ArithmeticException thrown during BigDecimal.divide

Question

I thought java.math.BigDecimal is supposed to be The Answer™ to the need of performing infinite precision arithmetic with decimal numbers.

Consider the following snippet:

import java.math.BigDecimal;
//...

final BigDecimal one = BigDecimal.ONE;
final BigDecimal three = BigDecimal.valueOf(3);
final BigDecimal third = one.divide(three);

assert third.multiply(three).equals(one); // this should pass, right?

I expect the assert to pass, but in fact the execution doesn't even get there: one.divide(three) causes ArithmeticException to be thrown!

Exception in thread "main" java.lang.ArithmeticException:
Non-terminating decimal expansion; no exact representable decimal result.
    at java.math.BigDecimal.divide

It turns out that this behavior is explicitly documented in the API:

In the case of divide, the exact quotient could have an infinitely long decimal expansion; for example, 1 divided by 3. If the quotient has a non-terminating decimal expansion and the operation is specified to return an exact result, an ArithmeticException is thrown. Otherwise, the exact result of the division is returned, as done for other operations.

Browsing around the API further, one finds that in fact there are various overloads of divide that performs inexact division, i.e.:

final BigDecimal third = one.divide(three, 33, RoundingMode.DOWN);
System.out.println(three.multiply(third));
// prints "0.999999999999999999999999999999999"

Of course, the obvious question now is "What's the point???". I thought BigDecimal is the solution when we need exact arithmetic, e.g. for financial calculations. If we can't even divide exactly, then how useful can this be? Does it actually serve a general purpose, or is it only useful in a very niche application where you fortunately just don't need to divide at all?

If this is not the right answer, what CAN we use for exact division in financial calculation? (I mean, I don't have a finance major, but they still use division, right???).

Endogamy

By the way, I'd really appreciate insight from people who've worked with financial software. I often heard BigDecimal being advocated over double, and I'd like to hear how they deal with divide. – Endogamy

I'm working on a very complex Java apps displaying lots of financial information in JTables (btw, JTables so complex and so often updated that I was forced to use several of the java.sun.com/products/jfc/tsc/articles/ChristmasTree optimizations to have my JTables performs acceptably on fast machines). The two things that stand out: the unit is typically the cent and I've got my own NumDiv rational abstraction. Note that it is not a shortcoming of the Java platform: at worst it's a shortcoming, if any, of the default JAVA APIs. – Cybil

Answer 1

If this is not the right answer, what CAN we use for exact division in financial calculation? (I mean, I don't have a finance major, but they still use division, right???).

Then I was in primary school¹, they taught me that when you divide by 1 by 3 you get a 0.33333... i.e. a recurring decimal. Division of numbers represented in decimal form is NOT exact. In fact for any fixed base there will be fractions (the result of dividing one integer by another) that cannot be represented exactly as a finite precision floating point number in that base. (The number will have a recurring part ...)

When you do financial calculations involving division, you have to consider the what to do with a recurring fraction. You can round it up, or down, or to the nearest whole number, or something else, but basically you cannot just forget about the issue.

The BigDecimal javadoc says this:

The BigDecimal class gives its user complete control over rounding behavior. If no rounding mode is specified and the exact result cannot be represented, an exception is thrown; otherwise, calculations can be carried out to a chosen precision and rounding mode by supplying an appropriate MathContext object to the operation.

In other words, it is your responsibility to tell BigDecimal what to do about rounding.

EDIT - in response to these followups from the OP.

How does BigDecimal detect infinite recurring decimal?

It does not explicitly detect the recurring decimal. It simply detects that the result of some operation cannot be represented exactly using the specified precision; e.g. too many digits are required after the decimal point for an exact representation.

It must keep track of and detect a cycle in the dividend. It COULD HAVE chosen to handle this another way, by marking where the recurring portion is, etc.

I suppose that BigDecimal could have been specified to represent a recurring decimal exactly; i.e. as a BigRational class. However, this would make the implementation more complicated and more expensive to use². And since most people expect numbers to be displayed in decimal, and the problem of recurring decimal recurs at that point.

The bottom line is that this extra complexity and runtime cost would be inappropriate for typical use-cases for BigDecimal. This includes financial calculations, where accounting conventions do not allow you to use recurring decimals.

---

^{1 - It was an excellent primary school. You may have been taught this in high school.}

^{2 - Either you try to remove common factors of the divisor and dividend (computationally expensive), or allow them to grow without bounds (expensive in space usage and computationally expensive for subsequent operations).}

Prudhoe

How does BigDecimal detect infinite recurring decimal? It must keep track of and detect a cycle in the dividend. It COULD HAVE chosen to handle this another way, by marking where the recurring portion is, etc. – Endogamy

@Stephen: Really appreciated your investigation into how it detects infinite expansion. I went and did it too after I made the comment, but I thought I should give you a chance to answer it yourself =) – Endogamy

Answer 2

The class is BigDecimal not BigFractional. From some of your comments it sounds like you just want to complain that someone didn't build in all possible number handling algorithms into this class. Financial apps do not need infinite decimal precision; just perfectly accurate values to the precision required (typically 0, 2, 4, or 5 decimal digits).

Actually I have dealt with many financial applications that use double. I don't like it but that was the way they are written (not in Java either). When there are exchange rates and unit conversions then there are both the potential of rounding and bruising problems. BigDecimal eliminates the later but there is still the former for division.

Wester

+1. You nailed it right with "BigDecimal not BigFractional". And your insight on double for financial application is appreciated. – Endogamy

Answer 3

If you want to work with decimals, not rational numbers, and you need exact arithmetics before the final rounding (rounding to cents or something), here's a little trick.

