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Floating-point numbers are represented in computer hardware as base 2 binary fractions. For example, the decimal fraction. These two fractions have identical values, the only real difference being that the first is written in base 10 fractional notation, and the second in base 2.

Unfortunately, most decimal fractions cannot be represented exactly as binary fractions. A consequence is that, in general, the decimal floating-point numbers you enter are only approximated by the binary floating-point numbers actually stored in the machine.

The problem is easier to understand at first in base You can approximate that as a base 10 fraction:. Stop at any finite number of bits, and you get an approximation. On most machines today, floats are approximated using a binary fraction with the numerator using the first 53 bits starting with the most significant bit and with the denominator as a power of two.

Many users are not aware of the approximation because of the way values are displayed. Python only prints a decimal approximation to the true decimal value of the binary approximation stored by the machine. On most machines, if Python were to print the true decimal value of the binary approximation stored for 0. That is more digits than most people find useful, so Python keeps the number of digits manageable by displaying a rounded value instead. Interestingly, there are many different decimal numbers that share the same nearest approximate binary fraction.

For example, the numbers 0. Historically, the Python prompt and built-in repr function would choose the one with 17 significant digits, 0. Starting with Python 3. Note that this is in the very nature of binary floating-point: For more pleasant output, you may wish to use string formatting to produce a limited number of significant digits:.

One illusion may beget another. For example, since 0. Also, since the 0. Though the numbers cannot be made closer to their intended exact values, the round function can be useful for post-rounding so that results with inexact values become comparable to one another:.

Binary floating-point arithmetic holds many surprises like this. See The Perils of Floating Point for a more complete account of other common surprises.

For use cases which require exact decimal representation, try using the decimal module which implements decimal arithmetic suitable for accounting applications and high-precision applications. If you are a heavy user of floating point operations you should take a look at the Numerical Python package and many other packages for mathematical and statistical operations supplied by the SciPy project.

Python provides tools that may help on those rare occasions when you really do want to know the exact value of a float. Since the representation is exact, it is useful for reliably porting values across different versions of Python platform independence and exchanging data with other languages that support the same format such as Java and C Another helpful tool is the math.

That can make a difference in overall accuracy so that the errors do not accumulate to the point where they affect the final total:. Basic familiarity with binary floating-point representation is assumed. Representation error refers to the fact that some most, actually decimal fractions cannot be represented exactly as binary base 2 fractions.

That is, 56 is the only value for N that leaves J with exactly 53 bits. The best possible value for J is then that quotient rounded:. Instead of displaying the full decimal value, many languages including older versions of Python , round the result to 17 significant digits:. The fractions and decimal modules make these calculations easy:. Interactive Input Editing and History Substitution. For example, the decimal fraction 0.

Table Of Contents Issues and Limitations Representation Error Previous topic The Python Software Foundation is a non-profit corporation. Last updated on Nov 16, Created using Sphinx 1.