# If 10 divided by 3 is equal to 3.3333-infinity, then why isn't 3.333333-infinity times 3 10?

Chris [Ninja]™ 2008/08/21 19:04:14
 Because... Well... Huh?
You!
Or you could say, if:
x = 10/3
x*3 = 9.9999999999-to infinity but does not every REALLY equal 10.

### Top Opinion

• Icedragon1969 2008/08/21 21:49:43
Well...
It does equal ten actually, but you have to remember that what you're doing is the sum of a series expansion. I'll try to write it out, but I don't know how well it's going to work without series notations and such.

3.333333...=3+(3/10)+(3/100)+...

Thus it is the sum of an infinite geometric series. We can calculate that sum rather easily by the formula:
S=a/(1-r), where in this case a=3 and r=1/10. There fore
S=3/(1-(1/10))=3/(9/10)=30/9=...

So not surprisingly 3.33333...=10/3, and if we multiply by 3 we get 10.

The problem is that if you leave it in decimal form, you're truncating, cutting off, the series, at some point. The more terms, or the farther along you make that truncation, that you use, the closer you'll get to 10 in this operation, but you still won't quite get 10 because you haven't summed over the entire series.

Does your head hurt yet? Mine sure does. I hate this series expansion crap. :(

## More polls by Chris [Ninja]™

### Sort By Most Raves Least Raves Oldest Newest text size Opinions

• leevi.mikkola.1 2015/03/28 10:58:08
Because...
if you had 10 atoms divided by 3 you could not do it because you can't divide atoms
same thing with apples you would eventually end up to atoms
• Jessie Colter 2013/09/13 23:55:26
Huh?
I didn't know you knew that you need teach to me how you did it any where place at ???????????
• Nicholas Nelson 2013/07/27 06:12:38
Because...
Because you always round up.
• terenceho 2010/10/08 07:26:49
Well...
Think of it this way: 10-9.99999...=0.00000...

No, you don't add a 1 at the end of the 0 because it's never ending, and if you added a 1 it wouldn't be.
• devils 2009/11/24 01:33:11
Because...
• eli 2009/04/04 04:27:15
Because...
3.333333333333iffinitlymultip... by 3 =10
because thing that are infinate are beyond human comprehension (explanation) 9.999999infinatly will go on infinatly evetualy the human mind canot umderstand it making 9.999=10 (this is pure opinion stated as fact)
• andrew 2009/03/28 23:07:57
Because...
It does not equal 10 it equals 9.999999999...
• KellyDew~YES WE CAN 2008/08/22 23:44:23
Because...
Im sorry, what was the question??? LOLOL!!!
• Huh?
thanks i lost my thought i was thinking about something...grr
• Erin<3=) 2008/08/22 13:41:02
Huh?
• Mikitty☆彡(SHARP) BN-0-WAWU 2008/08/22 12:04:16 (edited)
Huh?
u make my head hurt :( ???????????????????????????? i don't understand u
• ಌMiss Ranaಌ 2008/08/22 10:55:43
Because...
We just round it up to the whole number?

Man, I'm not going to think much more than that ~ my head will blow up otherwise..
• Captain America 2008/08/22 08:56:09 (edited)
Because...

because 3 .3333 infinity cant be multiplied or divided by any number no matter who tells you it can, you can only give a reference to an idea of an probablity of exactness by using a number like 3.333 and adding a bar - .

Good luck with it.

get it?
• lines 2008/08/22 03:59:14
Because...
...Seriously?
• runningintriangles 2008/08/22 03:29:41
Well...
...I'd like to answer this... but I had math first semester of last year... and it's summer... and the only thing on my mind is the current production I'm in... not math ;D
• Janie 2008/08/22 01:43:55
Well...
I'll do better if you give me the sale price at the mall!
• Luna 2008/08/21 23:10:21
• Icedragon1969 2008/08/21 21:49:43
Well...
It does equal ten actually, but you have to remember that what you're doing is the sum of a series expansion. I'll try to write it out, but I don't know how well it's going to work without series notations and such.

3.333333...=3+(3/10)+(3/100)+...

Thus it is the sum of an infinite geometric series. We can calculate that sum rather easily by the formula:
S=a/(1-r), where in this case a=3 and r=1/10. There fore
S=3/(1-(1/10))=3/(9/10)=30/9=...

So not surprisingly 3.33333...=10/3, and if we multiply by 3 we get 10.

The problem is that if you leave it in decimal form, you're truncating, cutting off, the series, at some point. The more terms, or the farther along you make that truncation, that you use, the closer you'll get to 10 in this operation, but you still won't quite get 10 because you haven't summed over the entire series.

Does your head hurt yet? Mine sure does. I hate this series expansion crap. :(
• Warren ... Icedrag... 2008/08/21 23:10:48
I've derived the proof to show that the infinite series of 9.9999.... converges to 10 below, but those still skeptical claim it is a "mathematical trick"!
• Icedrag... Warren ... 2008/08/22 04:05:52
I didn't go down that far, but I see it now.

