Show that 0.0223 repeating is a rational number. Let x = 0.0223

Answer 1

Rational numbers are numbers that can be written as fractions. If x=0.0223022302230223..., then we can multiply x so that it has an integer before the decimal and only the repeating decimal part after:

10,000x = 223.0223022302230223... Line this up with the original x and subtract:

- (x = .022302230223...)

9,999x = 223

divide both sides by 9999:

9,999x = 223

9,999 9,999

223 is prime, so you can't reduce the fraction, but x=223/9999.

I hope this helps! I think it's so cool that we can just subtract away an infinite decimal!

Have fun mathing,

Liz Z.