# PEP 327 -- Decimal Data Type

PEP: | 327 |
---|---|

Title: | Decimal Data Type |

Author: | Facundo Batista <facundo at taniquetil.com.ar> |

Status: | Final |

Type: | Standards Track |

Created: | 17-Oct-2003 |

Python-Version: | 2.4 |

Post-History: | 30-Nov-2003, 02-Jan-2004, 29-Jan-2004 |

Contents

# Abstract

The idea is to have a Decimal data type, for every use where decimals are needed but binary floating point is too inexact.

The Decimal data type will support the Python standard functions and operations, and must comply with the decimal arithmetic ANSI standard X3.274-1996 [1].

Decimal will be floating point (as opposed to fixed point) and will have bounded precision (the precision is the upper limit on the number of significant digits in a result). However, precision is user-settable, and a notion of significant trailing zeroes is supported so that fixed-point usage is also possible.

This work is based on code and test functions written by Eric Price, Aahz and Tim Peters. Just before Python 2.4a1, the decimal.py reference implementation was moved into the standard library; along with the documentation and the test suite, this was the work of Raymond Hettinger. Much of the explanation in this PEP is taken from Cowlishaw's work [2], comp.lang.python and python-dev.

# Motivation

Here I'll expose the reasons of why I think a Decimal data type is needed and why other numeric data types are not enough.

I wanted a Money data type, and after proposing a pre-PEP in comp.lang.python, the community agreed to have a numeric data type with the needed arithmetic behaviour, and then build Money over it: all the considerations about quantity of digits after the decimal point, rounding, etc., will be handled through Money. It is not the purpose of this PEP to have a data type that can be used as Money without further effort.

One of the biggest advantages of implementing a standard is that someone already thought out all the creepy cases for you. And to a standard GvR redirected me: Mike Cowlishaw's General Decimal Arithmetic specification [2]. This document defines a general purpose decimal arithmetic. A correct implementation of this specification will conform to the decimal arithmetic defined in ANSI/IEEE standard 854-1987, except for some minor restrictions, and will also provide unrounded decimal arithmetic and integer arithmetic as proper subsets.

## The problem with binary float

In decimal math, there are many numbers that can't be represented with a fixed number of decimal digits, e.g. 1/3 = 0.3333333333.......

In base 2 (the way that standard floating point is calculated), 1/2 = 0.1, 1/4 = 0.01, 1/8 = 0.001, etc. Decimal 0.2 equals 2/10 equals 1/5, resulting in the binary fractional number 0.001100110011001... As you can see, the problem is that some decimal numbers can't be represented exactly in binary, resulting in small roundoff errors.

So we need a decimal data type that represents exactly decimal numbers. Instead of a binary data type, we need a decimal one.

## Why floating point?

So we go to decimal, but why *floating point*?

Floating point numbers use a fixed quantity of digits (precision) to represent a number, working with an exponent when the number gets too big or too small. For example, with a precision of 5:

1234 ==> 1234e0 12345 ==> 12345e0 123456 ==> 12346e1

(note that in the last line the number got rounded to fit in five digits).

In contrast, we have the example of a `long` integer with infinite
precision, meaning that you can have the number as big as you want,
and you'll never lose any information.

In a fixed point number, the position of the decimal point is fixed. For a fixed point data type, check Tim Peter's FixedPoint at SourceForge [4]. I'll go for floating point because it's easier to implement the arithmetic behaviour of the standard, and then you can implement a fixed point data type over Decimal.

But why can't we have a floating point number with infinite precision? It's not so easy, because of inexact divisions. E.g.: 1/3 = 0.3333333333333... ad infinitum. In this case you should store a infinite amount of 3s, which takes too much memory, ;).

John Roth proposed to eliminate the division operator and force the
user to use an explicit method, just to avoid this kind of trouble.
This generated adverse reactions in comp.lang.python, as everybody
wants to have support for the `/` operator in a numeric data type.

With this exposed maybe you're thinking "Hey! Can we just store the 1 and the 3 as numerator and denominator?", which takes us to the next point.

## Why not rational?

Rational numbers are stored using two integer numbers, the numerator and the denominator. This implies that the arithmetic operations can't be executed directly (e.g. to add two rational numbers you first need to calculate the common denominator).

