Rational Versus Irrational Decimal Numbers

Number Theory Level pending

The decimal number 0.1010010000001... has the rule that the number of zeroes are consecutive factorial numbers, beginning with one factorial. This number is known to be irrational.

The following decimal number has this different up-and-down rule:

0.2345432345432345432...

Is it irrational?

Yes. No.

Linda Slovik by Linda Slovik , 31, USA

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2 solutions

Jordan Cahn
Nov 7, 2018

This decimal repeats! Any repeating decimal must be rational. More specifically, 0.2345432345432345432 … = 0. 234543 ‾ = 234543 999999 = 2113 9009 0.2345432345432345432\ldots = 0.\overline{234543} = \frac{234543}{999999} = \frac{2113}{9009}

6 Helpful 0 Interesting 0 Brilliant 0 Confused
Linda Slovik
Nov 7, 2018

If it is irrational, it will not have a repeating cycle of digits. One such cycle of digits is 234543. There are also different cycles. Therefore, it is not irrational. So, the answer is "no."

3 Helpful 0 Interesting 0 Brilliant 0 Confused

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