.9 repeating = 1? Is our numerical system flawed?

does anyone above a high school level of maths education think that 0.9 recurring equals 1?

Using your logic 9=10 - ah it doesnt matter - round it up

[edit on 31-3-2005 by Vanguard]

It is a property of real numbers that for two to be different, there must be an infinte number between them. For .9 and 1.0, there aren't any.

Vanguard, I'm sure many people here have a level of maths well beyond highschool, I for one have a 'beyond-highschool' level of maths.

What utrex said is absolutely true, in the set of real numbers for two numbers to be distinct there must exist a number between them. in the case of .9 and 1 no such number exists, this implies that the two are equal. I'm sure this can be proven by contradiction, i.e. assume such a number exists and show that it imples an impossibility or contradiction to the original statment.

[edit on 31/3/05 by cmdrpaddy]

There are several proofs you can find that are accepted by mathematicians as proof that 1 = .9repeating.

A simple one:

9999...


© Copyright 1999, Jim Loy


.999999... is equal to 1, exactly. Here is the simple proof:


1/9=.111111... (easy to prove using Geometric Series)
.999999...=(9)(1/9)=1



This fact (.999999...=1, along with similar equations like .1249999...=.125) slightly complicates a few proofs in set theory. When dealing with all possible decimal representations, you may have to show that you are accounting for the ones that are redundant. We deal with the redundancy by outlawing any decimal which ends in infinitely many nines (replacing them with their equivalent non-repeating decimal).


I received email suggesting that the above proof may be easier with 1/3=.333333... as the first step. It may be more obvious, as more people know that 1/3=.333333... But the proof is virtually identical then because, even though we all know that 1/3=.333333..., we still need to use a geometric series to prove that. In my opinion, 1/9=.111111... is slightly more esthetically pleasing, for some reason.

www.jimloy.com...

Ouch. I think my brain just imploded.
jk

I think it's just that the numbers are infinite... and therefore you can't use simple math to express them.

Wow. Lot of good posts here. Why can't we have a number: 0.00000~1 that would fit in between the 0.99999~ and 1.0? I know we shouldn't have any number after an infinity, but if they both go on forever,what does in matter? That is kind of my way with dealing with numbers like that, but perhaps I'm doing myself a disservice.

[edit on 2-4-2005 by Fig]

[edit on 2-4-2005 by Fig]

im sticking with 0.9 recurring tends to 1 rather than equals 1

Originally posted by Vanguard
im sticking with 0.9 recurring tends to 1 rather than equals 1

I'd like rather to say that 0.999... tends to 1 when you tend to an infinite number of 9's, but that it is equal to 1 when you have an infinite number of 9's.

That it obviously a theoretical position, as no one can ever write an infinity of 9's, and it is actually why we write several followed by '...', that actually means 'and so on until infinity'.

Practically, in maths theory, 0.999... = 1, in practical applications, 0.999... is not 1, but close to fit engouh given enough accuracy, depending on the use. Example, I go to the gas station today and fills up to € 27.499. I will pay actually 27.50 because there are no 0.1 cent! It is close enough.

[edit on 2-4-2005 by SpookyVince]

they are theoretically different, but practically identical. Which of the 2 numbers is larger? Which is smaller? Though the difference between the two is less than 1/googleplex by 1/googleplex, a difference is implied. It is like PI, it has been calculated to the 50 billionth decimal place. Is that an exact representation of PI? No. Is 1.31415 good enough, yes.

I had also another thought on that just now...

Our numerical system, to finally try to answer the initial question, is not flawed at all, but maybe, depending on the use and on the numbers, we should try to use a different base...

Explaining it:

We commonly use base 10 to write numbers (what is called decimal notation). Computers use base 2 to "write" numbers... Go and write things like 0.9 or 0.1 in binary. They're infinite numbers...

Example: 0.1 (dec) = 0.00011001100110011... (bin) so on. It is an infinitely repeating sequence (0.0 then a sequence of 0011), though in decimal it is finite.

Now, for instance, what if we would be writing in base 3? 1/3 in base three (well, 1/3 is base 10, it is written 1/10 in base 3!) is exactly 0.1! One third is a completely finite number to write then, using a different base. In base 9, it would be 1/3.

Got my point? It is only a representation of the same number, another way to write it down, just and only that.

This subject is like rediscussing if Zeno's arrow will hit the target or not.

A philosopher called Zeno of Elea who was born in 488 BC first came up with this issue. Google "Zeno's arrow" for more information.

simply there is 2X distance between his arrow and the target, but when his arrow reachs to half distance (1X) arrow starts to travel at half speed, and again when the arrow reachs to the remaining half distance it starts to travel at half speed. so the speed of arrow is like x, x/4, x/8, x/16 ....

mathematicians say it will hit the target in infinity because it always gains distance. Philosphers says it wont.

for 2500 years there is this paradox, because it is a paradox.

Zeno's Paradox is only a paradox because Zeno's Logic seemed to be correct, and yet arrows still hit their targets. However there was a flaw in his logic that took over 2000 years to fix.

Well my last statement is if you don't accept that .9 repeating is exactly, completely and utterly equal to 1, you must also accept that .1 repeating isn't equal to 1/9, .3 repeating isn't equal to 1/3, .6 repeating isn't equal to 2/3 etc....

Originally posted by djohnsto77
Well my last statement is if you don't accept that .9 repeating is exactly, completely and utterly equal to 1, you must also accept that .1 repeating isn't equal to 1/9, .3 repeating isn't equal to 1/3, .6 repeating isn't equal to 2/3 etc....

no, why must we?
If I divide 9 into 1 I get .1 repeating........Divide 1 into 1 = 1, not .9 repeating.

Originally posted by BlackGuardXIII
If I divide 9 into 1 I get .1 repeating........Divide 1 into 1 = 1, not .9 repeating.

multiply 1/9 * 9 you get 1
multiply .1 repeating * 9 you get .9 repeating
if you accept 1/9 equals .1 repeating, then you must accept that 1 = .9 repeating because of the Multiplicative Identity law of mathematics.

[edit on 4/3/2005 by djohnsto77]

Well sure, when you put it that way, it is hard to argue. I think you have convinced me. I see your point clearly now. thank you

Would people stop writing in on how fractions have equvilant decimals. some do like 1/4 or 1/8 but fractions(like 1/3) never terminate in base ten. A number such as pi does eventually end but 1/3 0r 1/9 never end. 1 divided by 3 equals 1/3 but doesNOT equal .33333........ Nomatter how how far you go there will always .0000.....1 left over. Our number system works well because we have different bases (1,2,8,10,16...) With these differences we can solve every problem

This isn't even high school math, .999..... DOES NOT equal 1. .33333..... DOES NOT equal 1/3 and any monkey that can divide could tell you that!!!!I hope we are done!

P.S..111...... does not equal 1/9(i forgot that one)


[edit on 7-4-2005 by Maggot Minion]

Wow, Maggot thanks for clearing that up...NOT!

What ignorance you bring to this thread!

1) Pi is an irrational number meaning that it neither terminates nor repeats

2) 1/9 does equal .1repeating and this is provable through a Geometric Series proof.

3) This is high school math

Pi is an irrational number

Bah not only is it irrational, it is transcendental!

Originally posted by djohnsto77
2) 1/9 does equal .1repeating and this is provable through a Geometric Series proof.

And, you know, division.

I'm laughing because this discussion is repeating itself.