Quoting Alex Martelli:

The performance implications of the fact that summing two rationals (which take O(M) and O(N) space respectively) gives a rational which takes O(M+N) memory space is just too troublesome. There are excellent Rational implementations in both pure Python and as extensions (e.g., gmpy), but they'll always be a "niche market" IMHO. Probably worth PEPping, not worth doing without Decimal -- which is the right way to represent sums of money, a truly major use case in the real world.

Anyway, if you're interested in this data type, you maybe will want to take a look at PEP 239: Adding a Rational Type to Python.

## So, what do we have?

The result is a Decimal data type, with bounded precision and floating point.

Will it be useful? I can't say it better than Alex Martelli:

Python (out of the box) doesn't let you have binary floating point numberswith whatever precision you specify: you're limited to what your hardware supplies. Decimal, be it used as a fixed or floating point number, should suffer from no such limitation: whatever bounded precision you may specify on number creation (your memory permitting) should work just as well. Most of the expense of programming simplicity can be hidden from application programs and placed in a suitable decimal arithmetic type. As per http://speleotrove.com/decimal/,a single data type can be used for integer, fixed-point, and floating-point decimal arithmetic-- and for money arithmetic which doesn't drive the application programmer crazy.

There are several uses for such a data type. As I said before, I will use it as base for Money. In this case the bounded precision is not an issue; quoting Tim Peters:

A precision of 20 would be way more than enough to account for total world economic output, down to the penny, since the beginning of time.

# General Decimal Arithmetic Specification

Here I'll include information and descriptions that are part of the specification [2] (the structure of the number, the context, etc.). All the requirements included in this section are not for discussion (barring typos or other mistakes), as they are in the standard, and the PEP is just for implementing the standard.

Because of copyright restrictions, I can not copy here explanations taken from the specification, so I'll try to explain it in my own words. I firmly encourage you to read the original specification document [2] for details or if you have any doubt.

## The Arithmetic Model

The specification is based on a decimal arithmetic model, as defined by the relevant standards: IEEE 854 [3], ANSI X3-274 [1], and the proposed revision [5] of IEEE 754 [6].

The model has three components:

- Numbers: just the values that the operation uses as input or output.
- Operations: addition, multiplication, etc.
- Context: a set of parameters and rules that the user can select and which govern the results of operations (for example, the precision to be used).

## Numbers

Numbers may be finite or special values. The former can be represented exactly. The latter are infinites and undefined (such as 0/0).

Finite numbers are defined by three parameters:

- Sign: 0 (positive) or 1 (negative).
- Coefficient: a non-negative integer.
- Exponent: a signed integer, the power of ten of the coefficient multiplier.

The numerical value of a finite number is given by:

(-1)**sign * coefficient * 10**exponent

Special values are named as following:

- Infinity: a value which is infinitely large. Could be positive or negative.
- Quiet NaN ("qNaN"): represent undefined results (
*Not a Number*). Does not cause an Invalid operation condition. The sign in a NaN has no meaning. - Signaling NaN ("sNaN"): also
*Not a Number*, but will cause an Invalid operation condition if used in any operation.

## Context

The context is a set of parameters and rules that the user can select and which govern the results of operations (for example, the precision to be used).

The context gets that name because it surrounds the Decimal numbers, with parts of context acting as input to, and output of, operations. It's up to the application to work with one or several contexts, but definitely the idea is not to get a context per Decimal number. For example, a typical use would be to set the context's precision to 20 digits at the start of a program, and never explicitly use context again.

These definitions don't affect the internal storage of the Decimal numbers, just the way that the arithmetic operations are performed.

The context is mainly defined by the following parameters (see Context Attributes for all context attributes):

- Precision: The maximum number of significant digits that can result from an arithmetic operation (integer > 0). There is no maximum for this value.
- Rounding: The name of the algorithm to be used when rounding is necessary, one of "round-down", "round-half-up", "round-half-even", "round-ceiling", "round-floor", "round-half-down", and "round-up". See Rounding Algorithms below.
- Flags and trap-enablers: Exceptional conditions are grouped into signals, controllable individually, each consisting of a flag (boolean, set when the signal occurs) and a trap-enabler (a boolean that controls behavior). The signals are: "clamped", "division-by-zero", "inexact", "invalid-operation", "overflow", "rounded", "subnormal" and "underflow".

