.9 Repeating

[Keefe]

I have heard people debating this all over the net... I want to know what you all think:

Does .9 repeating = 1 ?

Someone please explain!

[Aoshi]

.9 repitent doesnt equal 1, for all practical purposes in real life, it does. Mathematically it doesnt. also in real life since nothing is 100% efficient we have tolerances, so it wouldnt matter

[Keefe]

I think .9 repeated does equal 1 because 1/3 = .3 repeated and 1/3 x 3 = 3/3 = 1
.3 repeated x 3 = .9 repeated. Get it?

Here's another way:
let n = .9 repeated
10n = 9.9 repeated
10n - n = 9n 9.9 repeated - n = 9.0
Therefore 9n = 9
9n / 9 = n 9 / 9 = 1
n = 1I think .9 repeated does equal 1 because 1/3 = .3 repeated and 1/3 x 3 = 3/3 = 1
.3 repeated x 3 = .9 repeated. Get it?

Here's another way:
let n = .9 repeated
10n = 9.9 repeated
10n - n = 9n 9.9 repeated - n = 9.0
Therefore 9n = 9
9n / 9 = n 9 / 9 = 1
n = 1

[Aoshi]

see here is wehre your wrong keefe, .33333333 repitent * 3 equals .9 repitent on your calc, but nto in real life. cuz you can never get the true value of .9 repitent since it goes on forever, unless it is a fraction. 1/3 is teh true value of .3 repitent. SPAMif you take the true value of teh decimals you will get one, but if you take the approximations on your calc and multiply by 3, you will get and approximation answer, i.e. .9

[Keefe]

1/3 = .3 repeating, so 1/3 * 3 = 1

[Aoshi]

and thats it.....???
.any number repeating doesnt have a definate value, therfore you cannot multiply it. SPAMyou cannot calculate an infinite number, and that is what you are doing keefe.

[EnigmaticPhoenix]

Yes you can. SPAMHave you even taken a math course involving limits?

[Keefe]

EnigmaticPhoenix...can you explan who is right and who is wrong here?

Thanks!

[Grinnin Reaper]

Just try to tell a bank that the 99 cents in your account is a dollar. SPAM.99999 as far as you want to type it is not 1, only 1 = 1. SPAMThere can be only one. SPAM

[jinx]

I agree with Savage, .9 is .1 from 1, whether it be .1, .01, .001 etc. SPAMSome numbers are simply rounded off. When I budgeted multi-million dollar projects, the 4th or 6th decimal point was used. In large funds, the 4th decimal point will add up to considerable amounts. But .9 is .1 less than 1. Simple as that.

[Keefe]

not .9, .9 repeating

[Grinnin Reaper]

.9 repeating is still not 1 it is .0 repeating 1 away from being 1. SPAMIf you calculate to 10 decimal places and have .9999999999 you would need to add .0000000001 to it to have 1. SPAMIf .9 repeating was one it would be called one. SPAMOne number cannot equal another. SPAM1/3rd is actually .3 repeating. SPAMIt adds up to .9 repeating not 1 if you want to be technical. SPAMWe have always rounded it to keep things simple. SPAM

[thetechfreak]

same as if you do 2 / 3 = .6 repeating

.6 repeating x 3 = 1.8 repeating

repeating numbers suck in general

[Chef]

In general it's all crap ppl ;)
As my physics teacher used to say : "Throw away your calculators" (That's because he calculated anything in mind before the dumbs even pressed the = button)

stick to simple fractions

0.3~ x 3 = 9.9~ and not to 1

1/3 is not equal to 0.3~ !

because you can say 0.3~ is 0.33333
and you can say it's : 0.3333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333...
or you can even say it's :
0.333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333 3...


you cannot have one number equal to 2 or more different numbers at the same time ;)

and again 9.9~ - 0.9~ is not equal to 9

Simple is beautiful
Cheers
8)

[vee_ess]

yeah, 9.9~ - .9~ is not 9 but 9.0~9 cuz if you were to actually do it you'd have to keep the same number of sig figs. 1/3 is .3~, but you cannot show it as .3~ as .333 or .33333 or any number of 3's after that in mathematical equations because then, you not to add another3 at the end, and then another, then another. if you were to actuallly take .3~ it would add up to 1, but you cannot show it by hand cuz you will be doing it infinitely. .9~ is not 1, 1-.9~ = .0~1, cuz it is infinitely close to 1. and by all real world applications, we will use 1, like we use 3.14 or 22/7 for pi, which is another one that goes on infinitely.

so for those who say .9~ is 1 :o, then is 3.1415927 the same as 3.14 or is 3.14 the same as pi? :o he he he, tell people that, and they will laugh.

