From a simple mathematical calculation, anybody can say that 0.001 1, but when we try to execute 0.001 1 in Python or Java, the output is not equal to 0. This is not an error with your code or issue with compiler. Then why the output !=0. OK, let’s take it slowly and go back a little bit, start with Floating point numbers. Show Floatingpoint numbers are represented in computer hardware as base 2 (binary) fractions. For example, the decimal fraction 0.125 has value 1/10 + 2/100 + 5/1000, and in the same way the binary fraction 0.001 has value 0/2 + 0/4 + 1/8. 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 floatingpoint numbers you enter are only approximated by the binary floatingpoint numbers actually stored in the machine. For example, the decimal value 0.1 cannot be represented exactly as a base 2 fraction. In base 2, 1/10 is the infinitely repeating fraction 0.0001100110011001100110011001100110011001100110011... Stop at any finite number of bits, and you get an approximation. On a typical machine running Python, there are 53 bits of precision available for a Python float, so the value stored internally when you enter the decimal number 0.001 3 is the binary fraction0.00011001100110011001100110011001100110011001100110011010 which is close to, but not exactly equal to, 1/10. It’s easy to forget that the stored value is an approximation to the original decimal fraction, because of the way that floats are displayed at the interpreter prompt. Python only prints a decimal approximation to the true decimal value of the binary approximation stored by the machine. If Python were to print the true decimal value of the binary approximation stored for 0.1, it would have to display >>> 0.1 That is more digits than most people find useful, so Python keeps the number of digits manageable by displaying a rounded value instead >>> 0.1 It’s important to realize that this is, in a real sense, an illusion: the value in the machine is not exactly 1/10, you’re simply rounding the display of the true machine value. This fact becomes apparent as soon as you try to do arithmetic with these values >>> 0.1 + 0.2
Other surprises follow from this one. For example, if you try to round the value 2.675 to two decimal places, you get this >>> round(2.675, 2) The documentation for the builtin 0.001 4 function says that it rounds to the nearest value, rounding ties away from zero. Since the decimal fraction 0.001 5is exactly halfway between 0.001 6 and 0.001 7, you might expect the result here to be (a binary approximation to) 0.001 7. It’s not, because when the decimal string 0.001 5 is converted to a binary floatingpoint number, it’s again replaced with a binary approximation, whose exact value is2.67499999999999982236431605997495353221893310546875 Since this approximation is slightly closer to 0.001 6 than to 0.001 7, it’s rounded down.If you’re in a situation where you care which way your decimal halfwaycases are rounded, you should consider using the 0.0001100110011001100110011001100110011001100110011... 2 module. Incidentally, the 0.0001100110011001100110011001100110011001100110011... 2 module also provides a nice way to “see” the exact value that’s stored in any particular Python float>>> from decimal import Decimal Another consequence is that since 0.1 is not exactly 1/10, summing ten values of 0.1 may not yield exactly 1.0, either: 0.001 0Binary floatingpoint arithmetic holds many surprises like this. As that says near the end, “there are no easy answers.” Still, don’t be unduly wary of floatingpoint! The errors in Python float operations are inherited from the floatingpoint hardware, but you do need to keep in mind that it’s not decimal arithmetic, and that every float operation can suffer a new rounding error. While pathological cases do exist, for most casual use of floatingpoint arithmetic you’ll see the result you expect in the end if you simply round the display of your final results to the number of decimal digits you expect. So friends, be little careful while playing with floating point numbers. Happy Coding :)