## Default Contexts

The specification defines two default contexts, which should be easily selectable by the user.

Basic Default Context:

- flags: all set to 0
- trap-enablers: inexact, rounded, and subnormal are set to 0; all others are set to 1
- precision: is set to 9
- rounding: is set to round-half-up

Extended Default Context:

- flags: all set to 0
- trap-enablers: all set to 0
- precision: is set to 9
- rounding: is set to round-half-even

## Exceptional Conditions

The table below lists the exceptional conditions that may arise during the arithmetic operations, the corresponding signal, and the defined result. For details, see the specification [2].

Condition | Signal | Result |
---|---|---|

Clamped | clamped | see spec [2] |

Division by zero | division-by-zero | [sign,inf] |

Inexact | inexact | unchanged |

Invalid operation | invalid-operation | [0,qNaN] (or [s,qNaN] or [s,qNaN,d] when the cause is a signaling NaN) |

Overflow | overflow | depends on the rounding mode |

Rounded | rounded | unchanged |

Subnormal | subnormal | unchanged |

Underflow | underflow | see spec [2] |

Note: when the standard talks about "Insufficient storage", as long as this is implementation-specific behaviour about not having enough storage to keep the internals of the number, this implementation will raise MemoryError.

Regarding Overflow and Underflow, there's been a long discussion in python-dev about artificial limits. The general consensus is to keep the artificial limits only if there are important reasons to do that. Tim Peters gives us three:

...eliminating bounds on exponents effectively means overflow (and underflow) can never happen. But overflow

isa valuable safety net in real life fp use, like a canary in a coal mine, giving danger signs early when a program goes insane.Virtually all implementations of 854 use (and as IBM's standard even suggests) "forbidden" exponent values to encode non-finite numbers (infinities and NaNs). A bounded exponent can do this at virtually no extra storage cost. If the exponent is unbounded, then additional bits have to be used instead. This cost remains hidden until more time- and space- efficient implementations are attempted.

Big as it is, the IBM standard is a tiny start at supplying a complete numeric facility. Having no bound on exponent size will enormously complicate the implementations of, e.g., decimal sin() and cos() (there's then no a priori limit on how many digits of pi effectively need to be known in order to perform argument reduction).

Edward Loper give us an example of when the limits are to be crossed: probabilities.

That said, Robert Brewer and Andrew Lentvorski want the limits to be easily modifiable by the users. Actually, this is quite possible:

>>> d1 = Decimal("1e999999999") # at the exponent limit >>> d1 Decimal("1E+999999999") >>> d1 * 10 # exceed the limit, got infinity Traceback (most recent call last): File "<pyshell#3>", line 1, in ? d1 * 10 ... ... Overflow: above Emax >>> getcontext().Emax = 1000000000 # increase the limit >>> d1 * 10 # does not exceed any more Decimal("1.0E+1000000000") >>> d1 * 100 # exceed again Traceback (most recent call last): File "<pyshell#3>", line 1, in ? d1 * 100 ... ... Overflow: above Emax

## Rounding Algorithms

`round-down`: The discarded digits are ignored; the result is
unchanged (round toward 0, truncate):

1.123 --> 1.12 1.128 --> 1.12 1.125 --> 1.12 1.135 --> 1.13

`round-half-up`: If the discarded digits represent greater than or
equal to half (0.5) then the result should be incremented by 1;
otherwise the discarded digits are ignored:

1.123 --> 1.12 1.128 --> 1.13 1.125 --> 1.13 1.135 --> 1.14

`round-half-even`: If the discarded digits represent greater than
half (0.5) then the result coefficient is incremented by 1; if they
represent less than half, then the result is not adjusted; otherwise
the result is unaltered if its rightmost digit is even, or incremented
by 1 if its rightmost digit is odd (to make an even digit):

1.123 --> 1.12 1.128 --> 1.13 1.125 --> 1.12 1.135 --> 1.14

`round-ceiling`: If all of the discarded digits are zero or if the
sign is negative the result is unchanged; otherwise, the result is
incremented by 1 (round toward positive infinity):

1.123 --> 1.13 1.128 --> 1.13 -1.123 --> -1.12 -1.128 --> -1.12

`round-floor`: If all of the discarded digits are zero or if the
sign is positive the result is unchanged; otherwise, the absolute
value of the result is incremented by 1 (round toward negative
infinty):