[Chef]

Pi = ~ sqrt 10
Pi^2 ~ 10

just a tip for easier calculations

[vee_ess]

my calc says that is closer to 3.16

[rivergod]

Youz guys are making my head hurt! ;D

[vee_ess]

doesn't it feel good? :D

[CrazyProphet]

One cannot say that two different numbers are equal, but I claim that .9 repeating and 1 are the same number, and therefore equal. 1 - .9 repeating equals 0, not 0.(infinite zeroes)1, because you can't have a number after an infinite number of zeroes, because then they wouldn't be infinite. I'm not saying that pi terminates , I don't know what that has to do with anything. And if anyone cares to learn about limits, .9 repeating can be stated as a convergent series ((10^n) -1)/(10^n), and as n approaches infinity, it converges to the limit 1, which can be proved to exist, and therefore the series reaches 1 with enormous values n. Q.E.D.

[Jason425]

ok let's make this simple.. you have a big thing of grapes, a truckload, no a highway of trucks full of grapes. Say there are 1,000,000,000,000,001 grapes. Take one grape away and you have 1,000,000,000,000,000 grapes. Now tell me how there is the same number of grapes before and after? Think of it this way and you can see how rediculous it is to just move to another number for kicks.. 8)

[eviltechie]

i like being a scientific man so i will share my view the scientific way

due to our limitations of experiments, our instruments can only record accurately up to 6 significant figures

therefore, 0.99999999999999999 and repeating
would round up to be 1.00000

[Jason425]

yes, but rounding isn't exact... and since we're talking about .x1 , shouldn't it be exact? since we don't know exactely, we call them 2 seperate numbers and end it there

[james]

yeah, 9.9~ - .9~ is not 9 but 9.0~9 cuz if you were to actually do it you'd have to keep the same number of sig figs.

NOOO. that is SOO wrong. Sorry. You are totally misleading people. This is an abstract mathematical problem where sig figs DO NOT MATTER. infinite is infinite. therefore, muliplying 0.9~ by 10 does NOT shift everything to the left one space. There are the same number of 9's to the right of the decimali as there were before and after. therefore
a = { 10n = 9.9~
b = { n = 0.9~
a-b = { 9n = 9 => n = 1 but by by n = 0.9~ therefore 1 = 0.9~ is quite legitimate.

Some people here have definitely not taken calculus, and certainly haven't taken calc w/ theory behind it. If you haven't, i'd seriously consider STOP misleading people b/c of "simple" logic, that unfortunately doesn't hold shit on Newton. As others have indicated, you can do this in a much more elegant proof involving limits, but the above is enough. Now there is some confusion about how to deal with addition and subtraction of "infinity". THe problem here is, we have no infinity. We have infinitely many, so yea, we can add and subtract all day long. Very nice, if you DO want to learn how to add and subtract and compare the sizes of different kinds of infinity (among other things that are seemingly impossible w/ "simple" logic), I suggest learning advanced Set Theory.

For those who think that a number can't be two things at once, I suggest learning some more quantum physics. The problem of Schrodenger's cat, once you can wrap your mind around it, can be very enlightening. (hint: if your head doesn't hurt, you don't understand anything any quantum physicist has ever said. hint 2: if your head does hurt, you probably still don't understand most of what any quantum physicist has ever said.)


this

[Jason425]

well now i'm just upset and frustrated.. i'm in pre-cal.. grrr.. :mad:
oh well.. you can have your one number = 2 diff number bs.. :rolleyes:

[lithiumdeuteride]

I will now sum up (no pun intended) and refute the various invalid arguments people have made, then offer three valid proofs, in order of complexity, that zero-point-repeating-nine really does equal one.

Aoshi: Your arguments are flawed because we are not dealing with real life here. In laboratory measurements, there are error bands, written as "+/- 4%", for example. There is no uncertainty when dealing with pure math. It is valid to say that 6 +/- 2 is the same measured value as 7 +/- 2, but it's NOT valid to say that 6 = 7. Repeating numbers do have actual values, and CAN be multiplied. Consider 4/3. It is NOT an irrational number, because it CAN be expressed as a ratio of real, whole numbers, and therefore can be multiplied or divided or whatever.

