We're back after a server migration that caused effbot.org to fall over a bit harder than expected. Expect some glitches.

# decimal -- Decimal floating point arithmetic

New in version 2.4.

The decimal module provides support for decimal floating point arithmetic. It offers several advantages over the float datatype:

• Decimal numbers can be represented exactly. In contrast, numbers like `1.1` do not have an exact representation in binary floating point. End users typically would not expect ``` 1.1``` to display as `1.1000000000000001` as it does with binary floating point.

• The exactness carries over into arithmetic. In decimal floating point, “`0.1 + 0.1 + 0.1 - 0.3`” is exactly equal to zero. In binary floating point, result is ``` 5.5511151231257827e-017```. While near to zero, the differences prevent reliable equality testing and differences can accumulate. For this reason, decimal would be preferred in accounting applications which have strict equality invariants.

• The decimal module incorporates a notion of significant places so that “`1.30 + 1.20`” is `2.50`. The trailing zero is kept to indicate significance. This is the customary presentation for monetary applications. For multiplication, the “schoolbook” approach uses all the figures in the multiplicands. For instance, “```1.3 * 1.2```” gives `1.56` while “```1.30 * 1.20```” gives `1.5600`.

• Unlike hardware based binary floating point, the decimal module has a user settable precision (defaulting to 28 places) which can be as large as needed for a given problem:

```>>> getcontext().prec = 6
>>> Decimal(1) / Decimal(7)
Decimal("0.142857")
>>> getcontext().prec = 28
>>> Decimal(1) / Decimal(7)
Decimal("0.1428571428571428571428571429")
```
• Both binary and decimal floating point are implemented in terms of published standards. While the built-in float type exposes only a modest portion of its capabilities, the decimal module exposes all required parts of the standard. When needed, the programmer has full control over rounding and signal handling.

The module design is centered around three concepts: the decimal number, the context for arithmetic, and signals.

A decimal number is immutable. It has a sign, coefficient digits, and an exponent. To preserve significance, the coefficient digits do not truncate trailing zeroes. Decimals also include special values such as `Infinity`, ``` -Infinity```, and `NaN`. The standard also differentiates `-0` from `+0`.

The context for arithmetic is an environment specifying precision, rounding rules, limits on exponents, flags indicating the results of operations, and trap enablers which determine whether signals are treated as exceptions. Rounding options include `ROUND_CEILING`, `ROUND_DOWN`, ``` ROUND_FLOOR```, `ROUND_HALF_DOWN`, ``` ROUND_HALF_EVEN```, `ROUND_HALF_UP`, and ``` ROUND_UP```.

Signals are groups of exceptional conditions arising during the course of computation. Depending on the needs of the application, signals may be ignored, considered as informational, or treated as exceptions. The signals in the decimal module are: ``` Clamped```, `InvalidOperation`, ``` DivisionByZero```, `Inexact`, `Rounded`, `Subnormal`, `Overflow`, and ``` Underflow```.

For each signal there is a flag and a trap enabler. When a signal is encountered, its flag incremented from zero and, then, if the trap enabler is set to one, an exception is raised. Flags are sticky, so the user needs to reset them before monitoring a calculation.

## Quick-start Tutorial

The usual start to using decimals is importing the module, viewing the current context with getcontext and, if necessary, setting new values for precision, rounding, or enabled traps:

```>>> from decimal import *
>>> getcontext()
Context(prec=28, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
capitals=1, flags=[], traps=[Overflow, InvalidOperation,
DivisionByZero])

>>> getcontext().prec = 7       # Set a new precision
```

Decimal instances can be constructed from integers, strings or tuples. To create a Decimal from a float, first convert it to a string. This serves as an explicit reminder of the details of the conversion (including representation error). Decimal numbers include special values such as `NaN` which stands for “Not a number”, positive and negative `Infinity`, and `-0`.

```>>> Decimal(10)
Decimal("10")
>>> Decimal("3.14")
Decimal("3.14")
>>> Decimal((0, (3, 1, 4), -2))
Decimal("3.14")
>>> Decimal(str(2.0 ** 0.5))
Decimal("1.41421356237")
>>> Decimal("NaN")
Decimal("NaN")
>>> Decimal("-Infinity")
Decimal("-Infinity")
```

The significance of a new Decimal is determined solely by the number of digits input. Context precision and rounding only come into play during arithmetic operations.

```>>> getcontext().prec = 6
>>> Decimal('3.0')
Decimal("3.0")
>>> Decimal('3.1415926535')
Decimal("3.1415926535")
>>> Decimal('3.1415926535') + Decimal('2.7182818285')
Decimal("5.85987")
>>> getcontext().rounding = ROUND_UP
>>> Decimal('3.1415926535') + Decimal('2.7182818285')
Decimal("5.85988")
```

