### Math blogging: 0.999...

This recently featured article on Wikipedia brought back fond memories of impassioned, beer-fueled math arguments in college (yes, I'm a geek.) It's about the number 0.9999999... a quantity with an infinite string of 9's following the decimal (this quantity is usually denoted in math books with a bar over the 9, but I'm too lazy to figure out how to express that on a web page, I'll just stick with the trailing dots.)

We've all seen the number countless times on calculators, where they're typically the result of a rounding error. What you may not know, however, is that the number isn't merely a close approximation of the number one -- it's *identical* to one. The numbers 0.999... and 1.0 are exactly equal. Since there is no number small enough to represent the difference between the two, that difference must be zero.

Think about it another way. We all know the fraction 1/3 can be expressed in decimal notation as 0.333.... But multiply this quantity by 3, and what do you get? You get 0.999..., but we know that 3 * 1/3 is 1. See?

Here's another, somewhat more rigorous proof.

- Let
*x = 0.999...*

- Multiply both sides by 10:
*10x = 9.999...*

- Subtract x from both sides:
*9x = 9*

- Therefore
*x = 1*

Pretty cool, huh? This whole little diversion has reminded me that I really should wrap up by Banach-Tarski series sometime soon, for completeness sake if nothing else.

## Comments

OHMYGOD....I need to smoke a doober to figure this stuff out.

But 0.3 as a fraction isn't 1/3...I always look at it as 3/10. 0.3 = 30%; 1/3 = 33%...

Christ, I think the wallpaper in my office is now speaking to me....

Posted by: fred | October 25, 2006 09:35 AM

Too bad computer languages don't see them as equal. I've debugged this problem and coded around it many a time.

Interesting proof. I was on the Math Team in college. I like it.

Posted by: Bailey Hankins | October 25, 2006 09:46 AM

Hey Barry,

I take it that you are so disappointed about the state of the republican party and its ongoing implosion in these elections that you took a break from posting on political issues. I dont blame you. I think you should consider voting democratic this time. Too many things are in a disarray and the country desperately needs change. I really believe that rational republicans will end up voting democratic in 2 weeks, despite the tax issue. It is in the interest of the country.

Posted by: Blue Wind | October 25, 2006 10:27 AM

Blue:

1.) Of *course* I'm disappointed by the Republicans in office.

2.) I've *always* blogged about stuff other than politics.

3.) No, I will not be voting Democratic this year. Thank you for your suggestion, though. :)

Posted by: BNJ | October 25, 2006 11:16 AM

BNJ, I think you have a typo. Shouldn't step 3 read 9x=9?

Posted by: CRB | October 25, 2006 11:48 AM

Yes! Glad someone was paying attention. ;-)

Posted by: BNJ | October 25, 2006 11:54 AM

I've always blogged about stuff other than politicsI know that. It is just that recently the number of political posts is less.

No, I will not be voting Democratic this year.Will you be voting republican (after all that) or stay home (or got to work)? :)

Posted by: Blue Wind | October 25, 2006 11:56 AM

>

Will you be voting republican (after all that) or stay home (or got to work)? :)Sigh. Yet another American mind stuck in the two-party mindset.

I will do what I usually do in a mid-term election, and vote a straight Libertarian ticket.

Posted by: BNJ | October 25, 2006 12:00 PM

...beer-fueled math arguments in college...Does the beer make the math easier or harder?

Posted by: Roger | October 25, 2006 01:36 PM

It makes the arguments more fun.

Posted by: BNJ | October 25, 2006 02:16 PM

Of course, if you subtract 2x from each side, you get a different answer:

Let x = 0.999...

Multiply both sides by 10: 10x = 9.999...

Subtract 2x from both sides: 8x = 8.00000000000...01

Therefore x doesn't = 1, but it

almostdoes. ;-)Posted by: CRB | October 25, 2006 04:46 PM

(append to above post)

That's why engineers (among other reasons) use a factor of safety.

Posted by: CRB | October 25, 2006 04:48 PM

what's wrong with .001 ? i don't understand how you got from step 2 to 3 in that 9.999 became 9 by subtracting x ?

The little line above the numeral is very important part of this i think. when we see it on a calculator we know that it really means exactly .99 9/9 which IS 1, similar to 33 1/3 or 66 2/3. is that what you mean?

Otherwise .9999999999999999 is NOT 1.

But .9999... IS 1 I guess, since we can't find the symbol here on the internets it will have to do.

p.s. i may hav smoked a doober

Posted by: ortho | October 25, 2006 08:41 PM

p.s. Barry you are so obviously trying to distract us from the upcoming Dem takeover with deese maths

Posted by: ortho | October 25, 2006 08:43 PM

I gave up on math after we got to the "guzintas" (You know, 2 guzinta 6 etc.) But math geeks assure me that this joke is hilarious. A student presented the teacher with a problem that had puzzled him. "The answer is one," the teacher said without hesitation.

"How do you know?"

"Well, it had to be one, zero, or infinity, and obviously it wasn't zero or infinity."

Posted by: Paul Moore | October 26, 2006 04:21 AM

i don't understand how you got from step 2 to 3 in that 9.999 became 9 by subtracting x ?x = 0.999...

so

9.999... - 0.999... = 9

The proof is a mathmatical anomaly.

Posted by: CRB | October 26, 2006 10:29 AM

I hate to keep spoiling these conversations. I will try to be less professorish. In fact I had a student who was a seventh grade algebra teacher bring up this example and she was disappointed with the proper explanation. She used it on her advanced students to try and fuel their imagination. She was disappointed because when I explained it she realized she could not share the information with a seventh grader. It all hinges on two things 1: what is decimal notation and 2: what does the ... mean. Decimal notation is a compact way of representing sums. For instance 0.99 is a slick way of writing 0+9/10+9/100 or 234 is a slick way or writing 200+30+4. We use a base 10 notation in both directions. So 0.99999... is just a slick way of writing 0+9/10+9/100+9/1000... So what does the ... mean. It is a shorthand way of expressing the phrase "and so on" or better "the pattern continues forever".

In other words we are taking a limit in the sense of calculus. In fact 0.9999... is a slick way or representing an infinite sum. It is easy to show using the formula for finite geometric sums and some routine epsilon-delta arguments that the infinite sum in question 0.999... converges to 1. In other words 0.999... is an extremely convoluted way of talking about the number 1. So the demonstration which began this conversation is not a proof but rather a slick tautology. Follow it through and remember that saying x=0.9999... is just a complicated way of saying that x=1. So for instance in step 2 10x=9.9999... is just a strange way of saying that 10*1=10. A fact which is hard to doubt! That ... means a lot more than we usually want to think about.

Posted by: nathan | December 2, 2006 04:00 AM

The failure in this "proof" is that it ignores rules of significant digits. Multiplying any decimal by 10 leaves a 0 at the final significant digit, not a 9, so the subtraction step is incorrect.

Posted by: Significant Digit | October 26, 2007 05:12 PM

what is another way to writing 30%

Posted by: bilal | January 6, 2009 04:22 PM