Как округлить число в python

# Math Module

In addition to the built-in round function, the math module provides the floor , ceil , and trunc functions.

floor , ceil , trunc , and round always return a float .

round always breaks ties away from zero.

floor , ceil , and trunc always return an Integral value, while round returns an Integral value if called with one argument.

round breaks ties towards the nearest even number. This corrects the bias towards larger numbers when performing a large number of calculations.

# Warning!

As with any floating-point representation, some fractions cannot be represented exactly. This can lead to some unexpected rounding behavior.

# Warning about the floor, trunc, and integer division of negative numbers

Python (and C++ and Java) round away from zero for negative numbers. Consider:

# Trigonometry

# Calculating the length of the hypotenuse

# Converting degrees to/from radians

All math functions expect radians so you need to convert degrees to radians:

All results of the inverse trigonometic functions return the result in radians, so you may need to convert it back to degrees:

# Sine, cosine, tangent and inverse functions

Apart from the math.atan there is also a two-argument math.atan2 function, which computes the correct quadrant and avoids pitfalls of division by zero:

# Hyperbolic sine, cosine and tangent

# Logarithms

math.log(x) gives the natural (base e ) logarithm of x .

math.log can lose precision with numbers close to 1, due to the limitations of floating-point numbers. In order to accurately calculate logs close to 1, use math.log1p , which evaluates the natural logarithm of 1 plus the argument:

math.log10 can be used for logs base 10:

When used with two arguments, math.log(x, base) gives the logarithm of x in the given base (i.e. log(x) / log(base) .

# Constants

math modules includes two commonly used mathematical constants.

math.pi — The mathematical constant pi
math.e — The mathematical constant e (base of natural logarithm)

Python 3.5 and higher have constants for infinity and NaN ("not a number"). The older syntax of passing a string to float() still works.

# Infinity and NaN ("not a number")

In all versions of Python, we can represent infinity and NaN ("not a number") as follows:

In Python 3.5 and higher, we can also use the defined constants math.inf and math.nan :

The string representations display as inf and -inf and nan :

We can test for either positive or negative infinity with the isinf method:

We can test specifically for positive infinity or for negative infinity by direct comparison:

Python 3.2 and higher also allows checking for finiteness:

Comparison operators work as expected for positive and negative infinity:

But if an arithmetic expression produces a value larger than the maximum that can be represented as a float , it will become infinity:

However division by zero does not give a result of infinity (or negative infinity where appropriate), rather it raises a ZeroDivisionError exception.

Arithmetic operations on infinity just give infinite results, or sometimes NaN:

NaN is never equal to anything, not even itself. We can test for it is with the isnan method:

NaN always compares as "not equal", but never less than or greater than:

Arithmetic operations on NaN always give NaN. This includes multiplication by -1: there is no "negative NaN".

There is one subtle difference between the old float versions of NaN and infinity and the Python 3.5+ math library constants:

# Pow for faster exponentiation

Using the timeit module from the command line:

The built-in ** operator often comes in handy, but if performance is of the essence, use math.pow. Be sure to note, however, that pow returns floats, even if the arguments are integers:

# Copying signs

In Python 2.6 and higher, math.copysign(x, y) returns x with the sign of y . The returned value is always a float .

# Imaginary Numbers

Imaginary numbers in Python are represented by a "j" or "J" trailing the target number.

# Complex numbers and the cmath module

The cmath module is similar to the math module, but defines functions appropriately for the complex plane.

First of all, complex numbers are a numeric type that is part of the Python language itself rather than being provided by a library class. Thus we don’t need to import cmath for ordinary arithmetic expressions.

Note that we use j (or J ) and not i .

We must use 1j since j would be the name of a variable rather than a numeric literal.

We have the real part and the imag (imaginary) part, as well as the complex conjugate :

The built-in functions abs and complex are also part of the language itself and don’t require any import:

The complex function can take a string, but it can’t have spaces:

But for most functions we do need the module, for instance sqrt :

Naturally the behavior of sqrt is different for complex numbers and real numbers. In non-complex math the square root of a negative number raises an exception:

Functions are provided to convert to and from polar coordinates:

The mathematical field of complex analysis is beyond the scope of this example, but many functions in the complex plane have a "branch cut", usually along the real axis or the imaginary axis. Most modern platforms support "signed zero" as specified in IEEE 754, which provides continuity of those functions on both sides of the branch cut. The following example is from the Python documentation:

The cmath module also provides many functions with direct counterparts from the math module.

In addition to sqrt , there are complex versions of exp , log , log10 , the trigonometric functions and their inverses ( sin , cos , tan , asin , acos , atan ), and the hyperbolic functions and their inverses ( sinh , cosh , tanh , asinh , acosh , atanh ). Note however there is no complex counterpart of math.atan2 , the two-argument form of arctangent.

The constants pi and e are provided. Note these are float and not complex .

The cmath module also provides complex versions of isinf , and (for Python 3.2+) isfinite . See "Infinity and NaN

(opens new window) ". A complex number is considered infinite if either its real part or its imaginary part is infinite.

Likewise, the cmath module provides a complex version of isnan . See "Infinity and NaN

(opens new window) ". A complex number is considered "not a number" if either its real part or its imaginary part is "not a number".

Note there is no cmath counterpart of the math.inf and math.nan constants (from Python 3.5 and higher)

In Python 3.5 and higher, there is an isclose method in both cmath and math modules.

Built-in Functions¶

The Python interpreter has a number of functions and types built into it that are always available. They are listed here in alphabetical order.

Return the absolute value of a number. The argument may be an integer, a floating point number, or an object implementing __abs__() . If the argument is a complex number, its magnitude is returned.

Return an asynchronous iterator for an asynchronous iterable . Equivalent to calling x.__aiter__() .

Note: Unlike iter() , aiter() has no 2-argument variant.

New in version 3.10.

Return True if all elements of the iterable are true (or if the iterable is empty). Equivalent to:

When awaited, return the next item from the given asynchronous iterator , or default if given and the iterator is exhausted.

This is the async variant of the next() builtin, and behaves similarly.

This calls the __anext__() method of async_iterator, returning an awaitable . Awaiting this returns the next value of the iterator. If default is given, it is returned if the iterator is exhausted, otherwise StopAsyncIteration is raised.

New in version 3.10.

Return True if any element of the iterable is true. If the iterable is empty, return False . Equivalent to:

As repr() , return a string containing a printable representation of an object, but escape the non-ASCII characters in the string returned by repr() using \x , \u , or \U escapes. This generates a string similar to that returned by repr() in Python 2.

Convert an integer number to a binary string prefixed with “0b”. The result is a valid Python expression. If x is not a Python int object, it has to define an __index__() method that returns an integer. Some examples:

If the prefix “0b” is desired or not, you can use either of the following ways.

See also format() for more information.

Open file and return a corresponding file object . If the file cannot be opened, an OSError is raised. See Reading and Writing Files for more examples of how to use this function.

file is a path-like object giving the pathname (absolute or relative to the current working directory) of the file to be opened or an integer file descriptor of the file to be wrapped. (If a file descriptor is given, it is closed when the returned I/O object is closed unless closefd is set to False .)

mode is an optional string that specifies the mode in which the file is opened. It defaults to ‘r’ which means open for reading in text mode. Other common values are ‘w’ for writing (truncating the file if it already exists), ‘x’ for exclusive creation, and ‘a’ for appending (which on some Unix systems, means that all writes append to the end of the file regardless of the current seek position). In text mode, if encoding is not specified the encoding used is platform-dependent: locale.getencoding() is called to get the current locale encoding. (For reading and writing raw bytes use binary mode and leave encoding unspecified.) The available modes are:

open for reading (default)

open for writing, truncating the file first

open for exclusive creation, failing if the file already exists

open for writing, appending to the end of file if it exists

text mode (default)

open for updating (reading and writing)

The default mode is ‘r’ (open for reading text, a synonym of ‘rt’ ). Modes ‘w+’ and ‘w+b’ open and truncate the file. Modes ‘r+’ and ‘r+b’ open the file with no truncation.

As mentioned in the Overview , Python distinguishes between binary and text I/O. Files opened in binary mode (including ‘b’ in the mode argument) return contents as bytes objects without any decoding. In text mode (the default, or when ‘t’ is included in the mode argument), the contents of the file are returned as str , the bytes having been first decoded using a platform-dependent encoding or using the specified encoding if given.

