# Rounding Errors

Because floating-point numbers have a limited number of digits, they cannot represent all real numbers accurately: when there are more digits than the format allows, the leftover ones are omitted - the number is rounded. There are three reasons why this can be necessary:

• Large Denominators In any base, the larger the denominator of an (irreducible) fraction, the more digits it needs in positional notation. A sufficiently large denominator will require rounding, no matter what the base or number of available digits is. For example, 1/1000 cannot be accurately represented in less than 3 decimal digits, nor can any multiple of it (that does not allow simplifying the fraction).
• Periodical digits Any (irreducible) fraction where the denominator has a prime factor that does not occur in the base requires an infinite number of digits that repeat periodically after a certain point. For example, in decimal 1/4, 3/5 and 8/20 are finite, because 2 and 5 are the prime factors of 10. But 1/3 is not finite, nor is 2/3 or 1/7 or 5/6, because 3 and 7 are not factors of 10. Fractions with a prime factor of 5 in the denominator can be finite in base 10, but not in base 2 - the biggest source of confusion for most novice users of floating-point numbers.
• Non-rational numbers Non-rational numbers cannot be represented as a regular fraction at all, and in positional notation (no matter what base) they require an infinite number of non-recurring digits.

## Rounding modes

There are different methods to do rounding, and this can be very important in programming, because rounding can cause different problems in various contexts that can be addressed by using a better rounding mode. The most common rounding modes are:

• Rounding towards zero - simply truncate the extra digits. The simplest method, but it introduces larger errors than necessary as well as a bias towards zero when dealing with mainly positive or mainly negative numbers.
• Rounding half away from zero - if the truncated fraction is greater than or equal to half the base, increase the last remaining digit. This is the method generally taught in school and used by most people. It minimizes errors, but also introduces a bias (away from zero).
• Rounding half to even also known as banker’s rounding - if the truncated fraction is greater than half the base, increase the last remaining digit. If it is equal to half the base, increase the digit only if that produces an even result. This minimizes errors and bias, and is therefore preferred for bookkeeping.

Examples in base 10:

Towards zero Half away from zero Half to even
1.4 1 1 1
1.5 1 2 2
-1.6 -1 -2 -2
2.6 2 3 3
2.5 2 3 2
-2.4 -2 -2 -2

More rounding methods can be found at Wikipedia.