You can always manipulate your formulas so that there's only one final division. That way you won't lose precision during calculations and you'll always get the correctly rounded result. For instance

a/b + c

equals

(a + bc) / b.

Panhellenism

Oh what are all the engineers doing, We should cringe whenever we had to do manual manipulation for the computer in this time and age. – Silas

Answer 4

By the way, I'd really appreciate insight from people who've worked with financial software. I often heard BigDecimal being advocated over double

In financial reports we use alwasy BigDecimal with scale = 2 and ROUND_HALF_UP, since all printed values in a report must be lead to a reproducable result. If someone checks this using a simple calculator.

In switzerland they round to 0.05 since they no longer have 1 or 2 Rappen coins.

Semiramis

Answer 5

You should prefer BigDecimal for finance calculations. Rounding should be specified by the business. E.g. an amount (100,00$) has to be split equally across three accounts. There has to be a business rule which account takes the extra cent.

Double, floats are not approriate for use in financial applications because they can not represent fractions of 1 precisely that are not exponentials of 2. E.g. consider 0.6 = 6/10 = 1*1/2 + 0*1/4 + 0*1/8 + 1*1/16 + ... = 0.1001...b

For mathematic calculations you can use a symbolic number, e.g. storing denominator and numerator or even a whole expression (e.g. this number is sqrt(5)+3/4). As this is not the main use case of the java api you won' find it there.

Gorcock

Answer 6

Is there a need for

a=1/3;
b=a*3;

resulting in

b==1;

in financial systems? I guess not. In financial systems it is defined, which roundmode and scale has to be used, when doing calculations. In some situations, the roundmode and scale is defined in the law. All components can rely on such a defined behaviour. Returning b==1 would be a failure, because it would not fulfill the specified behaviour. This is very important when calculating prices etc.

It is like the IEEE 754 specifications for representing floats in binary digits. A component must not optimize a "better" representation without loss of information, because this will break the contract.

Considering

Does double satisfy the finance sector contract? Can you not control the rounding mode etc for double as well? – Endogamy

When working with quantities that must balance, the right approach for performing division is to keep both the quotient and remainder. Divide $1.00 among seven people will reveal that there's $0.14 for everyone, plus two extra pennies that can be allocated according to some policy. – Vennieveno

Answer 7

To divide save, you have to set the MATHcontext,

BigDecimal bd = new BigDecimal(12.12, MathContext.DECIMAL32).divide(new BigDecimal(2)).setScale(2, RoundingMode.HALF_UP);

Footfall

Answer 8

I accept that Java doesn't have great support for representing fractions, but you have to realise that it is impossible to keep things entirely precise when working with computers. At least in this case, the exception is telling you that precision is being lost.

As far as I know, "infinite precision arithmetic with decimal numbers" just isn't going to happen. If you have to work with decimals, what you're doing is probably fine, just catch the exceptions. Otherwise, a quick google search finds some interesting resources for working with fractions in Java:

http://commons.apache.org/math/userguide/fraction.html

http://www.merriampark.com/fractions.htm

Best way to represent a fraction in Java?

Gail

It is possible to keep things entirely precise. There is something called fractions. It's simply the case that the current libraries we have are limited. – Silas

@Silas - not really; consider irrational numbers. – Gail

For irrational numbers the library would store the number in the form of an expression. E.g. sqrt 2 is stored as sqrt 2, not 1.4142135..... etc. That's exactly how we do it in high school maths; if pre-resolving would give us imprecise results, then we only resolve the number in the last step when we need to display it. There's no reason why a computer couldn't do this. – Silas

@Silas given enough time, cpu and memory. It's just far too inefficient (although you may be able to blame computer architectures partially for this). Try it if you like; until then I wouldn't blame the libraries. – Gail

BigInteger is also "far too inefficient" compared to primitive int. That's the whole point of a code library: writing "far too inefficient" code which people use when the use case can sacrifice speed for functionality. "far too inefficient" is not a reason against building a dedicated BigRealNumber library. – Silas

@Silas Agreed; I thought you were arguing that the current libraries were wrong rather than that there was space for a new library. – Gail

I'm arguing for both. That there is a space for a BigRealNumber but Java currently doesn't provide one even after 20 years. And the same goes for many other languages. – Silas

Answer 9

Notice we are using a computer... A computer has a lot of ram and precision takes ram. So when you want an infinite precision you need
(infinite * infinite) ^ (infinite * Integer.MAX_VALUE) terrabyte ram...

I know 1 / 3 is 0.333333... and it should be possible to store it in ram like "one divided by three" and then you can multiply it back and you should have 1. But I don't think Java has something like that...
Maybe you have to win the Nobel Price for writing something doing that. ;-)

Amphitryon

"Maple supports both hardware (double) precision and infinite precision computations." maplesoft.com/products/maple/compare/numeric_computation.aspx – Endogamy

I wouldn't expect anything less from a system like Maple! :) Looking at a more "general purpose" language (like Java), Python does have built-in support for rational numbers: docs.python.org/library/fractions.html – Scoutmaster

@Martin: yes, the RAM is finite, but there are still ways to represent infiniteness symbolically. Just look at double: it has a POSITIVE_INFINITY and NEGATIVE_INFINITY. double only has 64 bits. The computer could've done something like storing .(3), where (number) is the repeating part. In fact, lots of implementations already do this (and no, they didn't all get Nobel Prizes for it). – Endogamy

No Nobel prizes for Maths, Computer Science, or any Engineering discipline. Sorry :-) – Prudhoe