A feature of rational numbers is that their decimal representations always have infinitely repeating sequences of digits. This is true not only for base 10, but also for every possible base of a place system.
Examples:
1 = 1.000000... = 0.999999...
1/3 = 0.333333...
1/6 = 0.166666...
1/7 = 0.142857142857142857...
1/9 = 0.111111...
In base 2:
1/3 (11) = 0.01010101...
1/5 (101) = 0.001100110011...
In base 3:
1/2 = 0.11111111...
That is why decimal representations are usually not exactly translated into floating-point binary representations. One only has a finite number of digits available, and one has to cut off an infinite sequence of them.
It is easy to prove that an infinite repeating sequence of digits gives a rational number.
It is more difficult to prove that every rational number can be represented with an infinite repeating sequence of digits, but it can be done.
Haven't thought of repeating decimals in any base.
1 = 1.000000... = 0.999999... This one I do not see or agree with.
I went through this argument on another thread. The utility of using a geometric series to a fractional number is to yield the nearest approximate fraction. For example converging a decimal to the nearest fractional drill size.
if you write it out the series produces the fraction 1/1 as an approximation to 0.999..
Fractional
approximations to repeating decimals. 0.999... does not equal 1.
0.777... 7/9
0.888... 8/9
0.999... 1/1
Rounding is context and accuracy driven. In real computations involving variables with varying number of significant digits whether to truncate 0.999... or round up to 1 depends on affect of calculation accuracIn 0.999... at what nth term does it roll over to 1?
In 0.777... at what nth term does it roll up to say 0.8?