This question arose out of something strange that I noticed after investigating this question further...

I always understood MATLAB variables to be double-precision by default. So, if I were to do something like declare a variable with 20 digits after the decimal point:

>> num = 2.71828182845904553488;
>> class(num)  % Display the variable type
ans =
double

I would expect the last 4 digits to be ignored, since the floating-point relative accuracy is on the order of 10^-16:

>> eps(num)
ans =
    4.440892098500626e-016

If I try to display the number with more than 16 digits after the decimal point (using either fprintf or sprintf), I get what I expect to see:

>> fprintf('%0.20f
', num)
2.71828182845904550000
>> sprintf('%0.20f', num)
ans =
2.71828182845904550000

In other words, digits 17 through 20 are all 0.

But things get weird when I pass num to the variable precision arithmetic function in the Symbolic Toolbox, telling it to represent the number using 21 digits of precision:

>> vpa(num, 21)
ans =
2.71828182845904553488

WHAT?! Those last 4 digits have reappeared! Shouldn't they have been lost when the original number I entered was stored as a double-precision variable num? Since num is a double-precision variable when it is passed to vpa, how did vpa know what they were?

My best guess as to what is happening is that MATLAB internally represents num with more precision than a double since I initialized it to a number with more digits past the decimal point than a double-precision variable could handle. Is this really what is happening, or is something else going on?

BONUS: And here's an additional source of confusion if you don't already have a migraine from the above...

>> num = 2.71828182845904553488;  % Declare with 20 digits past the decimal
>> num = 2.718281828459045531;    % Re-declare with 18 digits past the decimal
>> vpa(num, 21)
ans =
2.71828182845904553488  % It's the original 20-digit number!!!