Is 2.7177117771117777 irrational?

[JsStewartFan]

Is 2.71771177711177771111... irrational?

Homework Statement

I'm student teaching 8th graders Numbers and Operations. This is from an 8th grade activity I inherited with no "answer key." Is this decimal (with a pattern but not a repeating pattern) irrational? I am guessing it is, but I want to confirm it. Also, does anyone know if it is a "famous" number?

Homework Equations

The Attempt at a Solution

I know that all repeating decimals, such as 2.181818... or 3.156715671567... are rational because they can be represented by fractions a/b, where a and b are integers. And I have studied up on how to convert repeating decimals into fractions. I don't believe this "pattern" in the decimal number in question is "repeating," just a pattern. So I'm thinking it's irrational. But I'm curious about the number itself, also.

 

[gb7nash]

I don't believe this "pattern" in the decimal number in question is "repeating," just a pattern. So I'm thinking it's irrational. But I'm curious about the number itself, also.

Your intuition is correct. The decimal must be repeating the same pattern in order for it to be a rational number. Ergo, this is an irrational number.

 

[JsStewartFan]

Thanks! And you didn't recognize it, did you, as a well-known number?

 

[gb7nash]

I don't know. I've never seen it.

 

[ehild]

It is somewhat close to e=2.7182818284590...

ehild

 

[JsStewartFan]

Thanks, ehild and gb7nash! I wish I could think of how such a pattern could be generated. WTMTOMH, I guess.