Is this a rational number?

Printable View

October 6th 2011, 05:13 AM
Jman115

Is this a rational number?

2.202002000200002...

There is a pattern there so part of me wants to say yes. But there is technically no repeat or termination so a bigger part of me wants to say no.

I am making answer sheets for my students and initially I was going to say no, this is not a rational number due to the fact there is no repeat or termination of the decimal. But looking at the worksheet, it gives no options for the students to label a number as irrational. Just W, I or Q.

Typically the directions for these sheets are very specific, so that got me thinking that this must be a rational number and that the obvious pattern must allow it to be translated into a fraction somehow, but I am not sure how I would do that. I know how to turn repeating and terminating decimals into fractions but not this if it is in fact possible.

I know it is probably a basic question but I'm at a loss here.
October 6th 2011, 05:53 AM
dwsmith

Re: Is this a rational number?

I believe a theorem states that every rational number is either a repeating or terminal decimal. Therefore, if it doesn't repeat, it must be irrational.
October 6th 2011, 06:06 AM
Jman115

Re: Is this a rational number?

That is what I assumed, but the worksheet did not give an option for irrational numbers in the directions, which they are typically VERY specific. So just making sure I am not losing it. Thank you for the reply.
October 6th 2011, 06:29 AM
emakarov

Re: Is this a rational number?

And what are W and I, just for information? I assume that Q means rational numbers.
October 6th 2011, 07:08 AM
Jman115

Re: Is this a rational number?

Whole and Integers
October 6th 2011, 07:16 AM
emakarov

Re: Is this a rational number?

Well, this will probably be meddling in your affairs, but just in case. If W, I and Q are used as labels and are written in thin (regular) font, I guess it's OK. If, however, they denote sets of numbers and are written in thick font like this: $\mathbb{Q}$ , then the universal notation for integers is $\mathbb{Z}$ . Also, Wikipedia claims that "whole number" is a term with inconsistent definitions. It is better to say "natural numbers" and denote their set by $\mathbb{N}$ .
October 6th 2011, 07:36 AM
Jman115

Re: Is this a rational number?

In the instructions it says
W = whole
I = integers
and Q = Rational

In the book they also define Whole numbers as {0, 1, 2, 3...}
Natural as {1, 2, 3, 4...}

and Rational as any number that can be written in A over B form.

So I am going based off of the information this company gave me.
October 6th 2011, 08:40 AM
Soroban

Re: Is this a rational number?

Hello, Jman115!

Quote:

$\text{Is this a rational number? }\:2.202002000200002\hdots$

There is a pattern there so part of me wants to say yes.
But there is technically no repeat or termination
. . so a bigger part of me wants to say no.

You are right . . . there is a pattern.
. . But it does not have a repeating cycle.
As Jman115 suggested, it must be irrational.

We have: . $X \;=\;\frac{2}{10^0} + \frac{2}{10^1} + \frac{2}{10^3} + \frac{2}{10^6} + \frac{2}{10^{10}} + \frac{2}{10^{15}} + \hdots$

The exponents are Triangular Numbers.

. . That is: . $X \;=\;2\sum^{\infty}_{n=1}\frac{1}{10^{\frac{n(n-1)}{2}}}$

The series is neither arithmetic nor geometric, nor a combination thereof.
I have found no way to evaluate it.

It can be written as a recurrence: . $a_n \;=\;a_{n-1}\!\cdot\!\frac{1}{10^{n-1}},\;\;a_1\,=\,2$
. . but this doesn't help either . . .