Grinnin Reaper: Get used to the idea that common sense sometimes fails in the realm of math and science. Also, rounding does not simply things, it makes them more difficult to keep track of, because you have introduced error bounds where none previously existed.

vee_ess: There is little similarity between pi and 0.99999999~ because the former is a non-repeating decimal, and the latter has a simple pattern. Any ratio of two whole numbers can be expressed as a decimal with a repeating pattern. Since there are infinite combinations of numbers to make ratios with, it follows that any repeating decimal can be expressed as a ratio of two whole numbers. A ratio that is equal to 0.99999999~ is 1/1, or 1.

evil techie: You are certainly quite wrong about the accuracy of our instruments. Experiments are dependant upon the accuracy of instruments, and NOT the other way around (unless you count breaking the equipment). We also know the speed of light to at least eight significant digits. I therefore conclude that you have no idea what you're talking about, as I had previously suspected.

Here are those 3 proofs I promised:


PROOF 1:

1/3 = 0.33333333~
multiply both sides by 3:
1 = 0.99999999~


PROOF 2:

let x = 0.99999999~

multiply both sides by 10:
10x = 9.99999999~

subtract x = 0.99999999~ from 10x = 9.99999999~:
10x = 9.99999999~
- x = 0.99999999~
9x = 9.00000000~

divide both sides by 9:
x = 1


PROOF 3:

the series SUM(c*r^n,n,m,infinity) converges when |r|<1. read this as: "the riemann sum of c times r to the nth power where n equals m to infinity".

SUM(c*r^n,n,m,infinity) converges to this value: (c*r^m)/(1-r)
the proof that this what the series converges to is found in many calculus books.

setting variables:

let r = 1/10:
SUM(c*(1/10)^n,n,m,infinity) = (c*(1/10)^m)/(1-(1/10)) = c*(1/10)^m)/(9/10) = (10/9)*c*(1/10)^m

let c = 9:
SUM(9*(1/10)^n,n,m,infinity) = (10/9)*(9)*(1/10)^m = 10*(1/10)^m

let m = 1:
SUM(9*(1/10)^n,n,1,infinity) = 10*(1/10)^1 = 1

[kakomu]

it's you Primarscources!!!!

Ok, here's the refutation.

First of all, the use of a Calculator is invalid. It does not give exact answers when it comes to infinity (as it would infinitly loop and never come up with an answer). It actually uses series and sequence to calculate infinite numbers.

So, I'm going to explain this with Summation and series:

Sigma from 1->Infinity of 9 * .1^x

If you were to continually add this up, you will find that there is an asymptote of 1 (as a limit of 1). This clearly shows that repeating 9 does NOT equal 1.

If you were to try and do the integral of 9*.1^x from 1 to infinity, you will find that just plugging in infinity does NOT work.

Also, on the subject of Limits, if we have the Limit as X->infinity of 1/x, it equals 0, however, it's pretty obvious that there is NO number you can plug into 1/x to equal 0. With that being said, it's pretty simple to make the assessment that 0 is the ASYMPTOTE, much like 1 is the ASYMPTOTE of repeating 9.

I rest my case :D

[lithiumdeuteride]

Let's assume you are correct for a moment, and that 0.99999999~ actually IS a different number than 1, and get right down to the fundamentals of numbers in general. If the numbers are different, then there either ARE or ARE NOT other numbers between them. If there ARE other numbers between 0.99999999~ and 1, I challenge you to find one. If there ARE NOT any numbers between 0.99999999~ and 1, then 0.99999999~ must be the PRECEDING number, that is, it comes right before it. If it does come right before it, then a basic property of the Real Number System is invalid. That property is that there IS NO NEXT NUMBER. The Real Numbers are continuous, and often referred to as a Continuum.

I quote from the definition of Real Numbers from www.whatis.com :

"If x and z are real numbers such that x < z, then there always exists a real number y such that x < y < z."

This means that for any two numbers x and z where one of them is actually larger than the other, and thus different, as we are supposing 0.99999999~ and 1 to be, there is another number y inbetween them. Find that number y, I dare you!

[Jason425]

both arguments are invalid, as you two have proven, any argument involving the word infinity can not be proven with certainty (at least this one) both arguments are correct, but then who's right? Nobody, because it's infinity. (assuming that silly number rule is correct, or perhaps we just proved it wrong?)

[james]

both arguments are invalid, as you two have proven, any argument involving the word infinity can not be proven with certainty

what are you talking about?? Mathmaticians deal with infinity in very precise ways all the time. the people here are using the word infinity in imprecise ways because they don't know math from a hole in the wall, hence all the confusion. People have dealt with this issue in a variety of legitimate ways. Why do people keep dealing with this in such ignorant ways? I don't understand, if you've had any math at all, you know this answer, if you don't, why even bother TRYING to make an explanation

[eviltechie]

once some people here have learned to take the limits of x approaches to infinite for various functions, they will understand better...

for now, look up on L'Hopital's rule

also i forgot what is the question which this thread started upon
lol

[Jason425]

it was does .99 = 1.. just by looking at the question should give the answer. :/ This thread is just a bunch of overanalysis.

