stocks Off |journal=Mathematics Magazine |volume=73 |issue=1 |pages=47–48 |doi=10.1080/0025570X.2000.11996800 |jstor=2691491 }} {{pb}} {{cite news |last=Quinn |first=Kevin |title=Ever had problems rounding off figures? This stock exchange has |newspaper=Wall Street Journal |date=8 November 1983 |url=https://www5.in.tum.de/~huckle/Vancouv.pdf}} {{pb}} {{cite news |last=Lilley |first=Wayne |newspaper=The Toronto Star |date=November 29, 1983 |title=Vancouver stock index has right number at last |url=https://www5.in.tum.de/~huckle/Vancouv.pdf }} </ref>

For the examples below, {{math|sgn(''x'')}} refers to the [[sign function]] applied to the original number, {{mvar|x}}.

====Rounding down====
One may '''round down''' (or take the '''[[floor and ceiling functions|floor]]''', or '''round toward negative infinity'''): {{mvar|y}} is the largest integer that does not exceed {{mvar|x}}.

: <math>y = \mathrm{floor}(x) = \left\lfloor x \right\rfloor = -\left\lceil -x \right\rceil</math>

For example, 23.7 gets rounded to 23, and −23.2 gets rounded to −24.

====Rounding up====
One may also '''round up''' (or take the '''[[floor and ceiling functions|ceiling]]''', or '''round toward positive infinity'''): {{mvar|y}} is the smallest integer that is not less than {{mvar|x}}.

: <math>y = \operatorname{ceil}(x) = \left\lceil x \right\rceil = -\left\lfloor -x \right\rfloor</math>

For example, 23.2 gets rounded to 24, and −23.7 gets rounded to −23.

====Rounding toward zero====
One may also '''round toward zero''' (or '''[[truncation|truncate]]''', or '''round away from infinity'''): {{mvar|y}} is the integer that is closest to {{mvar|x}} such that it is between 0 and {{mvar|x}} (included); i.e. {{mvar|y}} is the integer part of {{mvar|x}}, without its fraction digits.

: <math>
y = \operatorname{truncate}(x)
= \sgn(x) \left\lfloor \left| x \right| \right\rfloor
= -\sgn(x) \left\lceil -\left| x \right| \right\rceil
= \begin{cases}
  \left\lfloor x \right\rfloor & x \ge 0 \\[5mu]
  \left\lceil x \right\rceil   & x < 0
\end{cases}
</math>

For example, 23.7 gets rounded to 23, and −23.7 gets rounded to −23.

====Rounding away from zero====
One may also '''round away from zero''' (or '''round toward infinity'''): {{mvar|y}} is the integer that is closest to 0 (or equivalently, to {{mvar|x}}) such that {{mvar|x}} is between 0 and {{mvar|y}} (included).

: <math>
y = \sgn(x) \left\lceil \left| x \right| \right\rceil
= -\sgn(x) \left\lfloor -\left| x \right| \right\rfloor
= \begin{cases}
  \left\lceil x \right\rceil   & x \ge 0 \\[5mu]
  \left\lfloor x \right\rfloor & x < 0
\end{cases}
</math>

For example, 23.2 gets rounded to 24, and −23.2 gets rounded to −24.