Is #1/42# a repeating or terminating decimal?

1 Answer
May 3, 2018

Answer:

It repeats:

#1/42 = 0.0bar(238095)#

Explanation:

Note that:

#42 = 2 * 3 * 7#

has factors which are not factors of #10#.

As a result, the decimal expansion of #1/42# does not terminate. It is a repeating decimal that we can find using long division.

Once the remainder in the long division repeats, so will the quotient.

#color(white)(0000")")underline(color(white)(0)0"."0color(white)(0)2color(white)(0)3color(white)(0)8color(white)(0)0color(white)(0)9color(white)(0)5#
#4color(white)(0)2color(white)(0))color(white)(0)1"."0color(white)(0)0color(white)(0)0color(white)(0)0color(white)(0)0color(white)(0)0color(white)(0)0#
#color(white)(0000")")color(white)(0)underline(color(white)(0".")8color(white)(0)4)#
#color(white)(0000")")color(white)(00".")1color(white)(0)6color(white)(0)0#
#color(white)(0000")")color(white)(00".")underline(1color(white)(0)2color(white)(0)6_#
#color(white)(0000")")color(white)(00"."00)3color(white)(0)4color(white)(0)0#
#color(white)(0000")")color(white)(00"."00)underline(3color(white)(0)3color(white)(0)6)#
#color(white)(0000")")color(white)(00"."000000)4color(white)(0)0color(white)(0)0#
#color(white)(0000")")color(white)(00"."000000)underline(3color(white)(0)7color(white)(0)8)#
#color(white)(0000")")color(white)(00"."00000000)2color(white)(0)2color(white)(0)0#
#color(white)(0000")")color(white)(00"."00000000)underline(2color(white)(0)1color(white)(0)0)#
#color(white)(0000")")color(white)(00"."0000000000)1color(white)(0)0#

So:

#1/42 = 0.0bar(238095)#