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#### WootBot

quality posts: 17 Private Messages

Staff

April has been Mathematics Awareness Month since 1999, making this the nineteenth time it's been observed. Wait, that's not right. The twentieth time. To celebrate our mathematical awareness, or lack thereof, Ken Jennings of Jeopardy! will be with us all month debunking all the popular misinformation about numbers that you thought once you could count on.

The Debunker: How Much Less is .9999… than One?

Like me, you may remember the first time you were asked to perform long division on two numbers that produced a repeating quotient. Do you recall the dreary, slowly dawning realization that these numbers were going to keep repeating in that pattern indefinitely? Even a respectable, friendly, familiar fraction like one-third turned out to be unending in its decimal expansion: 0.333333333333… We were taught to just put a horizontal line over the digit(s) that repeated and call it a day, but it sure wasn’t very satisfying.

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But the trouble really starts with .9 repeating: a decimal point followed by an endless string of nines. All levels of students, even including most undergraduate math majors, generally understand that this number is extremely close to 1—"infinitely close," some will allow—but still slightly less. After all, how could 0.999… be the same as 1.0? One begins with a zero and the other with a one. That difference in how they're expressed must reflect some tiny gap between them.

In fact, it's pretty easy to demonstrate intuitively using arithmetic that .9 repeating is exactly equal to 1, not slightly less. If .333… is one-third, and .666… is two-thirds, then obviously .999… is three-thirds, which is to say one, not a number slightly smaller than one. A more formal proof relies on the Archimedean property, which holds that the real numbers have no infinitely large or infinitely small elements, and is a little more involved but still fairly elementary. So why does our intuition insist that there must be a difference between .9 repeating and 1? Way down deep, apparently many of us still expect that .9999999 must stop somewhere; we have a hard time grappling with an endless series of nines. Or we incorrectly think of the series of nines as a process that is advanced as we do our long division, or zoom in on a number line. But that's not how limits work. 0.999999999… isn't infinitely nearing one as we reveal more digits of accuracy. It's already there.

Quick Quiz: Which actor has been Emmy-nominated three times for his role on Brooklyn Nine-Nine?

Ken Jennings is the author of eleven books, most recently his Junior Genius Guides, Because I Said So!, and Maphead. He's also the proud owner of an underwhelming Bag o' Crap. Follow him at ken-jennings.com or on Twitter as @KenJennings.

#### aardwolf64

quality posts: 57 Private Messages

My gut tells me that has to be wrong. An infinitely repeating 0.999999... will always be less than 1.

The difference between the two as the digits increase approaches 1/inifinity, but it will never reach zero (as 1/infinity is not meaningful). It will be an infinitely small difference.

Edit: And apparently it's generally accepted that it is right and I'm wrong... Hrmph...

http://www.purplemath.com/modules/howcan1.htm
https://en.wikipedia.org/wiki/0.999...

x30

#### moles1138

quality posts: 49 Private Messages

Gotta be Andre Braugher

#### moles1138

quality posts: 49 Private Messages

When at the store, Oogie will see something for \$13.99 and say it is \$13. I say it is \$14. Is that the same thing?

But at the Gas Pump it is \$2.34.9 and I always think \$2.34 not rounding up to \$2.35.

#### daveinwarshington

quality posts: 97 Private Messages

No matter how much .999999... wants to be '1', it will never achieve it.
It is less than 1 and will never be 1.
Mathematicians have concluded that it's equal to 1 just to make their lives a little more tidy.
Bullshit.
Scientists also thought the world is flat.

#### cklun

quality posts: 43 Private Messages

Volunteer Moderator

daveinwarshington wrote:No matter how much .999999... wants to be '1', it will achieve it. It is less than 1 and will never be 1.
Mathematicians have concluded that it's equal to 1 just to make their lives a little more tidy.
Bullshit.
Scientists also thought the world is flat.

although we all did it, doesn't it make you cringe when you see vinyl being roughed up?

"I NEVER EVER share garlic balls!"]

#### bestsportnascar

quality posts: 49 Private Messages
aardwolf64 wrote:My gut tells me that has to be wrong. An infinitely repeating 0.999999... will always be less than 1.

The difference between the two as the digits increase approaches 1/inifinity, but it will never reach zero (as 1/infinity is not meaningful). It will be an infinitely small difference.

Edit: And apparently it's generally accepted that it is right and I'm wrong... Hrmph...

http://www.purplemath.com/modules/howcan1.htm
https://en.wikipedia.org/wiki/0.999...

Nah you are right. If you cut a cake into thirds, some is left on the knife, but it is still considered a whole cake. Just think of that .00000000000000000.... as being the frosting you lick off the knife.

¯\_(ツ)_/¯

#### MrsKaren

quality posts: 0 Private Messages

Make a list of the decimal forms of the fraction ninths. Use your calculator if you don't want to divide.
1/9 = 0.111...
2/9 = 0.222...
3/9 = 0.333... Continue
and
8/9 = 0.888...
9/9 = 0.999... and 9/9 = 1

#### deanerino1

quality posts: 3 Private Messages

these community discussions no longer link to the bottom of our fancy friend's debunking posts. Makes it hard to reply to the "Quick Quiz"

Also, the polls no longer have discussion links.

#### peaceetc

quality posts: 72 Private Messages
deanerino1 wrote:these community discussions no longer link to the bottom of our fancy friend's debunking posts. Makes it hard to reply to the "Quick Quiz"

Also, the polls no longer have discussion links.

The polls are undergoing a time of deep reflection wherein they ponder the meaning of their existence and whether any of them (and us) are real. Also, they're broken.

I GOT A FRICKING LETTER (November, 2017)
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