1.123 --> 1.12 1.128 --> 1.12 -1.123 --> -1.13 -1.128 --> -1.13

`round-half-down`: If the discarded digits represent greater than
half (0.5) then the result is incremented by 1; otherwise the
discarded digits are ignored:

1.123 --> 1.12 1.128 --> 1.13 1.125 --> 1.12 1.135 --> 1.13

`round-up`: If all of the discarded digits are zero the result is
unchanged, otherwise the result is incremented by 1 (round away from
0):

1.123 --> 1.13 1.128 --> 1.13 1.125 --> 1.13 1.135 --> 1.14

# Rationale

I must separate the requirements in two sections. The first is to
comply with the ANSI standard. All the requirements for this are
specified in the Mike Cowlishaw's work [2]. He also provided a
**very large** suite of test cases.

The second section of requirements (standard Python functions support, usability, etc.) is detailed from here, where I'll include all the decisions made and why, and all the subjects still being discussed.

## Explicit construction

The explicit construction does not get affected by the context (there is no rounding, no limits by the precision, etc.), because the context affects just operations' results. The only exception to this is when you're Creating from Context.

### From int or long

There's no loss and no need to specify any other information:

Decimal(35) Decimal(-124)

### From string

Strings containing Python decimal integer literals and Python float literals will be supported. In this transformation there is no loss of information, as the string is directly converted to Decimal (there is not an intermediate conversion through float):

Decimal("-12") Decimal("23.2e-7")

Also, you can construct in this way all special values (Infinity and Not a Number):

Decimal("Inf") Decimal("NaN")

### From float

The initial discussion on this item was what should happen when passing floating point to the constructor:

`Decimal(1.1) == Decimal('1.1')``Decimal(1.1) == Decimal('110000000000000008881784197001252...e-51')`- an exception is raised

Several people alleged that (1) is the better option here, because
it's what you expect when writing `Decimal(1.1)`. And quoting John
Roth, it's easy to implement:

It's not at all difficult to find where the actual number ends and where the fuzz begins. You can do it visually, and the algorithms to do it are quite well known.

But If I *really* want my number to be
`Decimal('110000000000000008881784197001252...e-51')`, why can't I
write `Decimal(1.1)`? Why should I expect Decimal to be "rounding"
it? Remember that `1.1` *is* binary floating point, so I can
predict the result. It's not intuitive to a beginner, but that's the
way it is.

Anyway, Paul Moore showed that (1) can't work, because:

(1) says D(1.1) == D('1.1') but 1.1 == 1.1000000000000001 so D(1.1) == D(1.1000000000000001) together: D(1.1000000000000001) == D('1.1')

which is wrong, because if I write `Decimal('1.1')` it is exact, not
`D(1.1000000000000001)`. He also proposed to have an explicit
conversion to float. bokr says you need to put the precision in the
constructor and mwilson agreed:

d = Decimal (1.1, 1) # take float value to 1 decimal place d = Decimal (1.1) # gets `places` from pre-set context

But Alex Martelli says that:

Constructing with some specified precision would be fine. Thus, I think "construction from float with some default precision" runs a substantial risk of tricking naive users.

So, the accepted solution through c.l.p is that you can not call Decimal with a float. Instead you must use a method: Decimal.from_float(). The syntax:

Decimal.from_float(floatNumber, [decimal_places])

where `floatNumber` is the float number origin of the construction
and `decimal_places` are the number of digits after the decimal
point where you apply a round-half-up rounding, if any. In this way
you can do, for example:

Decimal.from_float(1.1, 2): The same as doing Decimal('1.1'). Decimal.from_float(1.1, 16): The same as doing Decimal('1.1000000000000001'). Decimal.from_float(1.1): The same as doing Decimal('1100000000000000088817841970012523233890533447265625e-51').

Based on later discussions, it was decided to omit from_float() from the API for Py2.4. Several ideas contributed to the thought process:

Interactions between decimal and binary floating point force the user to deal with tricky issues of representation and round-off. Avoidance of those issues is a primary reason for having the module in the first place.

The first release of the module should focus on that which is safe, minimal, and essential.

While theoretically nice, real world use cases for interactions between floats and decimals are lacking. Java included float/decimal conversions to handle an obscure case where calculations are best performed in decimal eventhough a legacy data structure requires the inputs and outputs to be stored in binary floating point.

