.9 (repeating) = 1

Yes, it's true

from WinCustomize Forums
In math class, we ventured upon something that I would like to show you all:
.9 (repeating) as in .9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 is actually equal to 1. Not approximately, but ACTUALLY.

Here's the proof:


x = .9 repeating
10x = 9.9 repeating
10x - x = 9.9 repeating - .9 repeating
9x = 9
x = 1
Therefore .9 repeating = x = 1.

I always assumed that a variable cannot stand for two numbers, so this must be true. Have fun pondering
16,675 views 50 replies
Reply #1 from WinCustomize Forums Top
Applying 'modifiers' to undefined numbers is the issue.
Reply #2 from WinCustomize Forums Top
There are several rounding errors in there, which is where the apparent ./9/ = 1 comes from.
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Reply #3 from WinCustomize Forums Top
To be precise:

x = ./9/
10x != 9./9/ --> 10x = 10*./9/
10x - x != 9./9/ - ./9/ --> 10x - x = 10*./9/ - ./9/
Reply #4 from WinCustomize Forums Top
First off I'd just like to say WRONG!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Second, although its awesome at 14 they're teaching you how to solve things at that level is awesome.
Thirdly, the teacher not knowing what she/he's talking about and yet probably has a Degree for teaching math is sad.

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.9 = .9 repeating, .9 repeating =/= 1 (=/= is the symbol for "does not equal")

your method for trying to solve an equation is right, but the technical part is the wrong point
_
.9 multiplied by 10 =/= 9.9 repeating, basically if you multiply it by anything other than 1 the repeating integer will either change or not repeat at all
_
.9 * 10 would = 9.9999999-----(a bunch of 9's) but wouldn't repeat forever, because moving that decimal 1 spot would signify it's 1 less than infinity. Now .999 * 10 would = 9.99 but in your statement of .9 repeating multiplied by 10 = 9.9 repeating you're stating that you're taking away one of the .9's from the infinite number.

now if you used .999 as the integer and did the same thing you tried you'd get this set of equations.

x = .999
10x = 9.99
10x - x or 9.99 - .999
9x =8.991
x=.999

See how it solved into itself as being THE EXACT SAME NUMBER YOU STARTED WITH??!! Absolutely every number will do that, everytime. The variable "x" doesn't have variation.

Don't worry, I will get to the part of showing why your equation is wrong, and how to correctly do it.


In "guesstimations" people round repeating decimals, in which case .9 repeating rounds to and is APPROXIMATELY 1, but it ISN'T 1.

When working with repeating decimals you cannot simplify it to a whole number like that. if you were given the equation of....
_
.9 * 10 the answer would be...
_
.9 * 10 *gasp*

...but horiz0n, you listed the problem as the answer. Indeed. it cannot be further simplified, no more than 1/2 can.

Now to disprove your formula all one needs to do is look at your answer, if you say

x = .9 repeating
10x = 9.9 repeating, and eventually you get to
x = 1, well if x = 1, then 10x would = 10, not 9.9 repeating.

So that proved yours wrong just using your own equation, without any other rules or logic. Now if you want to see how to actually/properly do it, I'll do the exact same thing you did, correctly.

x = .9 repeating
10x = 10 * .9 repeating
10x - x = 10 * .9 repeating - .9 repeating
9x = 9 * .9 repeating
x = .9 repeating

Therefore .9 repeating DOES NOT EQUAL 1
The variable x =.9 repeating exists because you cannot multiply .9 repeating, so if you had a problem like..
2 + 6 * 4 * .9 repeating, you define .9 repeating as x so that you can solve the problem. But once you get the answer you don't solve the answer too!
(BTW, the answer is 2+24x or 2+24*.9 repeating since x=.9 repeating, also note that the problem was not solved, IT WAS SIMPLIFIED.)

