Is 2.7177117771117777 irrational?

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Homework Help Overview

The discussion revolves around the classification of the number 2.7177117771117777... and whether it is irrational. The context is an 8th-grade activity related to Numbers and Operations, focusing on understanding rational versus irrational numbers.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to determine if the number is irrational based on its decimal representation, noting that it has a pattern but does not repeat. Some participants discuss the criteria for a number to be classified as rational versus irrational.

Discussion Status

Participants are exploring the nature of the number in question, with some suggesting that the lack of a repeating pattern indicates it is irrational. There is curiosity about whether the number is well-known, but no consensus has been reached regarding its classification.

Contextual Notes

The original poster mentions the absence of an answer key for the activity, which may contribute to the uncertainty in the discussion. There is also a reference to the number being close to the mathematical constant e, which may influence participants' reasoning.

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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.
 
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JsStewartFan said:
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.
 


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


I don't know. I've never seen it.
 


It is somewhat close to e=2.7182818284590...

ehild
 


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

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