Why does 1 - 2/3 - 1/3 not equal zero?

Hi Matt: Yes, this is an obvious one to get the ball rolling. By the way, I changed the formatting of your answer slightly to make the link a little clearer.

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Answer 2

Doug Hull 2011-1-21

6
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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/69-why-does-1-2-3-1-3-not-equal-zero#answer_271

在 MATLAB Online 中打开

As is mentioned frequently in the newsgroup, some floating point numbers can not be represented exactly in binary form. So that's why you see the very small but not zero result. See EPS.

The difference is that 0:0.1:0.4 increments by a number very close to but not exactly 0.1 for the reasons mentioned below. So after a few steps it will be off whereas [0 0.1 0.2 0.3 0.4] is forcing the the numbers to their proper value, as accurately as they can be represented anyway.

a=[0 0.1 0.2 0.3 0.4];
b=[0:.1:.4];
as=sprintf('%20.18f\n',a)
  as =
   0.000000000000000000 % ==
   0.100000000000000010 % ==
   0.200000000000000010 % ==
   0.299999999999999990 % ~= bs !
   0.400000000000000020 % ==
bs=sprintf('%20.18f\n',b)
  bs =
   0.000000000000000000 % ==
   0.100000000000000010 % ==
   0.200000000000000010 % ==
   0.300000000000000040 % ~= as !
   0.400000000000000020 % ==
% -and-
format hex;
hd=[a.',b.']
  hd =
   0000000000000000   0000000000000000 % ==
   3fb999999999999a   3fb999999999999a % ==
   3fc999999999999a   3fc999999999999a % ==
   3fd3333333333333   3fd3333333333334 % ~= !
   3fd999999999999a   3fd999999999999a % ==

If you're trying to compare two floating-point numbers, be very careful about using == to do so. An alternate comparison method is to check if the two numbers you're comparing are "close enough" (as expressed by a tolerance) to one another:

% instead of a == b
% use:
areEssentiallyEqual = abs(a-b) < tol
% for some small value of tol relative to a and b
% perhaps defined using eps(a) and/or eps(b)

You can see this same sort of behavior outside MATLAB. Using pencil and paper (or a chalkboard, or a whiteboard, etc.) compute x = 1/3 to as many decimal places as you want. The number of decimal places must be finite, however. Now compute y = 3*x. In exact arithmetic, y would be exactly 1; however, since x is not exactly one third but is a rounded approximation to one third, y will not be exactly 1.

For a readable introduction to floating point arithmetic, look at Cleve's Corner article from 1996: Floating Points (PDF)

For more rigorous and detailed information on floating point arithmetic, read the following paper: What Every Computer Scientist Should Know About Floating Point Arithmetic.

Another resource is Technical Note 1108 on the Support section of The MathWorks website.