Decimals interact well with much of the rest of python. Here is a small decimal floating point flying circus:

```
>>> data = map(Decimal, '1.34 1.87 3.45 2.35 1.00 0.03 9.25'.split())
>>> max(data)
Decimal("9.25")
>>> min(data)
Decimal("0.03")
>>> sorted(data)
[Decimal("0.03"), Decimal("1.00"), Decimal("1.34"), Decimal("1.87"),
Decimal("2.35"), Decimal("3.45"), Decimal("9.25")]
>>> sum(data)
Decimal("19.29")
>>> a,b,c = data[:3]
>>> str(a)
'1.34'
>>> float(a)
1.3400000000000001
>>> round(a, 1)     # round() first converts to binary floating point
1.3
>>> int(a)
1
>>> a * 5
Decimal("6.70")
>>> a * b
Decimal("2.5058")
>>> c % a
Decimal("0.77")
```

The quantize method rounds a number to a fixed exponent. This method is useful for monetary applications that often round results to a fixed number of places:

```
>>> Decimal('7.325').quantize(Decimal('.01'), rounding=ROUND_DOWN)
Decimal("7.32")
>>> Decimal('7.325').quantize(Decimal('1.'), rounding=ROUND_UP)
Decimal("8")
```

As shown above, the getcontext function accesses the current context and allows the settings to be changed. This approach meets the needs of most applications.

For more advanced work, it may be useful to create alternate contexts using the Context() constructor. To make an alternate active, use the setcontext function.

In accordance with the standard, the Decimal module provides two ready to use standard contexts, `BasicContext` and ``` ExtendedContext```. The former is especially useful for debugging because many of the traps are enabled:

```>>> myothercontext = Context(prec=60, rounding=ROUND_HALF_DOWN)
>>> setcontext(myothercontext)
>>> Decimal(1) / Decimal(7)
Decimal("0.142857142857142857142857142857142857142857142857142857142857")

>>> ExtendedContext
Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
capitals=1, flags=[], traps=[])
>>> setcontext(ExtendedContext)
>>> Decimal(1) / Decimal(7)
Decimal("0.142857143")
>>> Decimal(42) / Decimal(0)
Decimal("Infinity")

>>> setcontext(BasicContext)
>>> Decimal(42) / Decimal(0)
Traceback (most recent call last):
File "<pyshell#143>", line 1, in -toplevel-
Decimal(42) / Decimal(0)
DivisionByZero: x / 0
```

Contexts also have signal flags for monitoring exceptional conditions encountered during computations. The flags remain set until explicitly cleared, so it is best to clear the flags before each set of monitored computations by using the clear_flags method.

```>>> setcontext(ExtendedContext)
>>> getcontext().clear_flags()
>>> Decimal(355) / Decimal(113)
Decimal("3.14159292")
>>> getcontext()
Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
capitals=1, flags=[Inexact, Rounded], traps=[])
```

The flags entry shows that the rational approximation to `Pi` was rounded (digits beyond the context precision were thrown away) and that the result is inexact (some of the discarded digits were non-zero).

Individual traps are set using the dictionary in the traps field of a context:

```>>> Decimal(1) / Decimal(0)
Decimal("Infinity")
>>> getcontext().traps[DivisionByZero] = 1
>>> Decimal(1) / Decimal(0)
Traceback (most recent call last):
File "<pyshell#112>", line 1, in -toplevel-
Decimal(1) / Decimal(0)
DivisionByZero: x / 0
```

Most programs adjust the current context only once, at the beginning of the program. And, in many applications, data is converted to Decimal with a single cast inside a loop. With context set and decimals created, the bulk of the program manipulates the data no differently than with other Python numeric types.

## Decimal objects

class Decimal([value [, context]]) #|!|&

Constructs a new Decimal object based from value.

value can be an integer, string, tuple, or another Decimal object. If no value is given, returns `Decimal("0")`. If value is a string, it should conform to the decimal numeric string syntax:

```    sign           ::=  '+' | '-'
digit          ::=  '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
indicator      ::=  'e' | 'E'
digits         ::=  digit [digit]...
decimal-part   ::=  digits '.' [digits] | ['.'] digits
exponent-part  ::=  indicator [sign] digits
infinity       ::=  'Infinity' | 'Inf'
nan            ::=  'NaN' [digits] | 'sNaN' [digits]
numeric-value  ::=  decimal-part [exponent-part] | infinity
numeric-string ::=  [sign] numeric-value | [sign] nan
```

If value is a tuple, it should have three components, a sign (`0` for positive or `1` for negative), a tuple of digits, and an integer exponent. For example, “`Decimal((0, (1, 4, 1, 4), -3))`” returns `Decimal("1.414")`.