It's the reaction of a lot of people. It seems to me to come about because we live in a truncated, finite world, so many can't see past that. Try and explain to most people how there can be different sized infinities. ;)
• Warren ... Icedrag... 2008/08/22 16:34:22
I think some people don't understand the difference between the mathematical symbols we write on paper and the actual mathematical concepts. This is where the confusion comes from.
• Icedrag... Warren ... 2008/08/22 16:37:52
Hmmm...now that's a good point too. I'm always amazed, for instance, at the number of people who don't understand (or are amazed to find out) that a fraction is just another way of expressing division.

I also have experience with the fact that most people don't seem to understand that we humans invented math, that it is not intrinsic to the universe.
• Slug Diamond 2008/08/21 20:21:32
Huh?
Ouch, this makes my brain hurt...

• Another Benjamite :)~ 2008/08/21 20:02:56
Well...
I get where you are going with this. It is kind of a trick of mathematics when you use limits. Looking at this problem from a philisophical viewpoint, rather than a mathematical one, you have to consider that infinity is not a number, it is a concept. You can cancel infinity out of an equation, however you have not canceled out a real number, just a symbol that represents an endless enumeration. This kind of concept is difficult to represent with mathematics, so we basically just made up infinity to cover that which can not accurately be described. A rigid math whiz will have a difficult time looking at it from the philisophical side, however, at no fault of their own. This is how they have been trained.
• Warren ... Another... 2008/08/21 20:13:32
I disagree. All mathematics is a concept, whether it is the number 3 or infinity. There is no "philosophical viewpoint" in mathematics. The question was a mathematical one, and as such mathematics is the only tool that applies.Part of the problem arises because people are more comfortable with numbers in a calculator than mathematical symbols. In some cases (like this one) the numbers in the calculator (0.3333333...) are a less accurate representation than mathematical symbols, like 1/3.
• Another... Warren ... 2008/08/21 21:36:20
This is an ancient debate. I think there is a philosophical aspect to mathematics when dealing with subjects like infinity. Is "i" a reality, or was it created to "solve" a mathematical conundrum such as the square root of -1?
http://www.rpi.edu/~newbel/mi...
You make an excellent point that all numbers are concepts, however, which I did not give adequate thought too in my last post. I think engineers and others who use mathematics daily to describe the physical world would agree with your view, and those in the humanities or philosophy would agree with my "philosophical mathematics" viewpoint. I have to thank the author for this post. What an interesting thing to think about.
• Chris [... Another... 2008/08/21 21:37:35
You get it!
• Warren ... Another... 2008/08/21 21:43:55
Mathematics is a concept used to solve real world problems. The concepts may not be physically real, but the solutions determined by mathematics are real, and exact. I have used both imaginary numbers and infinite series in my engineering work and the results are applied to design real products in the real world.
• bubblegum 2008/08/21 19:51:47
Huh?
• Dave Sawyer ♥ Child of God ♥ 2008/08/21 19:35:18
Well...
It is. But you can't do it on a calculator because the calculator isn't infinite.

• Warren - Novus Ordo Seclorum 2008/08/21 19:20:32
Because...
9.9999999999......carried on to infinity does equal 10
• Warren ... Warren ... 2008/08/21 19:23:07
It is an infinite series:

9 + 9/10 + 9/100 + 9/1000 + ... 9/10^i

Converges to 10 in the limit when i goes to infinity.
• Kim [Ni... Warren ... 2008/08/21 19:24:36
doesn't it just approach infinity and never reach it?
• Warren ... Kim [Ni... 2008/08/21 19:26:26
Practically, yes, but in the mathematics of limits, it precisely equals 10 when the number of terms goes to infinity.
• Chris [... Warren ... 2008/08/21 19:25:30
But never ever truly approaches it. Carrying it on to what it approaches is just how we conceive of it. More like a patch in mathematics. If you go with the literal then it can never be 10. See where I'm going?
• Warren ... Chris [... 2008/08/21 19:28:12
No, it can be shown mathematically that it precisely equals 10. No rounding is necessary.
• Kim [Ni... Warren ... 2008/08/21 19:30:09
I want to take calc, but I'm afraid that I'll get owned.
• Warren ... Kim [Ni... 2008/08/21 19:37:40 (edited)
You don't need calc. Express x = 9.99999.... as an infinite series:

x = 9 + 9/10 + 9/100 + 9/1000 ... 9/10^i (i goes to infinity)

so: x/10 = 9/10 + 9/100 + 9/1000 ... 9/10^i (i goes to infinity)

x - x/10 = 9

so x = 10
• Chris [... Warren ... 2008/08/21 19:40:58
Yes, but that is our mathematical patch in reality though, it never truly does reach 10. Infinity goes on forever!
• Warren ... Chris [... 2008/08/21 19:44:39
Look at my proof. I didn't do any patches! I proved that x=10. I subtracted two infinite series to make all the terms go away except for the whole numbers.