Prove that a number is rational if and only if its decimal representation is eventually repeating.?

Question

I have shown that: If a number is rational then its decimal representation is eventually repeating.

I am having difficulties with proving: If a decimal representation is repeating then it is a rational number.

Answer 1

Just show that a decimal representation that is repeating can be written in a fraction and fractions are always rational.

Answer 2

Let x = a number with repeating decimals

Let n = number of digits in repeating part.

Let m = number of digits after decimal point but before repeating part



If we multiply x by 10^m, then we move decimal point m positions to right so that repeating part is right after decimal point.



If we then multiply x*10^m by 10^n, then we move decimal point another n positions to the right so that repeating part is still right after decimal point.



x * 10^m = number which has repeating decimals right after decimal point

x * 10^(m+n) = another number with repeating decimals right after decimal point



Now if we subtract x * 10^m from x * 10^(m+n), repeating decimals cancel out, leaving just the integer part.



x * 10^(m+n) - x * 10^m = k (some integer)

x (10^(m+n) - 10^m) = k

x = k / (10^(m+n) - 10^m)



Now since both k and (10^(m+n) - 10^m) are integers, then x is rational



-- Ματπmφm --

Answer 3

Let x be a number with a repeating decimal. Express x as:



x = z.(a1)(a2)...(an)(b1)(b2)...(bm)(b1)(b2)...(bm)...



where each (ai) is a non repeating decimal place, and each (bi) is in the repeating part of the decimal, and z>0 is an integer. To clarify, z could have many digits.



Now, multiply the equation by 10^n:



10^n * x = z(a1)(a2)...(an).(b1)(b2)...(bm)..., so the entire decimal portion repeats.



Now multiply by 10^m:



10^(n+m) * x = z(a1)(a2)...(an)(b1)(b2)...(bm).(b1)(b2) ... (bm)...



Subtract 10^n * x = z(a1)(a2)...(an).(b1)(b2)...(bm) from our last equation:



10^n * (10^m - 1) * x = z(a1)(a2)...(an)(b1)(b2)...(bm) - z(a1)(a2)...(an)



Notice that the right side no longer has any decimals. Now divide by 10^n * (10^m - 1):



x = ( z(a1)(a2)...(an)(b1)(b2)...(bm) - z(a1)(a2)...(an) ) / (10^n * (10^m - 1).



Therefore, x is a fraction with integer numerator and denominator, thus rational.