If the need arises, users can use string representations as an intermediate type. The advantage of this approach is that it makes explicit the assumptions about precision and representation (no wondering what is going on under the hood).

The Java docs for BigDecimal(double val) reflected their experiences with the constructor:

The results of this constructor can be somewhat unpredictable and its use is generally not recommended.

### From tuples

Aahz suggested to construct from tuples: it's easier
to implement `eval()`'s round trip and "someone who has numeric
values representing a Decimal does not need to convert them to a
string."

The structure will be a tuple of three elements: sign, number and exponent. The sign is 1 or 0, the number is a tuple of decimal digits and the exponent is a signed int or long:

Decimal((1, (3, 2, 2, 5), -2)) # for -32.25

Of course, you can construct in this way all special values:

Decimal( (0, (0,), 'F') ) # for Infinity Decimal( (0, (0,), 'n') ) # for Not a Number

### From Decimal

No mystery here, just a copy.

### Syntax for All Cases

Decimal(value1) Decimal.from_float(value2, [decimal_places])

where `value1` can be int, long, string, 3-tuple or Decimal,
`value2` can only be float, and `decimal_places` is an optional
non negative int.

### Creating from Context

This item arose in python-dev from two sources in parallel. Ka-Ping Yee proposes to pass the context as an argument at instance creation (he wants the context he passes to be used only in creation time: "It would not be persistent"). Tony Meyer asks from_string to honor the context if it receives a parameter "honour_context" with a True value. (I don't like it, because the doc specifies that the context be honored and I don't want the method to comply with the specification regarding the value of an argument.)

Tim Peters gives us a reason to have a creation that uses context:

In general number-crunching, literals may be given to high precision, but that precision isn't free andusuallyisn't needed

Casey Duncan wants to use another method, not a bool arg:

I find boolean arguments a general anti-pattern, especially given we have class methods. Why not use an alternate constructor like Decimal.rounded_to_context("3.14159265").

In the process of deciding the syntax of that, Tim came up with a better idea: he proposes not to have a method in Decimal to create with a different context, but having instead a method in Context to create a Decimal instance. Basically, instead of:

D.using_context(number, context)

it will be:

context.create_decimal(number)

From Tim:

While all operations in the spec except for the two to-string operations use context, no operations in the spec support an optional local context. That the Decimal() constructor ignores context by default is an extension to the spec. We must supply a context-honoring from-string operation to meet the spec. I recommend against any concept of "local context" in any operation -- it complicates the model and isn't necessary.

So, we decided to use a context method to create a Decimal that will use (only to be created) that context in particular (for further operations it will use the context of the thread). But, a method with what name?

Tim Peters proposes three methods to create from diverse sources
(from_string, from_int, from_float). I proposed to use one method,
`create_decimal()`, without caring about the data type. Michael
Chermside: "The name just fits my brain. The fact that it uses the
context is obvious from the fact that it's Context method".

The community agreed with that. I think that it's OK because a newbie will not be using the creation method from Context (the separate method in Decimal to construct from float is just to prevent newbies from encountering binary floating point issues).

So, in short, if you want to create a Decimal instance using a particular context (that will be used just at creation time and not any further), you'll have to use a method of that context:

# n is any datatype accepted in Decimal(n) plus float mycontext.create_decimal(n)

Example:

>>> # create a standard decimal instance >>> Decimal("11.2233445566778899") Decimal("11.2233445566778899") >>> >>> # create a decimal instance using the thread context >>> thread_context = getcontext() >>> thread_context.prec 28 >>> thread_context.create_decimal("11.2233445566778899") Decimal("11.2233445566778899") >>> >>> # create a decimal instance using other context >>> other_context = thread_context.copy() >>> other_context.prec = 4 >>> other_context.create_decimal("11.2233445566778899") Decimal("11.22")

## Implicit construction

As the implicit construction is the consequence of an operation, it will be affected by the context as is detailed in each point.

John Roth suggested that "The other type should be handled in the same way the decimal() constructor would handle it". But Alex Martelli thinks that

this total breach with Python tradition would be a terrible mistake. 23+"43" is NOT handled in the same way as 23+int("45"), and a VERY good thing that is too. It's a completely different thing for a user to EXPLICITLY indicate they want construction (conversion) and to just happen to sum two objects one of which by mistake could be a string.

