Inconsistencies when approximating to the nearest whole number

Question

Does 22.449 approximate to 22 or 23? If we see it one way $22.449≈22$ But on the other hand $22.449≈22.45≈22.5≈23$ Which one is correct?

Answer 1

Your problem seems to be that you implicitly expected that rounding behaves like equality, “less than”, and similar. Speaking in notation, you seem to assume that the $≈$ relation (meaning something like “the left-hand side rounded yields the right-hand side”) behaves like the $=$ relation (equality). I assume that from this you concluded:

$$22.449≈22.45≈22.5≈23 \qquad ⇒ \qquad 22.449 ≈ 23.$$

While you can make such a conclusion for other relation operators such as $=$, $≥$, $⊂$, this is not valid for $≈$ (in your sense). Since the symbol $≈$ suggests this, I do not consider it a good choice of notation here. Another argument against using $≈$ here is that it has a direction, even though its symbol suggests that it doesn’t. Finally, you use the same symbol for rounding to different digits.

Consider the alternative notation:

$$22.449 ~\stackrel{0.01}{⇝}~ 22.45 ~\stackrel{0.1 }{⇝}~ 22.5 ~\stackrel{1 }{⇝}~ 23; \qquad 22.449 ~\stackrel{1 }{⇝}~ 23.$$

Would you still assume that the left statement implies the right statement? And most importantly, would you still consider the right statement correct?

More abstractly speaking, the symbol $≈$ suggests an equivalence relation, though you used it for something that doesn’t fulfil the requirements of such a relation, in particular transitivity. (The others are not fulfilled either, but that’s not the cause of your problem.)

As much as we like all sorts of relations to have the nice properties of equivalence relations, rounding simply doesn’t have them. This sometimes leads to seemingly paradoxical behaviour – usually if you wrap everything up in a manner that suggests that rounding behaves like equality.

Answer 2

IEEE 754 is a standard that defines how to round a number to the nearest integer (at least for computers). There are many conventions, and in some of them 22.5 is rounded to either 22 or 23. However, in all the cases 22.449 is rounded to 22, because it is closer to 22 than to 23.