[crono]

if you ever go and calculate it by hand or calculator you'd find that 22/7 is in no way cloase to 3.14159265359.......
though 1/3 is very accurately 0.3 repeating.
I agree that 0.9 repeating is not so true for real life because you will never use that many numbers, but since we are talking about algebra here, I'd have to say 0.9 repeating=1 is algebratically true! :)

[dragontamer5788]

If you don't believe this last proof, then you have
1. Have disproven the value of the natural Logarithm "e"
http://www.jimloy.com/algebra/series.htm

2. Have disproven the value of PI, as defined by the infinite sum of a geometric sequence http://www.jimloy.com/algebra/seriez.htm

http://www.richland.cc.il.us/james/lecture/m116/sequences/geometric.html

As you can see, the infinite sum of a geometric sequence is equal to:
S= a1 / (1-r )

It's proof is shown here:
http://www.math.unl.edu/~gnorgard/calcres/gseries.html (just click "answer")

That such a thing exists.

Where S is the series. (an infinite sum of a geometric sequence)

Thus, we fill in the values. .99999~ is a series, moddled by .9+.09+.009+.0009 ... (9*10^-n)

a1 == .9 (the first value of the sequence)
r == .1 (a(k+1)/a(k) where k is a positive integer)

S = .9/(1-.1) == .9/.9 == 1

Thus, .9~ (the infinite geometric series, .9+.09+.009+.0009 ... (9*10^-n)) is equivalent to one.

Remember, both PI and E are defined by an infinite series/infinite geometric sequence. Thus, you disagree with this, you disagree that PI == 3.141592 ...

Again, for those who do not believe .3~ is equal to 1/3. The infinite sum of the sequence .3 + .03 + .003 ... (3*10^-n) is equal to

al == .3
r == .1

.333~ = .3/(1-.1) == .3/.9 == 1/3

[Jason425]

I'm in calculus, and I still don't believe it.. I don't think any formula or equation could convine me either... it's just one of those things...

[james]

I'm in calculus, and I still don't believe it.. I don't think any formula or equation could convine me either... it's just one of those things...

....called irrationality. The point of mathematics and science and logic is to overcome those things which we believe to be true to find what can be shown to be true. Without a doubt, this is one of those cases, except perhaps if you apply Godel's Incompleteness theorem, but I don't think that is necessary at this point.

[Prometheus]

*shrugs and continues studying permutations*

[vee_ess]

ok let's make this simple.. you have a big thing of grapes, a truckload, no a highway of trucks full of grapes. Say there are 1,000,000,000,000,001 grapes. Take one grape away and you have 1,000,000,000,000,000 grapes. Now tell me how there is the same number of grapes before and after? Think of it this way and you can see how rediculous it is to just move to another number for kicks.. 8)
Well, if you want to make that analogy, you have to make it completely. You aren't taking off an entire grape. Go down to a singular cell, then go down to a single molecule of it. Now go down to a single atom of that molecule. Within that molecule you have subatomic particles, choose one, if it's not an electron you go down another level to strings that make up subatomic particles, if you do go to the electron its a string already. Now take an infinitely small amount of the energy that makes up a string out. The string is still a string, because not all strings have exactly the same amount of energy, its the string equivalent of a wavelength that determines its properties anyways, so your real life example was just shut out by String Theory. :P

Some people here have definitely not taken calculus, and certainly haven't taken calc w/ theory behind it. If you haven't, i'd seriously consider STOP misleading people b/c of "simple" logic, that unfortunately doesn't hold shit on Newton.
James, I quite agree with you. The reference you made to me earlier in that post was referring to me before I took Calculus. While I agree that one cannot understand the concepts necessary to understand any truths of this without understanding the concepts one learns in Calculus, I don't think they shouldn't be a part of the discussion. They just won't be right, that's all ;).

Jason425, if you have seen L'Hopital's Rule (http://www.math.hmc.edu/calculus/tutorials/lhopital/), (lim f(x)/g(x) = lim f'(x)/g'(x) (http://www.newton.dep.anl.gov/askasci/math99/math99073.htm),)you will agree with me when I tell you that 1 (or any constant) divided by infiniti is equal to 0.
1 - .9~ = 1/(infiniti)
Deny that! :evilking:

[Jason425]

No, I haven't seen it.. but we're just getting into limits.. so i'm sure soon i'll be corrupted into believing you two ;)