So, here I define the behaviour again for each data type.

### From int or long

An int or long is a treated like a Decimal explicitly constructed from
Decimal(str(x)) in the current context (meaning that the to-string rules
for rounding are applied and the appropriate flags are set). This
guarantees that expressions like `Decimal('1234567') + 13579` match
the mental model of `Decimal('1234567') + Decimal('13579')`. That
model works because all integers are representable as strings without
representation error.

### From string

Everybody agrees to raise an exception here.

### From float

Aahz is strongly opposed to interact with float, suggesting an explicit conversion:

The problem is that Decimal is capable of greater precision, accuracy, and range than float.

The example of the valid python expression, `35 + 1.1`, seems to suggest
that `Decimal(35) + 1.1` should also be valid. However, a closer look
shows that it only demonstrates the feasibility of integer to floating
point conversions. Hence, the correct analog for decimal floating point
is `35 + Decimal(1.1)`. Both coercions, int-to-float and int-to-Decimal,
can be done without incurring representation error.

The question of how to coerce between binary and decimal floating point
is more complex. I proposed allowing the interaction with float,
making an exact conversion and raising ValueError if exceeds the
precision in the current context (this is maybe too tricky, because
for example with a precision of 9, `Decimal(35) + 1.2` is OK but
`Decimal(35) + 1.1` raises an error).

This resulted to be too tricky. So tricky, that c.l.p agreed to raise TypeError in this case: you could not mix Decimal and float.

### From Decimal

There isn't any issue here.

## Use of Context

In the last pre-PEP I said that "The Context must be omnipresent, meaning that changes to it affects all the current and future Decimal instances". I was wrong. In response, John Roth said:

The context should be selectable for the particular usage. That is, it should be possible to have several different contexts in play at one time in an application.

In comp.lang.python, Aahz explained that the idea is to have a "context per thread". So, all the instances of a thread belongs to a context, and you can change a context in thread A (and the behaviour of the instances of that thread) without changing nothing in thread B.

Also, and again correcting me, he said:

(the) Context applies only to operations, not to Decimal instances; changing the Context does not affect existing instances if there are no operations on them.

Arguing about special cases when there's need to perform operations with other rules that those of the current context, Tim Peters said that the context will have the operations as methods. This way, the user "can create whatever private context object(s) it needs, and spell arithmetic as explicit method calls on its private context object(s), so that the default thread context object is neither consulted nor modified".

## Python Usability

Decimal should support the basic arithmetic (

`+, -, *, /, //, **, %, divmod`) and comparison (`==, !=, <, >, <=, >=, cmp`) operators in the following cases (check Implicit Construction to see what types could OtherType be, and what happens in each case):- Decimal op Decimal
- Decimal op otherType
- otherType op Decimal
- Decimal op= Decimal
- Decimal op= otherType

Decimal should support unary operators (

`-, +, abs`).repr() should round trip, meaning that:

m = Decimal(...) m == eval(repr(m))

Decimal should be immutable.

Decimal should support the built-in methods:

- min, max
- float, int, long
- str, repr
- hash
- bool (0 is false, otherwise true)

There's been some discussion in python-dev about the behaviour of
`hash()`. The community agrees that if the values are the same, the
hashes of those values should also be the same. So, while Decimal(25)
== 25 is True, hash(Decimal(25)) should be equal to hash(25).

The detail is that you can NOT compare Decimal to floats or strings, so we should not worry about them giving the same hashes. In short:

hash(n) == hash(Decimal(n)) # Only if n is int, long, or Decimal

Regarding str() and repr() behaviour, Ka-Ping Yee proposes that repr() have the same behaviour as str() and Tim Peters proposes that str() behave like the to-scientific-string operation from the Spec.

This is possible, because (from Aahz): "The string form already contains all the necessary information to reconstruct a Decimal object".

And it also complies with the Spec; Tim Peters:

There's no requirement to have a methodnamed"to_sci_string", the only requirement is thatsomeway to spell to-sci-string's functionality be supplied. The meaning of to-sci-string is precisely specified by the standard, and is a good choice for both str(Decimal) and repr(Decimal).

# Documentation

This section explains all the public methods and attributes of Decimal and Context.

## Decimal Attributes

Decimal has no public attributes. The internal information is stored in slots and should not be accessed by end users.

