Numbers and Precision

### The float you started with is already wrong A Python `float` is an IEEE-754 binary64 value: a sign, an 11-bit exponent and a 53-bit significand, all in base two. Every value it can hold is some integer divided by a power of two. One tenth is not such a value — 1/10 in binary is an infinitely repeating expansion, exactly the way 1/3 repeats in decimal — so the literal `0.1` in your source is silently replaced by the nearest binary64 value, which written out in full is `0.1000000000000000055511151231257827021181583404541015625`. Nothing later in the program can recover the tenth you meant, because the tenth was never stored. ### The Decimal constructor is exact, and that is the trap `decimal.Decimal` is a base-ten type: it stores a sign, an integer coefficient and a decimal exponent, so one tenth is representable exactly as coefficient 1 and exponent -1. The catch is that the constructor never rounds and never guesses. Given a `str`, it parses the decimal digits you wrote. Given a `float`, it performs an exact base conversion of the binary value — and since that binary value is a dyadic rational, its exact decimal form is finite, which is why you get fifty-five digits of noise rather than a friendly `0.1`: ```pycon >>> from decimal import Decimal >>> Decimal(0.1) Decimal('0.1000000000000000055511151231257827021181583404541015625') >>> Decimal("0.1") Decimal('0.1') ``` People often read `Decimal(0.1)` as "convert this to a decimal type and clean it up". It does the opposite: it preserves, permanently and visibly, the error that the float already carried. Passing through a float on the way in defeats the entire point of using Decimal. ### Why it matters in practice Base ten is not merely prettier; it makes the sums people expect actually hold. Three tenths added as floats is not three tenths, while three tenths added as Decimals is: ```pycon >>> sum([0.1] * 3) == 0.3 False >>> sum([Decimal("0.1")] * 3) == Decimal("0.3") True ``` That is the property money, tax rates, invoice lines and any regulated arithmetic depend on. The usual production discipline is that a monetary value is never a float anywhere in the process: it arrives as a string (JSON text, an HTML form field, a CSV cell) or as an integer number of minor units, and is turned into a Decimal at the boundary. Watch the boundary carefully — a JSON parser will hand you a float for `1.15` unless you tell it not to, and a database driver may or may not hand back Decimals for a numeric column. ### Making the mistake loud The decimal module has a signal for exactly this. Turning on the `FloatOperation` trap in the current context makes building a Decimal from a float raise instead of quietly succeeding, which turns a class of silent data bugs into a stack trace during tests: ```python from decimal import Decimal, FloatOperation, localcontext with localcontext() as ctx: ctx.traps[FloatOperation] = True try: Decimal(0.1) except FloatOperation: print("refused to build a Decimal from a float") ``` ### What to do when a float is genuinely all you have Sometimes the float is upstream and unfixable. Then you must decide, explicitly, what the float was supposed to mean. If it came from human decimal data, round it back to the number of places that data has — `Decimal(str(x))` uses `repr`'s shortest round-tripping form, and `Decimal(x).quantize(Decimal("0.01"))` rounds under the context's rounding rule. If it came from a physical measurement, the extra digits are meaningless anyway. What you must not do is copy the float in exactly and then pretend the result is exact. ### Where floats sneak in The rule is easy to state and easy to violate, because the float usually appears before your code runs. `json.loads` produces a float for every JSON number with a point in it, so an amount is already damaged by the time you construct a Decimal from it; the fix is the parser hook, `json.loads(payload, parse_float=Decimal)`, which builds Decimals straight from the text the wire carried. A CSV reader hands you strings, which is safer, provided nobody helpfully calls `float()` on the column first. A database driver may return Decimals for a numeric column or floats for a double-precision one, so the column type upstream decides what you receive. Review these three boundaries and the problem largely disappears; review only the arithmetic and it will not. ### Two related details worth knowing First, the constructor is also exempt from the context's precision: `Decimal("1.23456789")` keeps all nine digits no matter what `prec` says, because only *operations* apply the context. If you want the context applied at construction time, use the context's own `create_decimal` method. Second, the constructor accepts `int`, `str`, `tuple` and `Decimal` — an int is always safe, which is why storing money as integer cents and dividing once at the edge is a legitimate alternative design to using Decimal end to end.