As pointed out by other answers, in python they return floats probably because of historical reasons to prevent overflow problems. However, they return integers in python 3.

>>> import math
>>> type(math.floor(3.1))
<class 'int'>
>>> type(math.ceil(3.1))
<class 'int'>

You can find more information in PEP 3141.

The math.ceil (ceiling) function returns the smallest integer higher or equal to x.

For Python 3:

import math
print(math.ceil(4.2))

For Python 2:

import math
print(int(math.ceil(4.2)))

Short answer

It was a bug. Well, not exactly a bug, but the behavior was changed based on a proposal for Python 3. Now, ceil and floor return integers (see also delnan's comment). Some details are here: http://www.afpy.org/doc/python/2.7/whatsnew/2.6.html

Why Python originally returned floats

This question has some nice answers about the behaviour before Python 3. Since the mathematical operators where wrappers around the C mathematical operators, it made sense to follow the convention of that language. Note that in C, the ceil function takes and returns a double. This makes sense because not all floats can be represented by integers (for values with a big exponent, there is no direct representation with integers).

Python was historically not explicitely designed to formally conform to some of the properties of mathematical operations (that would not happen by accident). Guido Von Rossum has acknowledged some early design mistakes and explained the rationale behind the types used in Python, notably why he preferred C types instead of reusing the ones in ABC. See for example:

• Early Language Design and Development

• The Problem with Integer Division. The division operator used to perform truncation when given integers or long, which was generally unexpected and error-prone: see Changing the Division Operator.

The language is supposed to evolve, though, and people tried to incorporate numeric type systems from other languages. For example, Reworking Python's Numeric Model and A Type Hierarchy for Numbers.

Why it should be an integer

The fact that integer 8 is also a real number does mean that we should return a floating point value after doing floor(8.2), exactly because we would not return a complex value with a zero imaginary part (8 is a complex number too).

This has to do with the mathematical definitions of the operations, not the possible machine representations of values: floor and ceiling mathematical functions are defined to return integers, whereas multiplication is a ring where we expect the product of x and y from set A to belong to set A too. Arguably, it would be surprising if 8.2 * 10 returned the integer 82 and not a floating point; similarly the are no good reasons for floor(8.2) to return 8.0 if we want to be conform to the mathematical meaning.

By the way, I disagree with some parts of Robert Harvey's answer.

• There are legitimate uses to return a value of a different type depending on an input parameter, especially with mathematical operations.

• I don't think the return type should be based on a presupposed common usage of the value and I don't see how convenient it would be. And if it was relevant, I'd probably expect to be given an integer: I generally do not combine the result of floor with a floating point.

Inconvenience of Python 3

Using the operations from C in Python could be seen as a leaky abstraction of mathematical operations, whereas Python generally tries to provide a high-level view of data-structures and functions. It can be argued that people programming in Python expect operations that just work (e.g. arbitrary precision integers) and prefer to avoid dealing with numeric types at the level of C (e.g. undefined behaviour of overflow for unsigned signed integers). That's why PEP-3141 was a sensible proposition.

However, with the resulting abstraction, there might be some cases where performance might degrade, especially if we want to take the ceiling or floor of big floats without converting them to big integers (see comment from Mark Dickinson). Some may argue that this is not a big deal if a conversion occurs because it does not impact the overall performance of your program (and this is probably true, in most cases). But unfortunately, the problem here is that the programmer cannot choose which behaviour suits the most her needs. Some languages define more expressive functions: for example Common Lisp provides fflor and fceiling, which return floating-point values. It would be preferable if Python could provide fceil too. Alternatively, a sufficiently smart compiler could detect float(math.ceil(x)) and do the right thing.

All integers that can be represented by floating point numbers have an exact representation. So you can safely use int on the result. Inexact representations occur only if you are trying to represent a rational number with a denominator that is not a power of two.

That this works is not trivial at all! It's a property of the IEEE floating point representation that int∘floor = ⌊⋅⌋ if the magnitude of the numbers in question is small enough, but different representations are possible where int(floor(2.3)) might be 1.

To quote from Wikipedia,

Any integer with absolute value less than or equal to 2²⁴ can be exactly represented in the single precision format, and any integer with absolute value less than or equal to 2⁵³ can be exactly represented in the double precision format.