Python doesn’t depend on the underlying operating system’s notion of text files; all the processing is done by Python itself, and is therefore platform-independent.

buffering is an optional integer used to set the buffering policy. Pass 0 to switch buffering off (only allowed in binary mode), 1 to select line buffering (only usable in text mode), and an integer > 1 to indicate the size in bytes of a fixed-size chunk buffer. Note that specifying a buffer size this way applies for binary buffered I/O, but TextIOWrapper (i.e., files opened with mode=’r+’ ) would have another buffering. To disable buffering in TextIOWrapper , consider using the write_through flag for io.TextIOWrapper.reconfigure() . When no buffering argument is given, the default buffering policy works as follows:

Binary files are buffered in fixed-size chunks; the size of the buffer is chosen using a heuristic trying to determine the underlying device’s “block size” and falling back on io.DEFAULT_BUFFER_SIZE . On many systems, the buffer will typically be 4096 or 8192 bytes long.

“Interactive” text files (files for which isatty() returns True ) use line buffering. Other text files use the policy described above for binary files.

encoding is the name of the encoding used to decode or encode the file. This should only be used in text mode. The default encoding is platform dependent (whatever locale.getencoding() returns), but any text encoding supported by Python can be used. See the codecs module for the list of supported encodings.

errors is an optional string that specifies how encoding and decoding errors are to be handled—this cannot be used in binary mode. A variety of standard error handlers are available (listed under Error Handlers ), though any error handling name that has been registered with codecs.register_error() is also valid. The standard names include:

‘strict’ to raise a ValueError exception if there is an encoding error. The default value of None has the same effect.

‘ignore’ ignores errors. Note that ignoring encoding errors can lead to data loss.

‘replace’ causes a replacement marker (such as ‘?’ ) to be inserted where there is malformed data.

‘surrogateescape’ will represent any incorrect bytes as low surrogate code units ranging from U+DC80 to U+DCFF. These surrogate code units will then be turned back into the same bytes when the surrogateescape error handler is used when writing data. This is useful for processing files in an unknown encoding.

‘xmlcharrefreplace’ is only supported when writing to a file. Characters not supported by the encoding are replaced with the appropriate XML character reference &#nnn; .

‘backslashreplace’ replaces malformed data by Python’s backslashed escape sequences.

‘namereplace’ (also only supported when writing) replaces unsupported characters with \N <. >escape sequences.

newline determines how to parse newline characters from the stream. It can be None , » , ‘\n’ , ‘\r’ , and ‘\r\n’ . It works as follows:

When reading input from the stream, if newline is None , universal newlines mode is enabled. Lines in the input can end in ‘\n’ , ‘\r’ , or ‘\r\n’ , and these are translated into ‘\n’ before being returned to the caller. If it is » , universal newlines mode is enabled, but line endings are returned to the caller untranslated. If it has any of the other legal values, input lines are only terminated by the given string, and the line ending is returned to the caller untranslated.

When writing output to the stream, if newline is None , any ‘\n’ characters written are translated to the system default line separator, os.linesep . If newline is » or ‘\n’ , no translation takes place. If newline is any of the other legal values, any ‘\n’ characters written are translated to the given string.

If closefd is False and a file descriptor rather than a filename was given, the underlying file descriptor will be kept open when the file is closed. If a filename is given closefd must be True (the default); otherwise, an error will be raised.

A custom opener can be used by passing a callable as opener. The underlying file descriptor for the file object is then obtained by calling opener with (file, flags). opener must return an open file descriptor (passing os.open as opener results in functionality similar to passing None ).

The newly created file is non-inheritable .

The following example uses the dir_fd parameter of the os.open() function to open a file relative to a given directory:

The type of file object returned by the open() function depends on the mode. When open() is used to open a file in a text mode ( ‘w’ , ‘r’ , ‘wt’ , ‘rt’ , etc.), it returns a subclass of io.TextIOBase (specifically io.TextIOWrapper ). When used to open a file in a binary mode with buffering, the returned class is a subclass of io.BufferedIOBase . The exact class varies: in read binary mode, it returns an io.BufferedReader ; in write binary and append binary modes, it returns an io.BufferedWriter , and in read/write mode, it returns an io.BufferedRandom . When buffering is disabled, the raw stream, a subclass of io.RawIOBase , io.FileIO , is returned.

See also the file handling modules, such as fileinput , io (where open() is declared), os , os.path , tempfile , and shutil .

Raises an auditing event open with arguments file , mode , flags .

The mode and flags arguments may have been modified or inferred from the original call.

Changed in version 3.3:

The opener parameter was added.

The ‘x’ mode was added.

IOError used to be raised, it is now an alias of OSError .

FileExistsError is now raised if the file opened in exclusive creation mode ( ‘x’ ) already exists.

Changed in version 3.4:

The file is now non-inheritable.

Changed in version 3.5:

If the system call is interrupted and the signal handler does not raise an exception, the function now retries the system call instead of raising an InterruptedError exception (see PEP 475 for the rationale).

The ‘namereplace’ error handler was added.

Changed in version 3.6:

Support added to accept objects implementing os.PathLike .

On Windows, opening a console buffer may return a subclass of io.RawIOBase other than io.FileIO .

Changed in version 3.11: The ‘U’ mode has been removed.

Given a string representing one Unicode character, return an integer representing the Unicode code point of that character. For example, ord(‘a’) returns the integer 97 and ord(‘€’) (Euro sign) returns 8364 . This is the inverse of chr() .

Return base to the power exp; if mod is present, return base to the power exp, modulo mod (computed more efficiently than pow(base, exp) % mod ). The two-argument form pow(base, exp) is equivalent to using the power operator: base**exp .

The arguments must have numeric types. With mixed operand types, the coercion rules for binary arithmetic operators apply. For int operands, the result has the same type as the operands (after coercion) unless the second argument is negative; in that case, all arguments are converted to float and a float result is delivered. For example, pow(10, 2) returns 100 , but pow(10, -2) returns 0.01 . For a negative base of type int or float and a non-integral exponent, a complex result is delivered. For example, pow(-9, 0.5) returns a value close to 3j .

For int operands base and exp, if mod is present, mod must also be of integer type and mod must be nonzero. If mod is present and exp is negative, base must be relatively prime to mod. In that case, pow(inv_base, -exp, mod) is returned, where inv_base is an inverse to base modulo mod.

Here’s an example of computing an inverse for 38 modulo 97 :

Changed in version 3.8: For int operands, the three-argument form of pow now allows the second argument to be negative, permitting computation of modular inverses.

Changed in version 3.8: Allow keyword arguments. Formerly, only positional arguments were supported.

Print objects to the text stream file, separated by sep and followed by end. sep, end, file, and flush, if present, must be given as keyword arguments.

All non-keyword arguments are converted to strings like str() does and written to the stream, separated by sep and followed by end. Both sep and end must be strings; they can also be None , which means to use the default values. If no objects are given, print() will just write end.

The file argument must be an object with a write(string) method; if it is not present or None , sys.stdout will be used. Since printed arguments are converted to text strings, print() cannot be used with binary mode file objects. For these, use file.write(. ) instead.

Whether the output is buffered is usually determined by file, but if the flush keyword argument is true, the stream is forcibly flushed.

Changed in version 3.3: Added the flush keyword argument.

Return a property attribute.

fget is a function for getting an attribute value. fset is a function for setting an attribute value. fdel is a function for deleting an attribute value. And doc creates a docstring for the attribute.

A typical use is to define a managed attribute x :

If c is an instance of C, c.x will invoke the getter, c.x = value will invoke the setter, and del c.x the deleter.

If given, doc will be the docstring of the property attribute. Otherwise, the property will copy fget’s docstring (if it exists). This makes it possible to create read-only properties easily using property() as a decorator :

The @property decorator turns the voltage() method into a “getter” for a read-only attribute with the same name, and it sets the docstring for voltage to “Get the current voltage.”

A property object has getter , setter , and deleter methods usable as decorators that create a copy of the property with the corresponding accessor function set to the decorated function. This is best explained with an example:

This code is exactly equivalent to the first example. Be sure to give the additional functions the same name as the original property ( x in this case.)

The returned property object also has the attributes fget , fset , and fdel corresponding to the constructor arguments.

Changed in version 3.5: The docstrings of property objects are now writeable.

Rather than being a function, range is actually an immutable sequence type, as documented in Ranges and Sequence Types — list, tuple, range .

Return a string containing a printable representation of an object. For many types, this function makes an attempt to return a string that would yield an object with the same value when passed to eval() ; otherwise, the representation is a string enclosed in angle brackets that contains the name of the type of the object together with additional information often including the name and address of the object. A class can control what this function returns for its instances by defining a __repr__() method. If sys.displayhook() is not accessible, this function will raise RuntimeError .

Return a reverse iterator . seq must be an object which has a __reversed__() method or supports the sequence protocol (the __len__() method and the __getitem__() method with integer arguments starting at 0 ).

round ( number , ndigits = None ) ¶

Return number rounded to ndigits precision after the decimal point. If ndigits is omitted or is None , it returns the nearest integer to its input.