The context precision does not affect how many digits are stored. That is determined exclusively by the number of digits in value. For example, “`Decimal("3.00000")`” records all five zeroes even if the context precision is only three.

The purpose of the context argument is determining what to do if value is a malformed string. If the context traps `InvalidOperation`, an exception is raised; otherwise, the constructor returns a new Decimal with the value of `NaN`.

Once constructed, Decimal objects are immutable.

Decimal floating point objects share many properties with the other builtin numeric types such as float and int. All of the usual math operations and special methods apply. Likewise, decimal objects can be copied, pickled, printed, used as dictionary keys, used as set elements, compared, sorted, and coerced to another type (such as float or long).

In addition to the standard numeric properties, decimal floating point objects also have a number of specialized methods:

Return the adjusted exponent after shifting out the coefficient’s rightmost digits until only the lead digit remains: `Decimal("321e+5").adjusted()` returns seven. Used for determining the position of the most significant digit with respect to the decimal point.

as_tuple() #|!|&

Returns a tuple representation of the number: “```(sign, digittuple, exponent)```“.

compare(other[, context]) #|!|&

Compares like __cmp__ but returns a decimal instance:

```        a or b is a NaN ==> Decimal("NaN")
a < b           ==> Decimal("-1")
a == b          ==> Decimal("0")
a > b           ==> Decimal("1")
```
max(other[, context]) #|!|&

Like “`max(self, other)`” except that the context rounding rule is applied before returning and that `NaN` values are either signalled or ignored (depending on the context and whether they are signaling or quiet).

min(other[, context]) #|!|&

Like “`min(self, other)`” except that the context rounding rule is applied before returning and that `NaN` values are either signalled or ignored (depending on the context and whether they are signaling or quiet).

normalize([context]) #|!|&

Normalize the number by stripping the rightmost trailing zeroes and converting any result equal to `Decimal("0")` to `Decimal("0e0")`. Used for producing canonical values for members of an equivalence class. For example, ``` Decimal("32.100")``` and `Decimal("0.321000e+2")` both normalize to the equivalent value ``` Decimal("32.1")```.

quantize(exp [, rounding[, context[, watchexp]]]) #|!|&

Quantize makes the exponent the same as exp. Searches for a rounding method in rounding, then in context, and then in the current context.

If watchexp is set (default), then an error is returned whenever the resulting exponent is greater than Emax or less than Etiny.

remainder_near(other[, context]) #|!|&

Computes the modulo as either a positive or negative value depending on which is closest to zero. For instance, “`Decimal(10).remainder_near(6)`” returns ``` Decimal("-2")``` which is closer to zero than ``` Decimal("4")```.

If both are equally close, the one chosen will have the same sign as self.

same_quantum(other[, context]) #|!|&

Test whether self and other have the same exponent or whether both are `NaN`.

sqrt([context]) #|!|&

Return the square root to full precision.

to_eng_string([context]) #|!|&

Convert to an engineering-type string.

Engineering notation has an exponent which is a multiple of 3, so there are up to 3 digits left of the decimal place. For example, converts `Decimal('123E+1')` to ``` Decimal("1.23E+3")```

to_integral([rounding[, context]]) #|!|&

Rounds to the nearest integer without signaling ``` Inexact``` or `Rounded`. If given, applies rounding; otherwise, uses the rounding method in either the supplied context or the current context.

## Context objects

Contexts are environments for arithmetic operations. They govern precision, set rules for rounding, determine which signals are treated as exceptions, and limit the range for exponents.

Each thread has its own current context which is accessed or changed using the getcontext and setcontext functions:

getcontext() #|!|&

Return the current context for the active thread.

setcontext(c) #|!|&

Set the current context for the active thread to c.

New contexts can formed using the Context constructor described below. In addition, the module provides three pre-made contexts:

class BasicContext #|!|&

This is a standard context defined by the General Decimal Arithmetic Specification. Precision is set to nine. Rounding is set to `ROUND_HALF_UP`. All flags are cleared. All traps are enabled (treated as exceptions) except `Inexact`, ``` Rounded```, and `Subnormal`.