## Decimal Methods

Following are the conversion and arithmetic operations defined in the Spec, and how that functionality can be achieved with the actual implementation.

to-scientific-string: Use builtin function

`str()`:>>> d = Decimal('123456789012.345') >>> str(d) '1.23456789E+11'

to-engineering-string: Use method

`to_eng_string()`:>>> d = Decimal('123456789012.345') >>> d.to_eng_string() '123.456789E+9'

to-number: Use Context method

`create_decimal()`. The standard constructor or`from_float()`constructor cannot be used because these do not use the context (as is specified in the Spec for this conversion).abs: Use builtin function

`abs()`:>>> d = Decimal('-15.67') >>> abs(d) Decimal('15.67')

add: Use operator

`+`:>>> d = Decimal('15.6') >>> d + 8 Decimal('23.6')

subtract: Use operator

`-`:>>> d = Decimal('15.6') >>> d - 8 Decimal('7.6')

compare: Use method

`compare()`. This method (and not the built-in function cmp()) should only be used when dealing with*special values*:>>> d = Decimal('-15.67') >>> nan = Decimal('NaN') >>> d.compare(23) '-1' >>> d.compare(nan) 'NaN' >>> cmp(d, 23) -1 >>> cmp(d, nan) 1

divide: Use operator

`/`:>>> d = Decimal('-15.67') >>> d / 2 Decimal('-7.835')

divide-integer: Use operator

`//`:>>> d = Decimal('-15.67') >>> d // 2 Decimal('-7')

max: Use method

`max()`. Only use this method (and not the built-in function max()) when dealing with*special values*:>>> d = Decimal('15') >>> nan = Decimal('NaN') >>> d.max(8) Decimal('15') >>> d.max(nan) Decimal('NaN')

min: Use method

`min()`. Only use this method (and not the built-in function min()) when dealing with*special values*:>>> d = Decimal('15') >>> nan = Decimal('NaN') >>> d.min(8) Decimal('8') >>> d.min(nan) Decimal('NaN')

minus: Use unary operator

`-`:>>> d = Decimal('-15.67') >>> -d Decimal('15.67')

plus: Use unary operator

`+`:>>> d = Decimal('-15.67') >>> +d Decimal('-15.67')

multiply: Use operator

`*`:>>> d = Decimal('5.7') >>> d * 3 Decimal('17.1')

normalize: Use method

`normalize()`:>>> d = Decimal('123.45000') >>> d.normalize() Decimal('123.45') >>> d = Decimal('120.00') >>> d.normalize() Decimal('1.2E+2')

quantize: Use method

`quantize()`:>>> d = Decimal('2.17') >>> d.quantize(Decimal('0.001')) Decimal('2.170') >>> d.quantize(Decimal('0.1')) Decimal('2.2')

remainder: Use operator

`%`:>>> d = Decimal('10') >>> d % 3 Decimal('1') >>> d % 6 Decimal('4')

remainder-near: Use method

`remainder_near()`:>>> d = Decimal('10') >>> d.remainder_near(3) Decimal('1') >>> d.remainder_near(6) Decimal('-2')

round-to-integral-value: Use method

`to_integral()`:>>> d = Decimal('-123.456') >>> d.to_integral() Decimal('-123')

same-quantum: Use method

`same_quantum()`:>>> d = Decimal('123.456') >>> d.same_quantum(Decimal('0.001')) True >>> d.same_quantum(Decimal('0.01')) False

square-root: Use method

`sqrt()`:>>> d = Decimal('123.456') >>> d.sqrt() Decimal('11.1110756')

power: User operator

`**`:>>> d = Decimal('12.56') >>> d ** 2 Decimal('157.7536')

Following are other methods and why they exist:

`adjusted()`: Returns the adjusted exponent. This concept is defined in the Spec: the adjusted exponent is the value of the exponent of a number when that number is expressed as though in scientific notation with one digit before any decimal point:>>> d = Decimal('12.56') >>> d.adjusted() 1

`from_float()`: Class method to create instances from float data types:>>> d = Decimal.from_float(12.35) >>> d Decimal('12.3500000')

`as_tuple()`: Show the internal structure of the Decimal, the triple tuple. This method is not required by the Spec, but Tim Peters proposed it and the community agreed to have it (it's useful for developing and debugging):>>> d = Decimal('123.4') >>> d.as_tuple() (0, (1, 2, 3, 4), -1) >>> d = Decimal('-2.34e5') >>> d.as_tuple() (1, (2, 3, 4), 3)

## Context Attributes

These are the attributes that can be changed to modify the context.

