Can a repeating decimal be equal to an integer?

2 Answers
Jun 11, 2015

Answer:

No, it will allways turn out to be a fraction.

Explanation:

I will not delve into how you turn a repeating decimal into a fraction, but just one example:

#0.333....=1/3#

There is one exeption though (see example above):

#0.999....=3*0.333....=3*1/3=1#

Dec 18, 2016

Answer:

Yes

Explanation:

The general term of a geometric series can be written:

#a_n = a*r^(n-1)#

where #a# is the initial term and #r# the common ratio.

When #abs(r) < 1# then its sum to infinity converges and is given by the formula:

#sum_(n=1)^oo ar^(n-1) = a/(1-r)#

So for example:

#0.999... = 9/10+9/100+9/1000+...#

is given by #a=9# and #r=1/10#

which has sum:

#sum_(n=1)^oo 9/10*(1/10)^(n-1) = (9/10)/(1-1/10) = (9/10)/(9/10) = 1#

So #0.bar(9) = 0.999... = 1#

In fact, any integer can be expressed as a repeating decimal using #9#'s.

For example:

#12345 = 12344.999... = 12344.bar(9)#

#-5 = -4.999... = -4.bar(9)#