Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one.
12.
If a fully reduced rational number's denominator has any prime factors other than 2 or 5, it cannot be expressed as a finite decimal fraction, and has a unique eventually repeating infinite decimal expansion.
13.
I think the problem you're having is with an infinite decimal expansion rather than irrationality; consider a line of length 1 / 9 = 0.111 . . . cm which you measure with a ruler.
14.
As part of Ed Dubinsky's APOS theory of mathematical learning, he and his collaborators ( 2005 ) propose that students who conceive of 0.999 & as a finite, indeterminate string with an infinitely small distance from 1 have " not yet constructed a complete process conception of the infinite decimal ".
15.
The current standard axiomatic definition is that real numbers form the unique field up to an isomorphism, whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, or infinite decimal representations, together with precise interpretations for the arithmetic operations and the order relation.
16.
A decimal may be a terminating decimal, which has a finite fractional part ( e . g . 15.600 ); a repeating decimal, which has an infinite ( non-terminating ) fractional part made up of a repeating sequence of digits ( e . g . 5.123 144 ); or an infinite decimal, which has a fractional part that neither terminates nor has an infinitely repeating pattern ( e . g . 3.14159265 . . . ).