Because many of the traps are enabled, this context is useful for debugging.

class ExtendedContext #|!|&

This is a standard context defined by the General Decimal Arithmetic Specification. Precision is set to nine. Rounding is set to `ROUND_HALF_EVEN`. All flags are cleared. No traps are enabled (so that exceptions are not raised during computations).

Because the trapped are disabled, this context is useful for applications that prefer to have result value of `NaN` or `Infinity` instead of raising exceptions. This allows an application to complete a run in the presence of conditions that would otherwise halt the program.

class DefaultContext #|!|&

This context is used by the Context constructor as a prototype for new contexts. Changing a field (such a precision) has the effect of changing the default for new contexts creating by the Context constructor.

This context is most useful in multi-threaded environments. Changing one of the fields before threads are started has the effect of setting system-wide defaults. Changing the fields after threads have started is not recommended as it would require thread synchronization to prevent race conditions.

In single threaded environments, it is preferable to not use this context at all. Instead, simply create contexts explicitly as described below.

The default values are precision=28, rounding=ROUND_HALF_EVEN, and enabled traps for Overflow, InvalidOperation, and DivisionByZero.

In addition to the three supplied contexts, new contexts can be created with the Context constructor.

class Context(prec=None, rounding=None, traps=None, flags=None, Emin=None, Emax=None, capitals=1) #|!|&

Creates a new context. If a field is not specified or is ``` None```, the default values are copied from the ``` DefaultContext```. If the flags field is not specified or is `None`, all flags are cleared.

The prec field is a positive integer that sets the precision for arithmetic operations in the context.

The rounding option is one of:

• `ROUND_CEILING` (towards `Infinity`),

• `ROUND_DOWN` (towards zero),

• `ROUND_FLOOR` (towards `-Infinity`),

• `ROUND_HALF_DOWN` (to nearest with ties going towards zero),

• `ROUND_HALF_EVEN` (to nearest with ties going to nearest even integer),

• `ROUND_HALF_UP` (to nearest with ties going away from zero), or

• `ROUND_UP` (away from zero).

The traps and flags fields list any signals to be set. Generally, new contexts should only set traps and leave the flags clear.

The Emin and Emax fields are integers specifying the outer limits allowable for exponents.

The capitals field is either `0` or ``` 1``` (the default). If set to `1`, exponents are printed with a capital `E`; otherwise, a lowercase `e` is used: `Decimal('6.02e+23')`.

The Context class defines several general purpose methods as well as a large number of methods for doing arithmetic directly in a given context.

clear_flags() #|!|&

Resets all of the flags to `0`.

copy() #|!|&

Return a duplicate of the context.

create_decimal(num) #|!|&

Creates a new Decimal instance from num but using self as context. Unlike the Decimal constructor, the context precision, rounding method, flags, and traps are applied to the conversion.

This is useful because constants are often given to a greater precision than is needed by the application. Another benefit is that rounding immediately eliminates unintended effects from digits beyond the current precision. In the following example, using unrounded inputs means that adding zero to a sum can change the result:

```    >>> getcontext().prec = 3
>>> Decimal("3.4445") + Decimal("1.0023")
Decimal("4.45")
>>> Decimal("3.4445") + Decimal(0) + Decimal("1.0023")
Decimal("4.44")
```
Etiny() #|!|&

Returns a value equal to “`Emin - prec + 1`” which is the minimum exponent value for subnormal results. When underflow occurs, the exponent is set to `Etiny`.

Etop() #|!|&

Returns a value equal to “`Emax - prec + 1`“.

The usual approach to working with decimals is to create Decimal instances and then apply arithmetic operations which take place within the current context for the active thread. An alternate approach is to use context methods for calculating within a specific context. The methods are similar to those for the Decimal class and are only briefly recounted here.

abs(x) #|!|&

Returns the absolute value of x.

Return the sum of x and y.

compare(x, y) #|!|&

Compares values numerically.

Like __cmp__ but returns a decimal instance:

```        a or b is a NaN ==> Decimal("NaN")
a < b           ==> Decimal("-1")
a == b          ==> Decimal("0")
a > b           ==> Decimal("1")
```
divide(x, y) #|!|&

Return x divided by y.

divmod(x, y) #|!|&

Divides two numbers and returns the integer part of the result.

max(x, y) #|!|&

Compare two values numerically and return the maximum.

If they are numerically equal then the left-hand operand is chosen as the result.

min(x, y) #|!|&

Compare two values numerically and return the minimum.