`prec`(int): the precision:>>> c.prec 9

`rounding`(str): rounding type (how to round):>>> c.rounding 'half_even'

`trap_enablers`(dict): if trap_enablers[exception] = 1, then an exception is raised when it is caused:>>> c.trap_enablers[Underflow] 0 >>> c.trap_enablers[Clamped] 0

`flags`(dict): when an exception is caused, flags[exception] is incremented (whether or not the trap_enabler is set). Should be reset by the user of Decimal instance:>>> c.flags[Underflow] 0 >>> c.flags[Clamped] 0

`Emin`(int): minimum exponent:>>> c.Emin -999999999

`Emax`(int): maximum exponent:>>> c.Emax 999999999

`capitals`(int): boolean flag to use 'E' (True/1) or 'e' (False/0) in the string (for example, '1.32e+2' or '1.32E+2'):>>> c.capitals 1

## Context Methods

The following methods comply with Decimal functionality from the Spec. Be aware that the operations that are called through a specific context use that context and not the thread context.

To use these methods, take note that the syntax changes when the operator is binary or unary, for example:

>>> mycontext.abs(Decimal('-2')) '2' >>> mycontext.multiply(Decimal('2.3'), 5) '11.5'

So, the following are the Spec operations and conversions and how to
achieve them through a context (where `d` is a Decimal instance and
`n` a number that can be used in an Implicit construction):

- to-scientific-string:
`to_sci_string(d)` - to-engineering-string:
`to_eng_string(d)` - to-number:
`create_decimal(number)`, see Explicit construction for`number`. - abs:
`abs(d)` - add:
`add(d, n)` - subtract:
`subtract(d, n)` - compare:
`compare(d, n)` - divide:
`divide(d, n)` - divide-integer:
`divide_int(d, n)` - max:
`max(d, n)` - min:
`min(d, n)` - minus:
`minus(d)` - plus:
`plus(d)` - multiply:
`multiply(d, n)` - normalize:
`normalize(d)` - quantize:
`quantize(d, d)` - remainder:
`remainder(d)` - remainder-near:
`remainder_near(d)` - round-to-integral-value:
`to_integral(d)` - same-quantum:
`same_quantum(d, d)` - square-root:
`sqrt(d)` - power:
`power(d, n)`

The `divmod(d, n)` method supports decimal functionality through
Context.

These are methods that return useful information from the Context:

`Etiny()`: Minimum exponent considering precision.>>> c.Emin -999999999 >>> c.Etiny() -1000000007

`Etop()`: Maximum exponent considering precision.>>> c.Emax 999999999 >>> c.Etop() 999999991

`copy()`: Returns a copy of the context.

# Reference Implementation

As of Python 2.4-alpha, the code has been checked into the standard library. The latest version is available from:

http://svn.python.org/view/python/trunk/Lib/decimal.py

The test cases are here:

http://svn.python.org/view/python/trunk/Lib/test/test_decimal.py

# References

[1] | (1, 2) ANSI standard X3.274-1996 (Programming Language REXX):
http://www.rexxla.org/Standards/ansi.html |

[2] | (1, 2, 3, 4, 5, 6, 7, 8) General Decimal Arithmetic specification (Cowlishaw):
http://speleotrove.com/decimal/decarith.html (related
documents and links at http://speleotrove.com/decimal/) |

[3] | ANSI/IEEE standard 854-1987 (Radix-Independent Floating-Point Arithmetic): http://www.cs.berkeley.edu/~ejr/projects/754/private/drafts/854-1987/dir.html (unofficial text; official copies can be ordered from http://standards.ieee.org/catalog/ordering.html) |

[4] | Tim Peter's FixedPoint at SourceForge: http://fixedpoint.sourceforge.net/ |

[5] | IEEE 754 revision: http://grouper.ieee.org/groups/754/revision.html |

[6] | IEEE 754 references: http://babbage.cs.qc.edu/courses/cs341/IEEE-754references.html |

# Copyright

This document has been placed in the public domain.