For the built-in types supporting round() , values are rounded to the closest multiple of 10 to the power minus ndigits; if two multiples are equally close, rounding is done toward the even choice (so, for example, both round(0.5) and round(-0.5) are 0 , and round(1.5) is 2 ). Any integer value is valid for ndigits (positive, zero, or negative). The return value is an integer if ndigits is omitted or None . Otherwise, the return value has the same type as number.

For a general Python object number , round delegates to number.__round__ .

The behavior of round() for floats can be surprising: for example, round(2.675, 2) gives 2.67 instead of the expected 2.68 . This is not a bug: it’s a result of the fact that most decimal fractions can’t be represented exactly as a float. See Floating Point Arithmetic: Issues and Limitations for more information.

Return a new set object, optionally with elements taken from iterable. set is a built-in class. See set and Set Types — set, frozenset for documentation about this class.

For other containers see the built-in frozenset , list , tuple , and dict classes, as well as the collections module.

This is the counterpart of getattr() . The arguments are an object, a string, and an arbitrary value. The string may name an existing attribute or a new attribute. The function assigns the value to the attribute, provided the object allows it. For example, setattr(x, ‘foobar’, 123) is equivalent to x.foobar = 123 .

name need not be a Python identifier as defined in Identifiers and keywords unless the object chooses to enforce that, for example in a custom __getattribute__() or via __slots__ . An attribute whose name is not an identifier will not be accessible using the dot notation, but is accessible through getattr() etc..

Since private name mangling happens at compilation time, one must manually mangle a private attribute’s (attributes with two leading underscores) name in order to set it with setattr() .

Return a slice object representing the set of indices specified by range(start, stop, step) . The start and step arguments default to None . Slice objects have read-only data attributes start , stop , and step which merely return the argument values (or their default). They have no other explicit functionality; however, they are used by NumPy and other third-party packages. Slice objects are also generated when extended indexing syntax is used. For example: a[start:stop:step] or a[start:stop, i] . See itertools.islice() for an alternate version that returns an iterator.

Return a new sorted list from the items in iterable.

Has two optional arguments which must be specified as keyword arguments.

key specifies a function of one argument that is used to extract a comparison key from each element in iterable (for example, key=str.lower ). The default value is None (compare the elements directly).

reverse is a boolean value. If set to True , then the list elements are sorted as if each comparison were reversed.

Use functools.cmp_to_key() to convert an old-style cmp function to a key function.

The built-in sorted() function is guaranteed to be stable. A sort is stable if it guarantees not to change the relative order of elements that compare equal — this is helpful for sorting in multiple passes (for example, sort by department, then by salary grade).

The sort algorithm uses only < comparisons between items. While defining an __lt__() method will suffice for sorting, PEP 8 recommends that all six rich comparisons be implemented. This will help avoid bugs when using the same data with other ordering tools such as max() that rely on a different underlying method. Implementing all six comparisons also helps avoid confusion for mixed type comparisons which can call reflected the __gt__() method.

For sorting examples and a brief sorting tutorial, see Sorting HOW TO .

Transform a method into a static method.

A static method does not receive an implicit first argument. To declare a static method, use this idiom:

The @staticmethod form is a function decorator – see Function definitions for details.

A static method can be called either on the class (such as C.f() ) or on an instance (such as C().f() ). Moreover, they can be called as regular functions (such as f() ).

Static methods in Python are similar to those found in Java or C++. Also, see classmethod() for a variant that is useful for creating alternate class constructors.

Like all decorators, it is also possible to call staticmethod as a regular function and do something with its result. This is needed in some cases where you need a reference to a function from a class body and you want to avoid the automatic transformation to instance method. For these cases, use this idiom:

For more information on static methods, see The standard type hierarchy .

Changed in version 3.10: Static methods now inherit the method attributes ( __module__ , __name__ , __qualname__ , __doc__ and __annotations__ ), have a new __wrapped__ attribute, and are now callable as regular functions.

Return a str version of object. See str() for details.

str is the built-in string class . For general information about strings, see Text Sequence Type — str .

Sums start and the items of an iterable from left to right and returns the total. The iterable’s items are normally numbers, and the start value is not allowed to be a string.

For some use cases, there are good alternatives to sum() . The preferred, fast way to concatenate a sequence of strings is by calling ».join(sequence) . To add floating point values with extended precision, see math.fsum() . To concatenate a series of iterables, consider using itertools.chain() .

Changed in version 3.8: The start parameter can be specified as a keyword argument.

Return a proxy object that delegates method calls to a parent or sibling class of type. This is useful for accessing inherited methods that have been overridden in a class.

The object_or_type determines the method resolution order to be searched. The search starts from the class right after the type.

For example, if __mro__ of object_or_type is D -> B -> C -> A -> object and the value of type is B , then super() searches C -> A -> object .

The __mro__ attribute of the object_or_type lists the method resolution search order used by both getattr() and super() . The attribute is dynamic and can change whenever the inheritance hierarchy is updated.

If the second argument is omitted, the super object returned is unbound. If the second argument is an object, isinstance(obj, type) must be true. If the second argument is a type, issubclass(type2, type) must be true (this is useful for classmethods).

There are two typical use cases for super. In a class hierarchy with single inheritance, super can be used to refer to parent classes without naming them explicitly, thus making the code more maintainable. This use closely parallels the use of super in other programming languages.

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The second use case is to support cooperative multiple inheritance in a dynamic execution environment. This use case is unique to Python and is not found in statically compiled languages or languages that only support single inheritance. This makes it possible to implement “diamond diagrams” where multiple base classes implement the same method. Good design dictates that such implementations have the same calling signature in every case (because the order of calls is determined at runtime, because that order adapts to changes in the class hierarchy, and because that order can include sibling classes that are unknown prior to runtime).

For both use cases, a typical superclass call looks like this:

In addition to method lookups, super() also works for attribute lookups. One possible use case for this is calling descriptors in a parent or sibling class.

Note that super() is implemented as part of the binding process for explicit dotted attribute lookups such as super().__getitem__(name) . It does so by implementing its own __getattribute__() method for searching classes in a predictable order that supports cooperative multiple inheritance. Accordingly, super() is undefined for implicit lookups using statements or operators such as super()[name] .

Also note that, aside from the zero argument form, super() is not limited to use inside methods. The two argument form specifies the arguments exactly and makes the appropriate references. The zero argument form only works inside a class definition, as the compiler fills in the necessary details to correctly retrieve the class being defined, as well as accessing the current instance for ordinary methods.

For practical suggestions on how to design cooperative classes using super() , see guide to using super().

class tuple class tuple ( iterable )

Rather than being a function, tuple is actually an immutable sequence type, as documented in Tuples and Sequence Types — list, tuple, range .

With one argument, return the type of an object. The return value is a type object and generally the same object as returned by object.__class__ .

The isinstance() built-in function is recommended for testing the type of an object, because it takes subclasses into account.

With three arguments, return a new type object. This is essentially a dynamic form of the class statement. The name string is the class name and becomes the __name__ attribute. The bases tuple contains the base classes and becomes the __bases__ attribute; if empty, object , the ultimate base of all classes, is added. The dict dictionary contains attribute and method definitions for the class body; it may be copied or wrapped before becoming the __dict__ attribute. The following two statements create identical type objects:

Keyword arguments provided to the three argument form are passed to the appropriate metaclass machinery (usually __init_subclass__() ) in the same way that keywords in a class definition (besides metaclass) would.

Changed in version 3.6: Subclasses of type which don’t override type.__new__ may no longer use the one-argument form to get the type of an object.

Return the __dict__ attribute for a module, class, instance, or any other object with a __dict__ attribute.

Objects such as modules and instances have an updateable __dict__ attribute; however, other objects may have write restrictions on their __dict__ attributes (for example, classes use a types.MappingProxyType to prevent direct dictionary updates).

Without an argument, vars() acts like locals() . Note, the locals dictionary is only useful for reads since updates to the locals dictionary are ignored.

A TypeError exception is raised if an object is specified but it doesn’t have a __dict__ attribute (for example, if its class defines the __slots__ attribute).

zip ( * iterables , strict = False ) ¶

Iterate over several iterables in parallel, producing tuples with an item from each one.

More formally: zip() returns an iterator of tuples, where the i-th tuple contains the i-th element from each of the argument iterables.

Another way to think of zip() is that it turns rows into columns, and columns into rows. This is similar to transposing a matrix.

zip() is lazy: The elements won’t be processed until the iterable is iterated on, e.g. by a for loop or by wrapping in a list .

One thing to consider is that the iterables passed to zip() could have different lengths; sometimes by design, and sometimes because of a bug in the code that prepared these iterables. Python offers three different approaches to dealing with this issue:

By default, zip() stops when the shortest iterable is exhausted. It will ignore the remaining items in the longer iterables, cutting off the result to the length of the shortest iterable:

zip() is often used in cases where the iterables are assumed to be of equal length. In such cases, it’s recommended to use the strict=True option. Its output is the same as regular zip() :

Unlike the default behavior, it raises a ValueError if one iterable is exhausted before the others:

Without the strict=True argument, any bug that results in iterables of different lengths will be silenced, possibly manifesting as a hard-to-find bug in another part of the program.

