Does 0.999999..... equal 1?

Factual Questions
1

I got into a long discussion with a close friend of mine recently, it was very interesting. Here goes:

1/3 = 0.3333…, right?

3 X (1/3)= 1, right?

3 X 0.33333… = 0.9999…, right?

so,

0.99999… = 1 RIGHT?

He argued that 0.99999… gets VERY close to but never reaches 1. I argue that the key is that the repeated decimal must be interpreted properly as an infinite series of 9s- his scenario has us stopping SOMEWHERE along this line to stop and see “where he is” ie, just how incredibly close (but not yet at and never will be) he is to 1. But my point is that it never stops, regardless of the fact that that makes it pretty much impossible to measure from a certain standpoint. That 0.99999… should equal 1 is just a quirk or shortcoming, oddity of our imperfect representation of mathematical theory, increasing its mystery and enhancing its elegance. So say I, he starts from the and round we go again, over and over. Im dizzy, please help us settle this for real.

Thank you in advance
TJ

2

Yes, they are equal. My favorite way to show it to people is to say, “Is there any number less than one that .99999… won’t be larger than?” No. “Is it ever going to be larger than one?” No. Squeezed right onto one’s lap. But the truth of it, without calculus, is the method you show: .333… is just a representation of one-third.

1/3 + 1/3 + 1/3 = 3/3. 3/3 isn’t very close to one, it is one. No matter how you write it. :slight_smile:

3

Short answer: Yes it is.

For a long answer have a look at this Thread.

4

And for a medium-length answer, check out this little number (heh) by the boss.

5

The factual answer to the question is “Yes, 0.999… does equal 1.”

Since this is covered by one of Cecil’s columns, I’ll close this. If you want to discuss why, 0.999… equals 1 (or why it does not), open a thread in Comments on Cecil’s Columns.

DrMatrix - GQ Moderator