double and precision

[Knackeback]

I use data type double at 32Bit platform and from
float.h I learn:

/* Number of decimal digits of precision in a double */
#undef DBL_DIG
#define DBL_DIG 15

My questions:
Are numbers like 3, 5.45, 1.23456789012345
i.e. numbers with siginificant count of digits <= 15
precise in a mathematical sense ?

If so I can add this numbers without rounding error as
long as I do not exceed the siginificant count of digits
limit 15.
Right?

To be utterly clear:
Must I expect rounding errors if I only add numbers n
with the property "siginificant count of digits <= 15"
and the resulting sum has also this limitation ?