Shorter iterables can be padded with a constant value to make all the iterables have the same length. This is done by itertools.zip_longest() .

Edge cases: With a single iterable argument, zip() returns an iterator of 1-tuples. With no arguments, it returns an empty iterator.

Tips and tricks:

The left-to-right evaluation order of the iterables is guaranteed. This makes possible an idiom for clustering a data series into n-length groups using zip(*[iter(s)]*n, strict=True) . This repeats the same iterator n times so that each output tuple has the result of n calls to the iterator. This has the effect of dividing the input into n-length chunks.

zip() in conjunction with the * operator can be used to unzip a list:

Changed in version 3.10: Added the strict argument.

This is an advanced function that is not needed in everyday Python programming, unlike importlib.import_module() .

This function is invoked by the import statement. It can be replaced (by importing the builtins module and assigning to builtins.__import__ ) in order to change semantics of the import statement, but doing so is strongly discouraged as it is usually simpler to use import hooks (see PEP 302) to attain the same goals and does not cause issues with code which assumes the default import implementation is in use. Direct use of __import__() is also discouraged in favor of importlib.import_module() .

The function imports the module name, potentially using the given globals and locals to determine how to interpret the name in a package context. The fromlist gives the names of objects or submodules that should be imported from the module given by name. The standard implementation does not use its locals argument at all and uses its globals only to determine the package context of the import statement.

level specifies whether to use absolute or relative imports. 0 (the default) means only perform absolute imports. Positive values for level indicate the number of parent directories to search relative to the directory of the module calling __import__() (see PEP 328 for the details).

When the name variable is of the form package.module , normally, the top-level package (the name up till the first dot) is returned, not the module named by name. However, when a non-empty fromlist argument is given, the module named by name is returned.

For example, the statement import spam results in bytecode resembling the following code:

The statement import spam.ham results in this call:

Note how __import__() returns the toplevel module here because this is the object that is bound to a name by the import statement.

On the other hand, the statement from spam.ham import eggs, sausage as saus results in

Here, the spam.ham module is returned from __import__() . From this object, the names to import are retrieved and assigned to their respective names.

If you simply want to import a module (potentially within a package) by name, use importlib.import_module() .

Changed in version 3.3: Negative values for level are no longer supported (which also changes the default value to 0).

Changed in version 3.9: When the command line options -E or -I are being used, the environment variable PYTHONCASEOK is now ignored.

Note that the parser only accepts the Unix-style end of line convention. If you are reading the code from a file, make sure to use newline conversion mode to convert Windows or Mac-style newlines.

Округление чисел

При выполнении различных арифметических операций важно, чтобы результат округлялся правильно. Часто требуется округлять в большую, меньшую сторону, до ближайшего целого или округлить до сотых.

Для этого программист может использовать различные инструменты, такие как встроенная функция round(), преобразование к типу int и функции из подключаемого модуля math.

Способы округления чисел

Для округления чисел придумано много способов, они не лишены недостатков, однако часто используются для решения задач. Разберёмся в тонкостях каждого из них.

Если используется стандартная библиотека math, то в начале кода её необходимо подключить. Сделать это можно, например, с помощью инструкции: import math .

math.ceil() — округление чисел в большую сторону

Функция получила своё имя от термина «ceiling», который используется в математике для описания числа, которое больше или равно заданному.

Любая дробь находится в целочисленном интервале, например, 1.2 лежит между 1 и 2. Функция ceil() определяет, какая из границ интервала наибольшая и записывает её в результат округления.

math.floor() — округление чисел в меньшую сторону

Функция округляет дробное число до ближайшего целого, которое меньше или равно исходному. Работает аналогично функции ceil() , но с округлением в противоположную сторону.

math.trunc() — отбрасывание дробной части

Возвращает целое число, не учитывая его дробную часть. То есть никакого округления не происходит, Python просто забывает о дробной части, приводя число к целочисленному виду.

Избавиться от дробной части можно с помощью обычного преобразования числа к типу int. Такой способ полностью эквивалентен использованию trunc() .

Нормальное округление

Python позволяет реализовать нормальное арифметическое округление, использовав функцию преобразования к типу int.

И хотя int() работает по другому алгоритму, результат её использования для положительных чисел полностью аналогичен выводу функции floor(), которая округляет числа «вниз». Для отрицательных аналогичен функции ceil().

Чтобы с помощью функции int() округлить число по математическим правилам, необходимо добавить к нему 0.5, если оно положительное, и -0.5, если оно отрицательное.

Тогда операция принимает такой вид: int(num + (0.5 if num > 0 else -0.5)). Чтобы каждый раз не писать условие, удобно сделать отдельную функцию:

Функция работает также, как стандартная функция округление во второй версии Python (арифметическое округление).

round() — округление чисел

round() — стандартная функция округления в языке Python. Она не всегда работает так, как ожидается, а её алгоритм различается в разных версиях Python.

В Python 2

Во второй версии Python используется арифметическое округление. Оно обладает постоянно растущей погрешностью, что приводит к появлению неточностей и ошибок.

Увеличение погрешности вызвано неравным количеством цифр, определяющих, в какую сторону округлять. Всего 4 цифры на конце приводят к округлению «вниз», и 5 цифр к округлению «вверх».

Помимо этого, могут быть неточности, например, если округлить число 2.675 до второго знака, получится число 2.67 вместо 2.68. Это происходит из-за невозможности точно представить десятичные числа типа «float» в двоичном коде.

В Python 3

В третьей версии Python используется банковское округление. Это значит, что округление происходит до самого близкого чётного.

Такой подход не избавляет от ошибок полностью, но уменьшает шанс их возникновения и позволяет программисту добиться большей точности при вычислениях.

Но если вам по каким то причинам нужно округление как в Python 2, то можно воспользоваться функцией написанной нами выше на основе приведения к целому числу.

Округление до сотых

У функции raund() есть ещё один аргумент. Он показывает до какого количества знаков после запятой следует округлять. Таким образом, если нам надо в Python округлить до сотых, этому параметру следует задать значение 2.

Пример округления до нужного знака:

Ошибки округления и модуль decimal

При округлении функцией round(), можно получить следующее:

Почему в одном случае округляется вниз, а в другом вверх? При переводе 2.85 в двоичную систему получается число, которое немного больше. Поэтому функция видит не «5», а «>5» и округляет вверх.

Проблему неточного представления чисел отлично иллюстрирует пример:

Из-за подобных ошибок числа типа «float» нельзя использовать там, где изменения значения на одну тысячную может привести к неверному результату. Решить данную проблему поможет модуль decimal.

How to Round Numbers in Python

It’s the era of big data, and every day more and more business are trying to leverage their data to make informed decisions. Many businesses are turning to Python’s powerful data science ecosystem to analyze their data, as evidenced by Python’s rising popularity in the data science realm.

One thing every data science practitioner must keep in mind is how a dataset may be biased. Drawing conclusions from biased data can lead to costly mistakes.

There are many ways bias can creep into a dataset. If you’ve studied some statistics, you’re probably familiar with terms like reporting bias, selection bias and sampling bias. There is another type of bias that plays an important role when you are dealing with numeric data: rounding bias.

In this article, you will learn:

Why the way you round numbers is important
How to round a number according to various rounding strategies, and how to implement each method in pure Python
How rounding affects data, and which rounding strategy minimizes this effect
How to round numbers in NumPy arrays and Pandas DataFrames
When to apply different rounding strategies

Take the Quiz: Test your knowledge with our interactive “Rounding Numbers in Python” quiz. Upon completion you will receive a score so you can track your learning progress over time:

This article is not a treatise on numeric precision in computing, although we will touch briefly on the subject. Only a familiarity with the fundamentals of Python is necessary, and the math involved here should feel comfortable to anyone familiar with the equivalent of high school algebra.

Let’s start by looking at Python’s built-in rounding mechanism.

Python’s Built-in round() Function

Python has a built-in round() function that takes two numeric arguments, n and ndigits , and returns the number n rounded to ndigits . The ndigits argument defaults to zero, so leaving it out results in a number rounded to an integer. As you’ll see, round() may not work quite as you expect.

The way most people are taught to round a number goes something like this:

Round the number n to p decimal places by first shifting the decimal point in n by p places by multiplying n by 10ᵖ (10 raised to the p th power) to get a new number m .

Then look at the digit d in the first decimal place of m . If d is less than 5, round m down to the nearest integer. Otherwise, round m up.

Finally, shift the decimal point back p places by dividing m by 10ᵖ.

It’s a straightforward algorithm! For example, the number 2.5 rounded to the nearest whole number is 3 . The number 1.64 rounded to one decimal place is 1.6 .