Show that to your "teacher"

When dealing with repeating decimals like that, it's often best to just turn that integer into a variable, such as y but it's already listed as x here. Which brings up the biggest point in this entire problem

THERE IS NO PROBLEM, What this in fact is, is just a Formula. And quite honestly a "useless formula" of proving that x = x, 1 = 1, that any given integer or number equals itself.

If x = .9 repeating, well, x = .9 repeating. Thats not an equation to solve, THATS THE FRIGGIN ANSWER. It's like saying 2+2 = 4, so what does 4 equal, UM HELLLOOOOOOO IT'S 4

A variable defines or takes the place of a number, and is always an absolute of what it is representing. When you define a variable you don't solve what that variable equals, cause you already found it. Say you have the problem 10 - x = 6, and solve that to be x = 4 AND THEN try to figure out what x is, YOU ALREADY DID.

Anyway, I'm starting to feel like I'm repeating myself, so I hope that helps you, and tell your teacher some random guy horiz0n said he/she should go back to school themselves.
Reply #5 from WinCustomize Forums Top
Jafo thinks he said it more simply....
Reply #6 from WinCustomize Forums Top
horiz0n totally agrees, but thought he'd explain why and went "overboard"
Reply #7 from WinCustomize Forums Top
craeonics provided the bridge between J-man and H-man

And the three of us are a bunch of smartasses.
Reply #8 from WinCustomize Forums Top
yeah you're real smart......*cough*
these are great things for kids.....like jokes but only for math...
and you come in and say....uhm...that's not correct...duh...
i highly doubt his teacher will think this is correct also...
like i said, these are fun things to show....you shouldn't get in to them
to deep....
oh and btw...i mean no disrespect to anybody here....i'm just sorry it's
been taken so serious.....does anybody remember laughter ??!!??
Reply #9 from WinCustomize Forums Top
.999 to infiniti does not equel 1

See, now I smart too!
Reply #10 from WinCustomize Forums Top
one third = 0.3333333333 repeating
two thirds = 0.6666666666 repeating
three thirds = 1
Reply #11 from WinCustomize Forums Top
moving that decimal 1 spot would signify it's 1 less than infinity


There is a logic flaw here. By definition, a infinite number is, uh, infinite. By multiplying it, you end up with another infinite number, that is, 9.9999... (repeating forever).

Anyway, there are lot of debates about it on the internet. See here: Link

See also the article Decimal on wikipedia: Link
Reply #13 from WinCustomize Forums Top
Here are some links:

http://www.newton.dep.anl.gov/newton/askasci/1995/math/MATH070.HTM
http://www.artofproblemsolving.com/Forum/viewtopic.php?p=147550
http://mathcentral.uregina.ca/QQ/database/QQ.09.00/sarah2.html
http://forums2.warcry.com/read.phtml?f=156&thread=540213&page=0

There's just a few of the people that agree with me, by the way, how do you prove Tiggz wrong???

Infinity is a difficult concept. You are asking: why is 0.999 . . . with
an infinite number of 9's, equal to the integer 1? Well, it is just a
different way of representing the same number. It even makes sense logical-
ly. Remember how you learned how to count using apples? Let us go back to
that concept. Say you have 9 apples and you give one to a friend. Now what
fraction of the total number of apples have you given away? Right, 1/9.
How do we represent that in decimals? Right, 0.11111 (repeating). Try it
by hand. You will quickly see that the 1's go on forever. Now, how many
apples does your friend have? She has:
(total number of apples) times (her fraction of the total number) =
= (9) times (0.11111 (repeating)) = .99999 (repeating) = 1 apple.
Reply #15 from WinCustomize Forums Top
Also,

9.9 repeating - .9 repeating isn't equal to 9 ( 9.9rep - .9rep =/= 9), because, then the repeating parts must be of the exact same length. But if they are of the exact same length, you can count them (countability). If you can count something, it's not repeated ad infinitum. Therefore, .9rep isn't repeated a.i. . But this contradicts with the statement made earlier that .9rep repeats a.i.. QED
Reply #16 from WinCustomize Forums Top
If you want to debate logic and theory, rant away people have been doing so forever, if you want to pass physics and trig, adhere to the afformentioned.
If thats just a riddle or clever concept, then uhh hardy har?

