Why is there an error in the calculation of 366.0/5.0?

I input the following code in Matlab and the result obtained has a small error, as shown below:

>> format long

>> 367/5

ans =

73.400000000000006

Answer 1

[Matt J]

All computers make imperfect calculations. It's a fact of life with finite precision floating point math.

[liao z]

thanks a lot ! get it ！

[Matt J]

Then please accept-click an answer.

Answer 2

[Walter Roberson]

By default, MATLAB calculates using IEEE 754 Double Precision representation of numbers. IEEE 754 is a binary floating point representation. One thing about binary floating point representations is that 1/10 is not exactly representable in any finite length binary floating point.

This is not a bug; it is an inherent limitation of binary floating point representation.

The mathematical reasons why 1/10 is not exactly representable in binary is the same reason why 1/3 and 1/6, 1/7, 1/9, and 1/13 are not exactly representable in finite decimal. Every finite base representation has numbers that cannot be exactly finitely represented.

[liao z]

I understand that any binary can only perfectly represent 2 ^ (N), and other numerical values can only be represented with finite precision. So what are the significant bits of the double variable?

[James Tursa]

The three closest numbers to 73.4 representable in IEEE double precision are:

format longg
x = [73.4-eps(73.4);73.4;73.4+eps(73.4)];

The binary floating point hex patterns of these three numbers are (they differ by 1 in the trailing bit):

num2hex(x)

ans = 3x16 char array
    '4052599999999999'
    '405259999999999a'
    '405259999999999b'

The exact decimal conversions of these binary floating point numbers are:

fprintf('%60.50f\n',x)

       73.39999999999999147348717087879776954650878906250000
       73.40000000000000568434188608080148696899414062500000
       73.40000000000001989519660128280520439147949218750000

MATLAB (or more precisely, the CPU floating point processor) picked the closest one for the result. There are no numbers (including 73.4 exactly) inbetween these numbers that are exactly representable in IEEE double precision.