If they are numerically equal then the left-hand operand is chosen as the result.

minus(x) #|!|&

Minus corresponds to the unary prefix minus operator in Python.

multiply(x, y) #|!|&

Return the product of x and y.

normalize(x) #|!|&

Normalize reduces an operand to its simplest form.

Essentially a plus operation with all trailing zeros removed from the result.

plus(x) #|!|&

Plus corresponds to the unary prefix plus operator in Python. This operation applies the context precision and rounding, so it is not an identity operation.

power(x, y[, modulo]) #|!|&

Return “`x ** y`” to the modulo if given.

The right-hand operand must be a whole number whose integer part (after any exponent has been applied) has no more than 9 digits and whose fractional part (if any) is all zeros before any rounding. The operand may be positive, negative, or zero; if negative, the absolute value of the power is used, and the left-hand operand is inverted (divided into 1) before use.

If the increased precision needed for the intermediate calculations exceeds the capabilities of the implementation then an `InvalidOperation` condition is signaled.

If, when raising to a negative power, an underflow occurs during the division into 1, the operation is not halted at that point but continues.

quantize(x, y) #|!|&

Returns a value equal to x after rounding and having the exponent of y.

Unlike other operations, if the length of the coefficient after the quantize operation would be greater than precision, then an `InvalidOperation` is signaled. This guarantees that, unless there is an error condition, the quantized exponent is always equal to that of the right-hand operand.

Also unlike other operations, quantize never signals Underflow, even if the result is subnormal and inexact.

remainder(x, y) #|!|&

Returns the remainder from integer division.

The sign of the result, if non-zero, is the same as that of the original dividend.

remainder_near(x, y) #|!|&

Computed the modulo as either a positive or negative value depending on which is closest to zero. For instance, “`Decimal(10).remainder_near(6)`” returns ``` Decimal("-2")``` which is closer to zero than ``` Decimal("4")```.

If both are equally close, the one chosen will have the same sign as self.

same_quantum(x, y) #|!|&

Test whether x and y have the same exponent or whether both are `NaN`.

sqrt() #|!|&

Return the square root to full precision.

subtract(x, y) #|!|&

Return the difference between x and y.

to_eng_string() #|!|&

Convert to engineering-type string.

Engineering notation has an exponent which is a multiple of 3, so there are up to 3 digits left of the decimal place. For example, converts `Decimal('123E+1')` to ``` Decimal("1.23E+3")```

to_integral(x) #|!|&

Rounds to the nearest integer without signaling ``` Inexact``` or `Rounded`.

to_sci_string() #|!|&

Converts a number to a string using scientific notation.

## Signals

Signals represent conditions that arise during computation. Each corresponds to one context flag and one context trap enabler.

The context flag is incremented whenever the condition is encountered. After the computation, flags may be checked for informational purposes (for instance, to determine whether a computation was exact). After checking the flags, be sure to clear all flags before starting the next computation.

If the context’s trap enabler is set for the signal, then the condition causes a Python exception to be raised. For example, if the DivisionByZero trap is set, then a DivisionByZero exception is raised upon encountering the condition.

class Clamped #|!|&

Altered an exponent to fit representation constraints.

Typically, clamping occurs when an exponent falls outside the context’s Emin and Emax limits. If possible, the exponent is reduced to fit by adding zeroes to the coefficient.

class DecimalException #|!|&

Base class for other signals and is a subclass of exceptions.ArithmeticError.

class DivisionByZero #|!|&

Signals the division of a non-infinite number by zero.

Can occur with division, modulo division, or when raising a number to a negative power. If this signal is not trapped, returns `Infinity` or `-Infinity` with the sign determined by the inputs to the calculation.

class Inexact #|!|&

Indicates that rounding occurred and the result is not exact.

Signals when non-zero digits were discarded during rounding. The rounded result is returned. The signal flag or trap is used to detect when results are inexact.

class InvalidOperation #|!|&

An invalid operation was performed.

Indicates that an operation was requested that does not make sense. If not trapped, returns `NaN`. Possible causes include:

```        Infinity - Infinity
0 * Infinity
Infinity / Infinity
x % 0
Infinity % x
x._rescale( non-integer )
sqrt(-x) and x > 0
0 ** 0
x ** (non-integer)
x ** Infinity
```
class Overflow #|!|&

Numerical overflow.

Indicates the exponent is larger than Emax after rounding has occurred. If not trapped, the result depends on the rounding mode, either pulling inward to the largest representable finite number or rounding outward to `Infinity`. In either case, Inexact and Rounded are also signaled.

class Rounded #|!|&

Rounding occurred though possibly no information was lost.