Now open up an interpreter session and round 2.5 to the nearest whole number using Python’s built-in round() function:

How does round() handle the number 1.5 ?

So, round() rounds 1.5 up to 2 , and 2.5 down to 2 !

Before you go raising an issue on the Python bug tracker, let me assure you that round(2.5) is supposed to return 2 . There is a good reason why round() behaves the way it does.

In this article, you’ll learn that there are more ways to round a number than you might expect, each with unique advantages and disadvantages. round() behaves according to a particular rounding strategy—which may or may not be the one you need for a given situation.

You might be wondering, “Can the way I round numbers really have that much of an impact?” Let’s take a look at just how extreme the effects of rounding can be.

How Much Impact Can Rounding Have?

Suppose you have an incredibly lucky day and find $100 on the ground. Rather than spending all your money at once, you decide to play it smart and invest your money by buying some shares of different stocks.

The value of a stock depends on supply and demand. The more people there are who want to buy a stock, the more value that stock has, and vice versa. In high volume stock markets, the value of a particular stock can fluctuate on a second-by-second basis.

Let’s run a little experiment. We’ll pretend the overall value of the stocks you purchased fluctuates by some small random number each second, say between $0.05 and -$0.05. This fluctuation may not necessarily be a nice value with only two decimal places. For example, the overall value may increase by $0.031286 one second and decrease the next second by $0.028476.

You don’t want to keep track of your value to the fifth or sixth decimal place, so you decide to chop everything off after the third decimal place. In rounding jargon, this is called truncating the number to the third decimal place. There’s some error to be expected here, but by keeping three decimal places, this error couldn’t be substantial. Right?

To run our experiment using Python, let’s start by writing a truncate() function that truncates a number to three decimal places:

The truncate() function works by first shifting the decimal point in the number n three places to the right by multiplying n by 1000 . The integer part of this new number is taken with int() . Finally, the decimal point is shifted three places back to the left by dividing n by 1000 .

Next, let’s define the initial parameters of the simulation. You’ll need two variables: one to keep track of the actual value of your stocks after the simulation is complete and one for the value of your stocks after you’ve been truncating to three decimal places at each step.

Start by initializing these variables to 100 :

Now let’s run the simulation for 1,000,000 seconds (approximately 11.5 days). For each second, generate a random value between -0.05 and 0.05 with the uniform() function in the random module, and then update actual and truncated :

The meat of the simulation takes place in the for loop, which loops over the range(1000000) of numbers between 0 and 999,999 . The value taken from range() at each step is stored in the variable _ , which we use here because we don’t actually need this value inside of the loop.

At each step of the loop, a new random number between -0.05 and 0.05 is generated using random.randn() and assigned to the variable randn . The new value of your investment is calculated by adding randn to actual_value , and the truncated total is calculated by adding randn to truncated_value and then truncating this value with truncate() .

As you can see by inspecting the actual_value variable after running the loop, you only lost about $3.55. However, if you’d been looking at truncated_value , you’d have thought that you’d lost almost all of your money!

Note: In the above example, the random.seed() function is used to seed the pseudo-random number generator so that you can reproduce the output shown here.

To learn more about randomness in Python, check out Real Python’s Generating Random Data in Python (Guide).

Ignoring for the moment that round() doesn’t behave quite as you expect, let’s try re-running the simulation. We’ll use round() this time to round to three decimal places at each step, and seed() the simulation again to get the same results as before:

What a difference!

Shocking as it may seem, this exact error caused quite a stir in the early 1980s when the system designed for recording the value of the Vancouver Stock Exchange truncated the overall index value to three decimal places instead of rounding. Rounding errors have swayed elections and even resulted in the loss of life.

How you round numbers is important, and as a responsible developer and software designer, you need to know what the common issues are and how to deal with them. Let’s dive in and investigate what the different rounding methods are and how you can implement each one in pure Python.

A Menagerie of Methods

There are a plethora of rounding strategies, each with advantages and disadvantages. In this section, you’ll learn about some of the most common techniques, and how they can influence your data.

Truncation

The simplest, albeit crudest, method for rounding a number is to truncate the number to a given number of digits. When you truncate a number, you replace each digit after a given position with 0. Here are some examples:

Value	Truncated To	Result
12.345	Tens place	10
12.345	Ones place	12
12.345	Tenths place	12.3
12.345	Hundredths place	12.34

You’ve already seen one way to implement this in the truncate() function from the How Much Impact Can Rounding Have? section. In that function, the input number was truncated to three decimal places by:

Multiplying the number by 1000 to shift the decimal point three places to the right
Taking the integer part of that new number with int()
Shifting the decimal place three places back to the left by dividing by 1000

You can generalize this process by replacing 1000 with the number 10ᵖ ( 10 raised to the pth power), where p is the number of decimal places to truncate to:

In this version of truncate() , the second argument defaults to 0 so that if no second argument is passed to the function, then truncate() returns the integer part of whatever number is passed to it.

The truncate() function works well for both positive and negative numbers:

You can even pass a negative number to decimals to truncate to digits to the left of the decimal point:

When you truncate a positive number, you are rounding it down. Likewise, truncating a negative number rounds that number up. In a sense, truncation is a combination of rounding methods depending on the sign of the number you are rounding.

Let’s take a look at each of these rounding methods individually, starting with rounding up.

Rounding Up

The second rounding strategy we’ll look at is called “rounding up.” This strategy always rounds a number up to a specified number of digits. The following table summarizes this strategy:

Value	Round Up To	Result
12.345	Tens place	20
12.345	Ones place	13
12.345	Tenths place	12.4
12.345	Hundredths place	12.35

To implement the “rounding up” strategy in Python, we’ll use the ceil() function from the math module.

The ceil() function gets its name from the term “ceiling,” which is used in mathematics to describe the nearest integer that is greater than or equal to a given number.

Every number that is not an integer lies between two consecutive integers. For example, the number 1.2 lies in the interval between 1 and 2 . The “ceiling” is the greater of the two endpoints of the interval. The lesser of the two endpoints in called the “floor.” Thus, the ceiling of 1.2 is 2 , and the floor of 1.2 is 1 .

In mathematics, a special function called the ceiling function maps every number to its ceiling. To allow the ceiling function to accept integers, the ceiling of an integer is defined to be the integer itself. So the ceiling of the number 2 is 2 .

In Python, math.ceil() implements the ceiling function and always returns the nearest integer that is greater than or equal to its input:

Notice that the ceiling of -0.5 is 0 , not -1 . This makes sense because 0 is the nearest integer to -0.5 that is greater than or equal to -0.5 .

Let’s write a function called round_up() that implements the “rounding up” strategy:

You may notice that round_up() looks a lot like truncate() . First, the decimal point in n is shifted the correct number of places to the right by multiplying n by 10 ** decimals . This new value is rounded up to the nearest integer using math.ceil() , and then the decimal point is shifted back to the left by dividing by 10 ** decimals .

This pattern of shifting the decimal point, applying some rounding method to round to an integer, and then shifting the decimal point back will come up over and over again as we investigate more rounding methods. This is, after all, the mental algorithm we humans use to round numbers by hand.

Let’s look at how well round_up() works for different inputs:

Just like truncate() , you can pass a negative value to decimals :

When you pass a negative number to decimals , the number in the first argument of round_up() is rounded to the correct number of digits to the left of the decimal point.

Take a guess at what round_up(-1.5) returns:

Is -1.0 what you expected?

If you examine the logic used in defining round_up() —in particular, the way the math.ceil() function works—then it makes sense that round_up(-1.5) returns -1.0 . However, some people naturally expect symmetry around zero when rounding numbers, so that if 1.5 gets rounded up to 2 , then -1.5 should get rounded up to -2 .

Let’s establish some terminology. For our purposes, we’ll use the terms “round up” and “round down” according to the following diagram:

Round up to the right and down to the left. (Image: David Amos)

Rounding up always rounds a number to the right on the number line, and rounding down always rounds a number to the left on the number line.

Rounding Down

The counterpart to “rounding up” is the “rounding down” strategy, which always rounds a number down to a specified number of digits. Here are some examples illustrating this strategy:

Value	Rounded Down To	Result
12.345	Tens place	10
12.345	Ones place	12
12.345	Tenths place	12.3
12.345	Hundredths place	12.34

To implement the “rounding down” strategy in Python, we can follow the same algorithm we used for both trunctate() and round_up() . First shift the decimal point, then round to an integer, and finally shift the decimal point back.

In round_up() , we used math.ceil() to round up to the ceiling of the number after shifting the decimal point. For the “rounding down” strategy, though, we need to round to the floor of the number after shifting the decimal point.

Lucky for us, the math module has a floor() function that returns the floor of its input:

Here’s the definition of round_down() :

That looks just like round_up() , except math.ceil() has been replaced with math.floor() .

