Strange result from comparison

Sorry for the naive question. Surely there is a trivial answer. Why does this happen? Thanks.

>> 12*10^-3==0.012

ans =

logical

1

>> 12*10^-4==0.0012

ans =

logical

0

[Sachi]

>> 12*10^-5==0.00012

ans =

logical

0

>> 12*10^-6==0.000012

ans =

logical

1

Answer 1

[Stephen23]

"Why does this happen?"

Because of the accumulated floating point error. Lets have a look at the values involved, to a higher precision:

fprintf('%.64f\n',12*10^-3,0.012) % these look the same...

0.0120000000000000002498001805406602215953171253204345703125000000
0.0120000000000000002498001805406602215953171253204345703125000000

fprintf('%.64f\n',12*10^-4,0.0012) % but are these the same?

0.0012000000000000001117161918529063768801279366016387939453125000
0.0011999999999999998948757573558054900786373764276504516601562500

Always remember that:

• binary floating point arithmetic has some similarities to the algebra and arithmetic that you learned at school, but it is definitely not the same thing.
• the values you see displayed by default in the command window are to a relatively low precision, in general they do not show numbers with their full precision (see NUM2STREXACT).

Learn more about binary floating point numbers:

https://www.mathworks.com/matlabcentral/answers/57444-faq-why-is-0-3-0-2-0-1-not-equal-to-zero

https://www.mathworks.com/matlabcentral/answers/316889-why-two-equal-numbers-are-not-equal

http://www.mathworks.com/matlabcentral/answers/102419-how-do-i-determine-if-the-error-in-my-answer-is-the-result-of-round-off-error-or-a-bug

http://www.mathworks.com/matlabcentral/answers/140656-about-numerical-difficulty-of-equality-comparison-operation-between-two-double-real-numbers-in-matlab

This is worth reading as well:

https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html

http://faculty.tarleton.edu/agapie/documents/cs_343_arch/papers/1991_Goldberg_FloatingPoint.pdf