Signaled whenever rounding discards digits; even if those digits are zero (such as rounding `5.00` to `5.0`). If not trapped, returns the result unchanged. This signal is used to detect loss of significant digits.

class Subnormal #|!|&

Exponent was lower than Emin prior to rounding.

Occurs when an operation result is subnormal (the exponent is too small). If not trapped, returns the result unchanged.

class Underflow #|!|&

Numerical underflow with result rounded to zero.

Occurs when a subnormal result is pushed to zero by rounding. Inexact and Subnormal are also signaled.

The following table summarizes the hierarchy of signals:

```
exceptions.ArithmeticError(exceptions.StandardError)
DecimalException
Clamped
DivisionByZero(DecimalException, exceptions.ZeroDivisionError)
Inexact
Overflow(Inexact, Rounded)
Underflow(Inexact, Rounded, Subnormal)
InvalidOperation
Rounded
Subnormal
```

## Floating Point Notes

### Mitigating round-off error with increased precision

The use of decimal floating point eliminates decimal representation error (making it possible to represent ``` 0.1``` exactly); however, some operations can still incur round-off error when non-zero digits exceed the fixed precision.

The effects of round-off error can be amplified by the addition or subtraction of nearly offsetting quantities resulting in loss of significance. Knuth provides two instructive examples where rounded floating point arithmetic with insufficient precision causes the breakdown of the associative and distributive properties of addition:

```# Examples from Seminumerical Algorithms, Section 4.2.2.
>>> from decimal import *
>>> getcontext().prec = 8

>>> u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111')
>>> (u + v) + w
Decimal("9.5111111")
>>> u + (v + w)
Decimal("10")

>>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
>>> (u*v) + (u*w)
Decimal("0.01")
>>> u * (v+w)
Decimal("0.0060000")
```

The decimal module makes it possible to restore the identities by expanding the precision sufficiently to avoid loss of significance:

```>>> getcontext().prec = 20
>>> u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111')
>>> (u + v) + w
Decimal("9.51111111")
>>> u + (v + w)
Decimal("9.51111111")
>>>
>>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
>>> (u*v) + (u*w)
Decimal("0.0060000")
>>> u * (v+w)
Decimal("0.0060000")
```

### Special values

The number system for the decimal module provides special values including ``` NaN```, `sNaN`, `-Infinity`, ``` Infinity```, and two zeroes, `+0` and ``` -0```.

Infinities can be constructed directly with: ``` Decimal('Infinity')```. Also, they can arise from dividing by zero when the DivisionByZero signal is not trapped. Likewise, when the Overflow signal is not trapped, infinity can result from rounding beyond the limits of the largest representable number.

The infinities are signed (affine) and can be used in arithmetic operations where they get treated as very large, indeterminate numbers. For instance, adding a constant to infinity gives another infinite result.

Some operations are indeterminate and return `NaN`, or if the InvalidOperation signal is trapped, raise an exception. For example, `0/0` returns `NaN` which means “not a number”. This variety of `NaN` is quiet and, once created, will flow through other computations always resulting in another `NaN`. This behavior can be useful for a series of computations that occasionally have missing inputs — it allows the calculation to proceed while flagging specific results as invalid.

A variant is `sNaN` which signals rather than remaining quiet after every operation. This is a useful return value when an invalid result needs to interrupt a calculation for special handling.

The signed zeros can result from calculations that underflow. They keep the sign that would have resulted if the calculation had been carried out to greater precision. Since their magnitude is zero, both positive and negative zeros are treated as equal and their sign is informational.

In addition to the two signed zeros which are distinct yet equal, there are various representations of zero with differing precisions yet equivalent in value. This takes a bit of getting used to. For an eye accustomed to normalized floating point representations, it is not immediately obvious that the following calculation returns a value equal to zero:

```>>> 1 / Decimal('Infinity')
Decimal("0E-1000000026")
```

The getcontext function accesses a different Context object for each thread. Having separate thread contexts means that threads may make changes (such as ``` getcontext.prec=10```) without interfering with other threads.

Likewise, the setcontext function automatically assigns its target to the current thread.

If setcontext has not been called before getcontext, then getcontext will automatically create a new context for use in the current thread.

