On Thu, Oct 29, 2009 at 12:43 AM, Christopher Dicely <cmdicely / gmail.com> wrote: > BigDecimal actually works with decimal numbers, which are a subset of > rational numbers; Rational does precise math with rational numbers. I'm afraid that this statement might be confusing some folks, because, em, it ain't exactly true. Mathematically we have Integers which have an infinite number of values between -Infinity, -1, 0, 1, .. Infinity Real Numbers which include the integers but have an infinite number of values between each integer Rational Numbers which are a subset of the real numbers which can be expressed as a fraction with integer numerator and divisor Irrational Numbers which are the other Real numbers, this includes numbers like Pi, and e. In computers we have integers which represent some subset of the mathematical integers which can be represented in some small number of bits floating point numbers which represent numbers in a form of scientific notation, a number in some base (usually 2 or 10), which usually is assumed to have a radix point preceding it), and an exponent represented by a signed integer using a small number of bits which represents how may digits/bits to shift the radix point right or left. Most floating points representations use a binary base, so the radix point marks the division between the bits which represent the part of the number greater than 0, and those which represent a binary fraction. But the representation can also be decimal, with each digit taking four bits, the radix point represents the decimal point, and moves left and right in multiples of a digit. Now floats have a few problems: 1) They trade off precision for range. For values near zero (when the exponent is zero), the last bit represents a rather small increment, but If I need to represent say 100.0002, then for a binary float I need at least 4 bits to represent that 100 part, so the least significant bit has a value which is 2^4 times bigger than in the representation of 0.0002. So as the exponent increases, the smallest difference I can represent gets bigger, and if I add two floating point number I can only preserve as many fractional digits as the larger number can represent. 2) Depending on the base, certain fractional values can't be exactly represented, this is easier to describe for a base 10 float. For example the value 1/3, even though it is a Rational, can't be exactly represented as a decimal float, since the fractional part is .333333333.... with an infinite number of 3 digits needed to exactly represent the value. So floating point numbers, whether binary, decimal or some other base have an infinite number of un-representable real numbers, both rationals and irrationals. You can change the parameters of this problem by changing the base and increasing the number of digits, but you can't get away from it completely. Ruby tackes the integer vs Integer problem by having Fixnums produce Bignums when necessary, Bignums have an alternative representation without a fixed number of bits. It also has the Rational class which represents a mathematical rational number as a numerator and demominator as a reduced Fraction. This allows mathematical rationals to be represented at some cost in performance. Whenever a Rational is involved in an arithmetic operation the result needs to be reduced, which involves calculating the greatest common divisor. The DateTime uses a Rational to represent a point in time as a julian 'day' with the fraction representing the part of the day since midnight, and benchmarking code doing DateTime manipulations almost invariable reveals that 99% of the time is spend doing gcd calculations. But floats are floats, and BigDecimals are just Ruby's implementation of base 10 floats. For monetary calculations, the best approach is usually to use integers and scale them, so for US currency you might use the number of cents as an integer (Fixnum/Bignum depending on the budget <G>). Or in cases where it's needed in some binary fraction of a cent. -- Rick DeNatale Blog: http://talklikeaduck.denhaven2.com/ Twitter: http://twitter.com/RickDeNatale WWR: http://www.workingwithrails.com/person/9021-rick-denatale LinkedIn: http://www.linkedin.com/in/rickdenatale