Can we write bar35 or (353535.....) In form of p/q where q is not equal to 0 ? Thanks

1 Answer
Apr 22, 2018

# 0.overline{35} = 35/99 #

Explanation:

In Socratic we can write 0.overline{35} between hashtag signs and it renders as

# 0.overline{35} #

That's pretty nice. I found it by trying some LaTeX, whose features have been partially adapted.

Turning to the problem, whenever we have a purely repeating decimal like this, where the repeating part starts right after the decimal point, we quickly get the fraction by just adding a denominator of nines whose length is the length of the repeating part.

# 0.overline{35} = 35/99 #

That's nice for quickly getting the answer, but why does it work? It works because a purely repeating decimal is really a geometric series in disguise, and the same trick for adding up a geometric series works for a repeating decimal:

# x = 0.overline{35} #

# 100 x = 35.overline{35} #

# 100 x - x = 35 #

#x = 35/99 #