You can test round_down() on a few different values:

The effects of round_up() and round_down() can be pretty extreme. By rounding the numbers in a large dataset up or down, you could potentially remove a ton of precision and drastically alter computations made from the data.

Before we discuss any more rounding strategies, let’s stop and take a moment to talk about how rounding can make your data biased.

Interlude: Rounding Bias

You’ve now seen three rounding methods: truncate() , round_up() , and round_down() . All three of these techniques are rather crude when it comes to preserving a reasonable amount of precision for a given number.

There is one important difference between truncate() and round_up() and round_down() that highlights an important aspect of rounding: symmetry around zero.

Recall that round_up() isn’t symmetric around zero. In mathematical terms, a function f(x) is symmetric around zero if, for any value of x, f(x) + f(-x) = 0. For example, round_up(1.5) returns 2 , but round_up(-1.5) returns -1 . The round_down() function isn’t symmetric around 0, either.

On the other hand, the truncate() function is symmetric around zero. This is because, after shifting the decimal point to the right, truncate() chops off the remaining digits. When the initial value is positive, this amounts to rounding the number down. Negative numbers are rounded up. So, truncate(1.5) returns 1 , and truncate(-1.5) returns -1 .

The concept of symmetry introduces the notion of rounding bias, which describes how rounding affects numeric data in a dataset.

The “rounding up” strategy has a round towards positive infinity bias, because the value is always rounded up in the direction of positive infinity. Likewise, the “rounding down” strategy has a round towards negative infinity bias.

The “truncation” strategy exhibits a round towards negative infinity bias on positive values and a round towards positive infinity for negative values. Rounding functions with this behavior are said to have a round towards zero bias, in general.

Let’s see how this works in practice. Consider the following list of floats:

Let’s compute the mean value of the values in data using the statistics.mean() function:

Now apply each of round_up() , round_down() , and truncate() in a list comprehension to round each number in data to one decimal place and calculate the new mean:

After every number in data is rounded up, the new mean is about -1.033 , which is greater than the actual mean of about 1.108 . Rounding down shifts the mean downwards to about -1.133 . The mean of the truncated values is about -1.08 and is the closest to the actual mean.

This example does not imply that you should always truncate when you need to round individual values while preserving a mean value as closely as possible. The data list contains an equal number of positive and negative values. The truncate() function would behave just like round_up() on a list of all positive values, and just like round_down() on a list of all negative values.

What this example does illustrate is the effect rounding bias has on values computed from data that has been rounded. You will need to keep these effects in mind when drawing conclusions from data that has been rounded.

Typically, when rounding, you are interested in rounding to the nearest number with some specified precision, instead of just rounding everything up or down.

For example, if someone asks you to round the numbers 1.23 and 1.28 to one decimal place, you would probably respond quickly with 1.2 and 1.3 . The truncate() , round_up() , and round_down() functions don’t do anything like this.

What about the number 1.25 ? You probably immediately think to round this to 1.3 , but in reality, 1.25 is equidistant from 1.2 and 1.3 . In a sense, 1.2 and 1.3 are both the nearest numbers to 1.25 with single decimal place precision. The number 1.25 is called a tie with respect to 1.2 and 1.3 . In cases like this, you must assign a tiebreaker.

The way that most people are taught break ties is by rounding to the greater of the two possible numbers.

Rounding Half Up

The “rounding half up” strategy rounds every number to the nearest number with the specified precision, and breaks ties by rounding up. Here are some examples:

Value	Round Half Up To	Result
13.825	Tens place	10
13.825	Ones place	14
13.825	Tenths place	13.8
13.825	Hundredths place	13.83

To implement the “rounding half up” strategy in Python, you start as usual by shifting the decimal point to the right by the desired number of places. At this point, though, you need a way to determine if the digit just after the shifted decimal point is less than or greater than or equal to 5 .

One way to do this is to add 0.5 to the shifted value and then round down with math.floor() . This works because:

If the digit in the first decimal place of the shifted value is less than five, then adding 0.5 won’t change the integer part of the shifted value, so the floor is equal to the integer part.

If the first digit after the decimal place is greater than or equal to 5 , then adding 0.5 will increase the integer part of the shifted value by 1 , so the floor is equal to this larger integer.

Here’s what this looks like in Python:

Notice that round_half_up() looks a lot like round_down() . This might be somewhat counter-intuitive, but internally round_half_up() only rounds down. The trick is to add the 0.5 after shifting the decimal point so that the result of rounding down matches the expected value.

Let’s test round_half_up() on a couple of values to see that it works:

Since round_half_up() always breaks ties by rounding to the greater of the two possible values, negative values like -1.5 round to -1 , not to -2 :

Great! You can now finally get that result that the built-in round() function denied to you:

Before you get too excited though, let’s see what happens when you try and round -1.225 to 2 decimal places:

Wait. We just discussed how ties get rounded to the greater of the two possible values. -1.225 is smack in the middle of -1.22 and -1.23 . Since -1.22 is the greater of these two, round_half_up(-1.225, 2) should return -1.22 . But instead, we got -1.23 .

Is there a bug in the round_half_up() function?

When round_half_up() rounds -1.225 to two decimal places, the first thing it does is multiply -1.225 by 100 . Let’s make sure this works as expected:

Well… that’s wrong! But it does explain why round_half_up(-1.225, 2) returns -1.23. Let’s continue the round_half_up() algorithm step-by-step, utilizing _ in the REPL to recall the last value output at each step:

Even though -122.00000000000001 is really close to -122 , the nearest integer that is less than or equal to it is -123 . When the decimal point is shifted back to the left, the final value is -1.23 .

Well, now you know how round_half_up(-1.225, 2) returns -1.23 even though there is no logical error, but why does Python say that -1.225 * 100 is -122.50000000000001 ? Is there a bug in Python?

Aside: In a Python interpreter session, type the following:

Seeing this for the first time can be pretty shocking, but this is a classic example of floating-point representation error. It has nothing to do with Python. The error has to do with how machines store floating-point numbers in memory.

Most modern computers store floating-point numbers as binary decimals with 53-bit precision. Only numbers that have finite binary decimal representations that can be expressed in 53 bits are stored as an exact value. Not every number has a finite binary decimal representation.

For example, the decimal number 0.1 has a finite decimal representation, but infinite binary representation. Just like the fraction 1/3 can only be represented in decimal as the infinitely repeating decimal 0.333. , the fraction 1/10 can only be expressed in binary as the infinitely repeating decimal 0.0001100110011. .

A value with an infinite binary representation is rounded to an approximate value to be stored in memory. The method that most machines use to round is determined according to the IEEE-754 standard, which specifies rounding to the nearest representable binary fraction.

The Python docs have a section called Floating Point Arithmetic: Issues and Limitations which has this to say about the number 0.1:

On most machines, if Python were to print the true decimal value of the binary approximation stored for 0.1 , it would have to display

That is more digits than most people find useful, so Python keeps the number of digits manageable by displaying a rounded value instead

Just remember, even though the printed result looks like the exact value of 1/10 , the actual stored value is the nearest representable binary fraction. (Source)

For a more in-depth treatise on floating-point arithmetic, check out David Goldberg’s article What Every Computer Scientist Should Know About Floating-Point Arithmetic, originally published in the journal ACM Computing Surveys, Vol. 23, No. 1, March 1991.

The fact that Python says that -1.225 * 100 is -122.50000000000001 is an artifact of floating-point representation error. You might be asking yourself, “Okay, but is there a way to fix this?” A better question to ask yourself is “Do I need to fix this?”

Floating-point numbers do not have exact precision, and therefore should not be used in situations where precision is paramount. For applications where the exact precision is necessary, you can use the Decimal class from Python’s decimal module. You’ll learn more about the Decimal class below.

If you have determined that Python’s standard float class is sufficient for your application, some occasional errors in round_half_up() due to floating-point representation error shouldn’t be a concern.

Now that you’ve gotten a taste of how machines round numbers in memory, let’s continue our discussion on rounding strategies by looking at another way to break a tie.

Rounding Half Down

The “rounding half down” strategy rounds to the nearest number with the desired precision, just like the “rounding half up” method, except that it breaks ties by rounding to the lesser of the two numbers. Here are some examples:

Value	Round Half Down To	Result
13.825	Tens place	10
13.825	Ones place	14
13.825	Tenths place	13.8
13.825	Hundredths place	13.82

You can implement the “rounding half down” strategy in Python by replacing math.floor() in the round_half_up() function with math.ceil() and subtracting 0.5 instead of adding:

Let’s check round_half_down() against a few test cases:

Both round_half_up() and round_half_down() have no bias in general. However, rounding data with lots of ties does introduce a bias. For an extreme example, consider the following list of numbers:

Let’s compute the mean of these numbers:

Next, compute the mean on the data after rounding to one decimal place with round_half_up() and round_half_down() :

Every number in data is a tie with respect to rounding to one decimal place. The round_half_up() function introduces a round towards positive infinity bias, and round_half_down() introduces a round towards negative infinity bias.