"There is a logic flaw here. By definition, a infinite number is, uh, infinite. By multiplying it, you end up with another infinite number, that is, 9.9999... (repeating forever)."
You cannot mulitply it because you cannot understand it well enough to factor, which is why when working with numbers you cannot calculate, you round it, or you exchange with a variable. One way is accurate, one way is precise.

"i highly doubt his teacher will think this is correct also..."
If they "ventured" about it in class, I'm pretty sure the teacher is included.
And if the teacher knew about it and was just trying to make the students think, which is more than possible if not likely as the good teachers make you think instead of just telling it like it is all day, then yeah that would be a good thing. Otherwise, yeah thats sad for someone with a degree.

"how do you prove Tiggz wrong???"
You don't, he's right =p

If you really just want some puzzling concept, try to understand how infinity works. In the most simple form positively, you just think, no matter how big something is, you can make it bigger.
In a negative form, no matter how small or microscopic something is, you can cut it in half.

So if I'm walking to a door, and the space between my hand the the doorknob can keep being cut in half infinetly, how can I ever reach it?
Reply #17 from WinCustomize Forums Top
If you want logic and theory rant away, if you want to pass physics and trig, adhere to the afformentioned.




Proving infinity has a limit undoes time and erases existance in all its magnificent forms.
Reply #18 from WinCustomize Forums Top
Um, no, tiggz is wrong as well. 0./3/ is an approximation of 1/3 and not the exact same number. If you can't grasp that, try comparing it to asymptotes. It's the same thing.

As for infinity, it's not a bunch of nines, it's infinity. It's not really a number, so you can't really do math with it: infinity + 1 = infinity, while infinity / infinity = undefined.
Reply #19 from WinCustomize Forums Top
This reminds me of something else I heard once... No 2 objects will ever touch, technically. If an object continues to cut it's distance from another object in half, it can, in theory continually move closer and closer to that object for eternity and never touch it.
Reply #20 from WinCustomize Forums Top
You all, just made my head assplode... I am sitting her typing and look like a (bad) Joecartoon, right now.
Reply #21 from WinCustomize Forums Top
Tiggz is right(or right enough), you're just going beyond as to what an infinite number is doing, (and if you want to elaborate feel free cause parabola's aren't my strong point)
In effect it would be almost better to look at it like....
1/3 = .333333 repeating
2/3 = .666667
3/3 = 1

However you just explained "infinity" better than I was able to. The integer itself goes on infinetly, but the number isn't infinity, no number can define infinity.

"No 2 objects will ever touch, technically."

You win the prize for the "how can I reach the doorknob" question.
Reply #22 from WinCustomize Forums Top
crea, so what your saying is that if I split something 3 ways and then put it back together its not whole?

so 3 thirds or 3/3 is not 1

1/3 + 1/3 + 1/3 = 0.999999999999?
even as 1 split 3 ways is .3333333?


See you guys, I knew 1 + 1 = 3!!!
joeKnowledge head hurts now that math does not make sense anymore
Reply #23 from WinCustomize Forums Top
This reminds me of something else I heard once... No 2 objects will ever touch, technically. If an object continues to cut it's distance from another object in half, it can, in theory continually move closer and closer to that object for eternity and never touch it.



Damn... How does that work with living Cells? While most split, some merge to form new life. Wouldn't these 2 object have to touch in order to create?

You know, this is why I love math. Its the God science. Just when it means this, it can also mean that as well.

If 2 objects can never tough then what make the reaction when one object hits the other? What makes the force? Gravity? Mass?
Reply #24 from WinCustomize Forums Top
joe: No, what I'm saying is that 1/3 is unequal to 0.3, 0.33, 0.333 and so on. That's just an approximation used in calculator displays and such. You can get really close (infinitely close) if you keep adding 3's at the end of the number, but it will never be exactly 1/3.