Why does dividing and multiplying by the same number cause a numerical error?

What is is that causes a numerical error when dividing and multiplying by the same number? For instance,

format long
13/1e5*1e5

gives

12.999999999999998

I'm shouldn't be running out of double precision floating point room, should I?

Incidentally,

   13/sqrt(2)*sqrt(2)

works fine.

For 1e5, let's display a list of all the numbers that don't work between 1 and 100:

myNumbers = [];
sr = 1e5
for i = 1:100
    if (i/sr*sr-i) ~= 0 
        myNumbers(end+1) = i;
    end
end
myNumbers

this returns:

     7    13    14    15    26    28    30    51    52    56    60

I'll live with it, but I'm just curious if it's something I should have expected...

cheers,

Will

Answer 1

[Azzi Abdelmalek]

Read this http://matlab.wikia.com/wiki/FAQ#Why_is_0.3_-_0.2_-_0.1_.28or_similar.29_not_equal_to_zero.3F

Answer 2

[Stephen23]

Standard rules for operator precedence, also used by MATLAB, state that multiplication and division are on the same precedence level, and that within each precedence level operators have equal precedence and are evaluated from left to right_So

13 / 1e5 * 1e5

will be evaluated as

(13 / 1e5) * 1e5

Lets have look at the result of the first part (the intermediate result):

>> fprintf('%.30f\n', 13/1e5)
0.000129999999999999990000000000

Which should answer your question: the value of 13/1e5 is not exactly representable by a double data type. Welcome to the world of numeric mathematics!

http://matlab.wikia.com/wiki/FAQ#Why_is_0.3_-_0.2_-_0.1_.28or_similar.29_not_equal_to_zero.3F

[James Tursa]

FYI, fprintf and sprintf (at least on a PC) do not show the exact decimal conversion of a floating point number, they only show a rounded version of it. To see the exact decimal conversion one has to use other methods, e.g. using the symbolic toolbox or using the num2strexact utility:

>> fprintf('%.60f\n',13/1e5)
0.000129999999999999990000000000000000000000000000000000000000 <-- rounded
>> num2strexact(13/1e5)
ans =
1.2999999999999998861154038021226142518571577966213226318359375e-4
>> num2strexact(13/1e5*1e5)
ans =
1.29999999999999982236431605997495353221893310546875e1

You can find num2strexact here:

http://www.mathworks.com/matlabcentral/fileexchange/22239-num2strexact--exact-version-of-num2str-