Why does 7*(0.1) - 0.7 = 1.1102e-16 whereas 9*(0.1)-0.9 =0 in MATLAB?

Why does

7*(0.1)-0.7 = 1.1102e-16,

3*(0.1)-0.3 = 5.55112e-17,

6*(0.1)-0.6 = 1.1102e-16

whereas

9*(0.1)-0.9 = 0,

2*(0.1)-0.2 = 0,

4*(0.1)-0.4 = 0 in MATLAB?

Answer 1

[Stephen23]

Those values that you are computing with cannot be exactly represented using binary floating point numbers. What you see printed in the command window is the closest representation to 5 or 16 significant digits, depending on your current format setting. To see the "real" value download James Tursa's FEX submission:

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

Use James Tursa's num2strexact and you will see that none of those values really have the exact value 0.1, or whatever value you use. All you are looking at is a representation of those floating point numbers displayed in the command window, to the precision defined by your format setting. Just because you see 0.1 displayed tells you nothing about the "real" floating point number's value.

Note that you can change how the value is displayed by changing the format.

You need to learn about the limits of floating point numbers. Start by reading these:

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

Answer 2

[madhan ravi]

https://www.mathworks.com/matlabcentral/answers/428577-very-odd-error-with-simple-multiplication-ok-not-so-odd#answer_345716

Answer 3

[the cyclist]

See the section "Accuracy of Floating-Point Data" in this documentation page.

Some of those combinations are represented by the same floating point value, and some are not.