The remaining rounding strategies we’ll discuss all attempt to mitigate these biases in different ways.

Rounding Half Away From Zero

If you examine round_half_up() and round_half_down() closely, you’ll notice that neither of these functions is symmetric around zero:

One way to introduce symmetry is to always round a tie away from zero. The following table illustrates how this works:

Value	Round Half Away From Zero To	Result
15.25	Tens place	20
15.25	Ones place	15
15.25	Tenths place	15.3
-15.25	Tens place	-20
-15.25	Ones place	-15
-15.25	Tenths place	-15.3

To implement the “rounding half away from zero” strategy on a number n , you start as usual by shifting the decimal point to the right a given number of places. Then you look at the digit d immediately to the right of the decimal place in this new number. At this point, there are four cases to consider:

If n is positive and d >= 5 , round up
If n is positive and d < 5 , round down
If n is negative and d >= 5 , round down
If n is negative and d < 5 , round up

After rounding according to one of the above four rules, you then shift the decimal place back to the left.

Given a number n and a value for decimals , you could implement this in Python by using round_half_up() and round_half_down() :

That’s easy enough, but there’s actually a simpler way!

If you first take the absolute value of n using Python’s built-in abs() function, you can just use round_half_up() to round the number. Then all you need to do is give the rounded number the same sign as n . One way to do this is using the math.copysign() function.

math.copysign() takes two numbers a and b and returns a with the sign of b :

Notice that math.copysign() returns a float , even though both of its arguments were integers.

Using abs() , round_half_up() and math.copysign() , you can implement the “rounding half away from zero” strategy in just two lines of Python:

In round_half_away_from_zero() , the absolute value of n is rounded to decimals decimal places using round_half_up() and this result is assigned to the variable rounded_abs . Then the original sign of n is applied to rounded_abs using math.copysign() , and this final value with the correct sign is returned by the function.

Checking round_half_away_from_zero() on a few different values shows that the function behaves as expected:

The round_half_away_from_zero() function rounds numbers the way most people tend to round numbers in everyday life. Besides being the most familiar rounding function you’ve seen so far, round_half_away_from_zero() also eliminates rounding bias well in datasets that have an equal number of positive and negative ties.

Let’s check how well round_half_away_from_zero() mitigates rounding bias in the example from the previous section:

The mean value of the numbers in data is preserved almost exactly when you round each number in data to one decimal place with round_half_away_from_zero() !

However, round_half_away_from_zero() will exhibit a rounding bias when you round every number in datasets with only positive ties, only negative ties, or more ties of one sign than the other. Bias is only mitigated well if there are a similar number of positive and negative ties in the dataset.

How do you handle situations where the number of positive and negative ties are drastically different? The answer to this question brings us full circle to the function that deceived us at the beginning of this article: Python’s built-in round() function.

Rounding Half To Even

One way to mitigate rounding bias when rounding values in a dataset is to round ties to the nearest even number at the desired precision. Here are some examples of how to do that:

Value	Round Half To Even To	Result
15.255	Tens place	20
15.255	Ones place	15
15.255	Tenths place	15.3
15.255	Hundredths place	15.26

The “rounding half to even strategy” is the strategy used by Python’s built-in round() function and is the default rounding rule in the IEEE-754 standard. This strategy works under the assumption that the probabilities of a tie in a dataset being rounded down or rounded up are equal. In practice, this is usually the case.

Now you know why round(2.5) returns 2 . It’s not a mistake. It is a conscious design decision based on solid recommendations.

To prove to yourself that round() really does round to even, try it on a few different values:

The round() function is nearly free from bias, but it isn’t perfect. For example, rounding bias can still be introduced if the majority of the ties in your dataset round up to even instead of rounding down. Strategies that mitigate bias even better than “rounding half to even” do exist, but they are somewhat obscure and only necessary in extreme circumstances.

Finally, round() suffers from the same hiccups that you saw in round_half_up() thanks to floating-point representation error:

You shouldn’t be concerned with these occasional errors if floating-point precision is sufficient for your application.

When precision is paramount, you should use Python’s Decimal class.

The Decimal Class

Python’s decimal module is one of those “batteries-included” features of the language that you might not be aware of if you’re new to Python. The guiding principle of the decimal module can be found in the documentation:

Decimal “is based on a floating-point model which was designed with people in mind, and necessarily has a paramount guiding principle – computers must provide an arithmetic that works in the same way as the arithmetic that people learn at school.” – excerpt from the decimal arithmetic specification. (Source)

The benefits of the decimal module include:

Exact decimal representation: 0.1 is actually 0.1 , and 0.1 + 0.1 + 0.1 — 0.3 returns 0 , as you’d expect.
Preservation of significant digits: When you add 1.20 and 2.50 , the result is 3.70 with the trailing zero maintained to indicate significance.
User-alterable precision: The default precision of the decimal module is twenty-eight digits, but this value can be altered by the user to match the problem at hand.

Let’s explore how rounding works in the decimal module. Start by typing the following into a Python REPL:

decimal.getcontext() returns a Context object representing the default context of the decimal module. The context includes the default precision and the default rounding strategy, among other things.

As you can see in the example above, the default rounding strategy for the decimal module is ROUND_HALF_EVEN . This aligns with the built-in round() function and should be the preferred rounding strategy for most purposes.

Let’s declare a number using the decimal module’s Decimal class. To do so, create a new Decimal instance by passing a string containing the desired value:

Note: It is possible to create a Decimal instance from a floating-point number, but doing so introduces floating-point representation error right off the bat. For example, check out what happens when you create a Decimal instance from the floating-point number 0.1 :

In order to maintain exact precision, you must create Decimal instances from strings containing the decimal numbers you need.

Just for fun, let’s test the assertion that Decimal maintains exact decimal representation:

Ahhh. That’s satisfying, isn’t it?

Rounding a Decimal is done with the .quantize() method:

Okay, that probably looks a little funky, so let’s break that down. The Decimal(«1.0») argument in .quantize() determines the number of decimal places to round the number. Since 1.0 has one decimal place, the number 1.65 rounds to a single decimal place. The default rounding strategy is “rounding half to even,” so the result is 1.6 .

Recall that the round() function, which also uses the “rounding half to even strategy,” failed to round 2.675 to two decimal places correctly. Instead of 2.68 , round(2.675, 2) returns 2.67 . Thanks to the decimal modules exact decimal representation, you won’t have this issue with the Decimal class:

Another benefit of the decimal module is that rounding after performing arithmetic is taken care of automatically, and significant digits are preserved. To see this in action, let’s change the default precision from twenty-eight digits to two, and then add the numbers 1.23 and 2.32 :

To change the precision, you call decimal.getcontext() and set the .prec attribute. If setting the attribute on a function call looks odd to you, you can do this because .getcontext() returns a special Context object that represents the current internal context containing the default parameters used by the decimal module.

The exact value of 1.23 plus 2.32 is 3.55 . Since the precision is now two digits, and the rounding strategy is set to the default of “rounding half to even,” the value 3.55 is automatically rounded to 3.6 .

To change the default rounding strategy, you can set the decimal.getcontect().rounding property to any one of several flags. The following table summarizes these flags and which rounding strategy they implement:

Flag	Rounding Strategy
decimal.ROUND_CEILING	Rounding up
decimal.ROUND_FLOOR	Rounding down
decimal.ROUND_DOWN	Truncation
decimal.ROUND_UP	Rounding away from zero
decimal.ROUND_HALF_UP	Rounding half away from zero
decimal.ROUND_HALF_DOWN	Rounding half towards zero
decimal.ROUND_HALF_EVEN	Rounding half to even
decimal.ROUND_05UP	Rounding up and rounding towards zero

The first thing to notice is that the naming scheme used by the decimal module differs from what we agreed to earlier in the article. For example, decimal.ROUND_UP implements the “rounding away from zero” strategy, which actually rounds negative numbers down.

Secondly, some of the rounding strategies mentioned in the table may look unfamiliar since we haven’t discussed them. You’ve already seen how decimal.ROUND_HALF_EVEN works, so let’s take a look at each of the others in action.

The decimal.ROUND_CEILING strategy works just like the round_up() function we defined earlier:

Notice that the results of decimal.ROUND_CEILING are not symmetric around zero.

The decimal.ROUND_FLOOR strategy works just like our round_down() function:

Like decimal.ROUND_CEILING , the decimal.ROUND_FLOOR strategy is not symmetric around zero.