The new context is copied from a prototype context called DefaultContext. To control the defaults so that each thread will use the same values throughout the application, directly modify the DefaultContext object. This should be done before any threads are started so that there won’t be a race condition between threads calling getcontext. For example:

```# Set applicationwide defaults for all threads about to be launched
DefaultContext.prec = 12
DefaultContext.rounding = ROUND_DOWN
DefaultContext.traps = ExtendedContext.traps.copy()
DefaultContext.traps[InvalidOperation] = 1
setcontext(DefaultContext)

# Afterwards, the threads can be started
t1.start()
t2.start()
t3.start()
. . .
```

## Recipes

Here are a few recipes that serve as utility functions and that demonstrate ways to work with the Decimal class:

```def moneyfmt(value, places=2, curr='', sep=',', dp='.',
pos='', neg='-', trailneg=''):
"""Convert Decimal to a money formatted string.

places:  required number of places after the decimal point
curr:    optional currency symbol before the sign (may be blank)
sep:     optional grouping separator (comma, period, space, or blank)
dp:      decimal point indicator (comma or period)
only specify as blank when places is zero
pos:     optional sign for positive numbers: '+', space or blank
neg:     optional sign for negative numbers: '-', '(', space or blank
trailneg:optional trailing minus indicator:  '-', ')', space or blank

>>> d = Decimal('-1234567.8901')
>>> moneyfmt(d, curr='\$')
'-\$1,234,567.89'
>>> moneyfmt(d, places=0, sep='.', dp='', neg='', trailneg='-')
'1.234.568-'
>>> moneyfmt(d, curr='\$', neg='(', trailneg=')')
'(\$1,234,567.89)'
>>> moneyfmt(Decimal(123456789), sep=' ')
'123 456 789.00'
>>> moneyfmt(Decimal('-0.02'), neg='<', trailneg='>')
'<.02>'

"""
q = Decimal((0, (1,), -places))    # 2 places --> '0.01'
sign, digits, exp = value.quantize(q).as_tuple()
assert exp == -places
result = []
digits = map(str, digits)
build, next = result.append, digits.pop
if sign:
build(trailneg)
for i in range(places):
if digits:
build(next())
else:
build('0')
build(dp)
i = 0
while digits:
build(next())
i += 1
if i == 3 and digits:
i = 0
build(sep)
build(curr)
if sign:
build(neg)
else:
build(pos)
result.reverse()
return ''.join(result)

def pi():
"""Compute Pi to the current precision.

>>> print pi()
3.141592653589793238462643383

"""
getcontext().prec += 2  # extra digits for intermediate steps
three = Decimal(3)      # substitute "three=3.0" for regular floats
lasts, t, s, n, na, d, da = 0, three, 3, 1, 0, 0, 24
while s != lasts:
lasts = s
n, na = n+na, na+8
d, da = d+da, da+32
t = (t * n) / d
s += t
getcontext().prec -= 2
return +s               # unary plus applies the new precision

def exp(x):
"""Return e raised to the power of x.  Result type matches input type.

>>> print exp(Decimal(1))
2.718281828459045235360287471
>>> print exp(Decimal(2))
7.389056098930650227230427461
>>> print exp(2.0)
7.38905609893
>>> print exp(2+0j)
(7.38905609893+0j)

"""
getcontext().prec += 2
i, lasts, s, fact, num = 0, 0, 1, 1, 1
while s != lasts:
lasts = s
i += 1
fact *= i
num *= x
s += num / fact
getcontext().prec -= 2
return +s

def cos(x):
"""Return the cosine of x as measured in radians.

>>> print cos(Decimal('0.5'))
0.8775825618903727161162815826
>>> print cos(0.5)
0.87758256189
>>> print cos(0.5+0j)
(0.87758256189+0j)

"""
getcontext().prec += 2
i, lasts, s, fact, num, sign = 0, 0, 1, 1, 1, 1
while s != lasts:
lasts = s
i += 2
fact *= i * (i-1)
num *= x * x
sign *= -1
s += num / fact * sign
getcontext().prec -= 2
return +s

def sin(x):
"""Return the sine of x as measured in radians.

>>> print sin(Decimal('0.5'))
0.4794255386042030002732879352
>>> print sin(0.5)
0.479425538604
>>> print sin(0.5+0j)
(0.479425538604+0j)

"""
getcontext().prec += 2
i, lasts, s, fact, num, sign = 1, 0, x, 1, x, 1
while s != lasts:
lasts = s
i += 2
fact *= i * (i-1)
num *= x * x
sign *= -1
s += num / fact * sign
getcontext().prec -= 2
return +s
```

## Decimal FAQ

Q. It is cumbersome to type ``` decimal.Decimal('1234.5')```. Is there a way to minimize typing when using the interactive interpreter?

