Does 0.999… Equal 1 ?

For those who want immediate gratification, YES, 0.999… = 1

The dot dot dot (…) means 9’s repeating infinitely, past 2012, past the end of time, FOREVER. That is a lot of nines. It can also be written with a bar over the 9, as 0.9.

So what is going on here with 0.999…

  • 0.9 != 1
  • 0.9999 != 1
  • 0.9999999 != 1
  • 0.9999999999 != 1
  • 0. 9999999999999 != 1
  • 0. 9999999999999999 != 1
  • 0. 9999999999999999999 != 1
  • 0. 9999999999999999999999 != 1
  • 0. 9999999999999999999999999 != 1
  • 0. 9999999999999999999999999999 != 1
  • 0. 9999999999999999999999999999999 != 1
  • 0. 9999999999999999999999999999999999 != 1
  • 0. 9999999999999999999999999999999999999 != 1
  • 0. 9999999999999999999999999999999999999999 != 1
  • Wheeeeeeeeeeeeeeeeeeeeeeeeeeeeee! ………………………………..

According to the hill of ASCII art, sliding down farther and farther never gets you there.  The pattern never reaches a value of 1. The sequence will always be just a hair under 1.

This is along the same lines as Zeno’s Dichotomy paradox:
To move onto an elevator, you must first get half way there, and then get a quarter of the way there, and then one eighth, and one sixteenth, and one 32nd, and 1/64th and 1/128th, and so on…. such that you never reach the elevator because there is always an additional half step to travel.  Zeno was good at driving people crazy with ideas like that.

Guess what? It is a riddle. Calculus is the answer.

Using the concept of infinity (positive infinity in this case) and the concept of a limit equation, calculus shows that indeed, 0.999… equals 1.

YES!

Think along conventional lines and get conventional answers.

Information in this article and the calculus equation graphic are attributed to http://en.wikipedia.org/wiki/0.999….

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