The decimal.ROUND_DOWN and decimal.ROUND_UP strategies have somewhat deceptive names. Both ROUND_DOWN and ROUND_UP are symmetric around zero:

The decimal.ROUND_DOWN strategy rounds numbers towards zero, just like the truncate() function. On the other hand, decimal.ROUND_UP rounds everything away from zero. This is a clear break from the terminology we agreed to earlier in the article, so keep that in mind when you are working with the decimal module.

There are three strategies in the decimal module that allow for more nuanced rounding. The decimal.ROUND_HALF_UP method rounds everything to the nearest number and breaks ties by rounding away from zero:

Notice that decimal.ROUND_HALF_UP works just like our round_half_away_from_zero() and not like round_half_up() .

There is also a decimal.ROUND_HALF_DOWN strategy that breaks ties by rounding towards zero:

The final rounding strategy available in the decimal module is very different from anything we have seen so far:

In the above examples, it looks as if decimal.ROUND_05UP rounds everything towards zero. In fact, this is exactly how decimal.ROUND_05UP works, unless the result of rounding ends in a 0 or 5 . In that case, the number gets rounded away from zero:

In the first example, the number 1.49 is first rounded towards zero in the second decimal place, producing 1.4 . Since 1.4 does not end in a 0 or a 5 , it is left as is. On the other hand, 1.51 is rounded towards zero in the second decimal place, resulting in the number 1.5 . This ends in a 5 , so the first decimal place is then rounded away from zero to 1.6 .

In this section, we have only focused on the rounding aspects of the decimal module. There are a large number of other features that make decimal an excellent choice for applications where the standard floating-point precision is inadequate, such as banking and some problems in scientific computing.

For more information on Decimal , check out the Quick-start Tutorial in the Python docs.

Next, let’s turn our attention to two staples of Python’s scientific computing and data science stacks: NumPy and Pandas.

Rounding NumPy Arrays

In the domains of data science and scientific computing, you often store your data as a NumPy array . One of NumPy’s most powerful features is its use of vectorization and broadcasting to apply operations to an entire array at once instead of one element at a time.

Let’s generate some data by creating a 3×4 NumPy array of pseudo-random numbers:

First, we seed the np.random module so that you can easily reproduce the output. Then a 3×4 NumPy array of floating-point numbers is created with np.random.randn() .

Note: You’ll need to pip3 install numpy before typing the above code into your REPL if you don’t already have NumPy in your environment. If you installed Python with Anaconda, you’re already set!

If you haven’t used NumPy before, you can get a quick introduction in the Getting Into Shape section of Brad Solomon’s Look Ma, No For-Loops: Array Programming With NumPy here at Real Python.

To round all of the values in the data array, you can pass data as the argument to the np.around() function. The desired number of decimal places is set with the decimals keyword argument. The round half to even strategy is used, just like Python’s built-in round() function.

For example, the following rounds all of the values in data to three decimal places:

np.around() is at the mercy of floating-point representation error, just like round() is.

For example, the value in the third row of the first column in the data array is 0.20851975 . When you round this to three decimal places using the “rounding half to even” strategy, you expect the value to be 0.208 . But you can see in the output from np.around() that the value is rounded to 0.209 . However, the value 0.3775384 in the first row of the second column rounds correctly to 0.378 .

If you need to round the data in your array to integers, NumPy offers several options:

The np.ceil() function rounds every value in the array to the nearest integer greater than or equal to the original value:

Hey, we discovered a new number! Negative zero!

Actually, the IEEE-754 standard requires the implementation of both a positive and negative zero. What possible use is there for something like this? Wikipedia knows the answer:

Informally, one may use the notation “ −0 ” for a negative value that was rounded to zero. This notation may be useful when a negative sign is significant; for example, when tabulating Celsius temperatures, where a negative sign means below freezing. (Source)

To round every value down to the nearest integer, use np.floor() :

You can also truncate each value to its integer component with np.trunc() :

Finally, to round to the nearest integer using the “rounding half to even” strategy, use np.rint() :

You might have noticed that a lot of the rounding strategies we discussed earlier are missing here. For the vast majority of situations, the around() function is all you need. If you need to implement another strategy, such as round_half_up() , you can do so with a simple modification:

Thanks to NumPy’s vectorized operations, this works just as you expect:

Now that you’re a NumPy rounding master, let’s take a look at Python’s other data science heavy-weight: the Pandas library.

Rounding Pandas Series and DataFrame

The Pandas library has become a staple for data scientists and data analysts who work in Python. In the words of Real Python’s own Joe Wyndham:

Pandas is a game-changer for data science and analytics, particularly if you came to Python because you were searching for something more powerful than Excel and VBA. (Source)

Note: Before you continue, you’ll need to pip3 install pandas if you don’t already have it in your environment. As was the case for NumPy, if you installed Python with Anaconda, you should be ready to go!

The two main Pandas data structures are the DataFrame , which in very loose terms works sort of like an Excel spreadsheet, and the Series , which you can think of as a column in a spreadsheet. Both Series and DataFrame objects can also be rounded efficiently using the Series.round() and DataFrame.round() methods:

The DataFrame.round() method can also accept a dictionary or a Series , to specify a different precision for each column. For instance, the following examples show how to round the first column of df to one decimal place, the second to two, and the third to three decimal places:

If you need more rounding flexibility, you can apply NumPy’s floor() , ceil() , and rint() functions to Pandas Series and DataFrame objects:

The modified round_half_up() function from the previous section will also work here:

Congratulations, you’re well on your way to rounding mastery! You now know that there are more ways to round a number than there are taco combinations. (Well… maybe not!) You can implement numerous rounding strategies in pure Python, and you have sharpened your skills on rounding NumPy arrays and Pandas Series and DataFrame objects.

There’s just one more step: knowing when to apply the right strategy.

Applications and Best Practices

The last stretch on your road to rounding virtuosity is understanding when to apply your newfound knowledge. In this section, you’ll learn some best practices to make sure you round your numbers the right way.

Store More and Round Late

When you deal with large sets of data, storage can be an issue. In most relational databases, each column in a table is designed to store a specific data type, and numeric data types are often assigned precision to help conserve memory.

For example, a temperature sensor may report the temperature in a long-running industrial oven every ten seconds accurate to eight decimal places. The readings from this are used to detect abnormal fluctuations in temperature that could indicate the failure of a heating element or some other component. So, there might be a Python script running that compares each incoming reading to the last to check for large fluctuations.

The readings from this sensor are also stored in a SQL database so that the daily average temperature inside the oven can be computed each day at midnight. The manufacturer of the heating element inside the oven recommends replacing the component whenever the daily average temperature drops .05 degrees below normal.

For this calculation, you only need three decimal places of precision. But you know from the incident at the Vancouver Stock Exchange that removing too much precision can drastically affect your calculation.

If you have the space available, you should store the data at full precision. If storage is an issue, a good rule of thumb is to store at least two or three more decimal places of precision than you need for your calculation.

Finally, when you compute the daily average temperature, you should calculate it to the full precision available and round the final answer.

Obey Local Currency Regulations

When you order a cup of coffee for $2.40 at the coffee shop, the merchant typically adds a required tax. The amount of that tax depends a lot on where you are geographically, but for the sake of argument, let’s say it’s 6%. The tax to be added comes out to $0.144. Should you round this up to $0.15 or down to $0.14? The answer probably depends on the regulations set forth by the local government!

Situations like this can also arise when you are converting one currency to another. In 1999, the European Commission on Economical and Financial Affairs codified the use of the “rounding half away from zero” strategy when converting currencies to the Euro, but other currencies may have adopted different regulations.

Another scenario, “Swedish rounding”, occurs when the minimum unit of currency at the accounting level in a country is smaller than the lowest unit of physical currency. For example, if a cup of coffee costs $2.54 after tax, but there are no 1-cent coins in circulation, what do you do? The buyer won’t have the exact amount, and the merchant can’t make exact change.

How situations like this are handled is typically determined by a country’s government. You can find a list of rounding methods used by various countries on Wikipedia.

If you are designing software for calculating currencies, you should always check the local laws and regulations in your users’ locations.

When In Doubt, Round Ties To Even

When you are rounding numbers in large datasets that are used in complex computations, the primary concern is limiting the growth of the error due to rounding.

Of all the methods we’ve discussed in this article, the “rounding half to even” strategy minimizes rounding bias the best. Fortunately, Python, NumPy, and Pandas all default to this strategy, so by using the built-in rounding functions you’re already well protected!

Summary

Whew! What a journey this has been!

In this article, you learned that:

There are various rounding strategies, which you now know how to implement in pure Python.

Every rounding strategy inherently introduces a rounding bias, and the “rounding half to even” strategy mitigates this bias well, most of the time.

The way in which computers store floating-point numbers in memory naturally introduces a subtle rounding error, but you learned how to work around this with the decimal module in Python’s standard library.

You can round NumPy arrays and Pandas Series and DataFrame objects.

There are best practices for rounding with real-world data.