A. Some users abbreviate the constructor to just a single letter:

```>>> D = decimal.Decimal
>>> D('1.23') + D('3.45')
Decimal("4.68")
```

Q. In a fixed-point application with two decimal places, some inputs have many places and need to be rounded. Others are not supposed to have excess digits and need to be validated. What methods should be used?

A. The quantize method rounds to a fixed number of decimal places. If the ``` Inexact``` trap is set, it is also useful for validation:

```>>> TWOPLACES = Decimal(10) ** -2       # same as Decimal('0.01')

>>> # Round to two places
>>> Decimal("3.214").quantize(TWOPLACES)
Decimal("3.21")

>>> # Validate that a number does not exceed two places
>>> Decimal("3.21").quantize(TWOPLACES, context=Context(traps=[Inexact]))
Decimal("3.21")

>>> Decimal("3.214").quantize(TWOPLACES, context=Context(traps=[Inexact]))
Traceback (most recent call last):
...
Inexact: Changed in rounding
```

Q. Once I have valid two place inputs, how do I maintain that invariant throughout an application?

A. Some operations like addition and subtraction automatically preserve fixed point. Others, like multiplication and division, change the number of decimal places and need to be followed-up with a quantize step.

Q. There are many ways to express the same value. The numbers `200`, `200.000`, `2E2`, and `.02E+4` all have the same value at various precisions. Is there a way to transform them to a single recognizable canonical value?

A. The normalize method maps all equivalent values to a single representive:

```>>> values = map(Decimal, '200 200.000 2E2 .02E+4'.split())
>>> [v.normalize() for v in values]
[Decimal("2E+2"), Decimal("2E+2"), Decimal("2E+2"), Decimal("2E+2")]
```

Q. Some decimal values always print with exponential notation. Is there a way to get a non-exponential representation?

A. For some values, exponential notation is the only way to express the number of significant places in the coefficient. For example, expressing `5.0E+3` as `5000` keeps the value constant but cannot show the original’s two-place significance.

Q. Is there a way to convert a regular float to a Decimal?

A. Yes, all binary floating point numbers can be exactly expressed as a Decimal. An exact conversion may take more precision than intuition would suggest, so trapping `Inexact` will signal a need for more precision:

```def floatToDecimal(f):
"Convert a floating point number to a Decimal with no loss of information"
# Transform (exactly) a float to a mantissa (0.5 <= abs(m) < 1.0) and an
# exponent.  Double the mantissa until it is an integer.  Use the integer
# mantissa and exponent to compute an equivalent Decimal.  If this cannot
# be done exactly, then retry with more precision.

mantissa, exponent = math.frexp(f)
while mantissa != int(mantissa):
mantissa *= 2.0
exponent -= 1
mantissa = int(mantissa)

oldcontext = getcontext()
setcontext(Context(traps=[Inexact]))
try:
while True:
try:
return mantissa * Decimal(2) ** exponent
except Inexact:
getcontext().prec += 1
finally:
setcontext(oldcontext)
```

Q. Why isn’t the floatToDecimal routine included in the module?

A. There is some question about whether it is advisable to mix binary and decimal floating point. Also, its use requires some care to avoid the representation issues associated with binary floating point:

```>>> floatToDecimal(1.1)
Decimal("1.100000000000000088817841970012523233890533447265625")
```

Q. Within a complex calculation, how can I make sure that I haven’t gotten a spurious result because of insufficient precision or rounding anomalies.

A. The decimal module makes it easy to test results. A best practice is to re-run calculations using greater precision and with various rounding modes. Widely differing results indicate insufficient precision, rounding mode issues, ill-conditioned inputs, or a numerically unstable algorithm.

Q. I noticed that context precision is applied to the results of operations but not to the inputs. Is there anything to watch out for when mixing values of different precisions?

A. Yes. The principle is that all values are considered to be exact and so is the arithmetic on those values. Only the results are rounded. The advantage for inputs is that “what you type is what you get”. A disadvantage is that the results can look odd if you forget that the inputs haven’t been rounded:

```>>> getcontext().prec = 3
>>> Decimal('3.104') + D('2.104')
Decimal("5.21")
>>> Decimal('3.104') + D('0.000') + D('2.104')
Decimal("5.20")
```

The solution is either to increase precision or to force rounding of inputs using the unary plus operation:

```>>> getcontext().prec = 3
>>> +Decimal('1.23456789')      # unary plus triggers rounding
Decimal("1.23")
```

Alternatively, inputs can be rounded upon creation using the Context.create_decimal method:

```>>> Context(prec=5, rounding=ROUND_DOWN).create_decimal('1.2345678')
Decimal("1.2345")
```