1 = 0.9999999...

[KingJoe]

Good point Hal_9 million

 

I think we might need a maths graduate or something to give a definitive answer.

 

You rang?

Don't worry if you're totally confused by this, lots of people are (at least half of the people on my teacher training course got this 'wrong' and decided that 0.99... wasn't the same as 1).

0.999...=1 (the key to this is in the '...' part)

Most of my usual strategies for explaining this have been used. The main reason for not acceptoing it is that the 'infinity' that most people come across in maths (or their lives) is the POTENTIAL infinity (i.e. an infinity that comes from repeating a process forever) the infinity that we deal with here is an actual infinity (which is the headwrecking one).

For those of you familiar with 'sequences and series' at a level, try making the sequence of numbers that goes 0.9, 0.99, 0.999, 0.9999,... and find a value for the limi of the sequence (or the series 0.9+0.09+0.009+0.0009+...)

There is no approximation involved, by the way.

it is also true for any number with a finite decimal expansionr: 1.0000000...=0.9999999... and 2.63000000...=2.629999999....

 

It's to do with the density of the real numbers (guven any two distinct real numbers, there's another one between them- I'm not going into the proof of that one!).

 

Taking the potential infinity route, 1-0.9=0.1, 1-0.99=0.01, 1-0.999=0.001.

as we add more 9's, we get closer to 1. If we add an INFINITE number of 9's...

The point is that there is no LAST '9' in 0.999... so there is no difference between 1 and 0.999...

 

Also, someone mentioned this being a fault of the decimal system, but it isnt:

in binary 0.1111111...=1 in trinary 0.2222222...=1 etc

 

If anyone is really confused (and desperate to know the correct answer (or prove me wrong...) then feel free to PM me or write it on the thread (but it seems to be annoying enough people to get locked pretty soon...

 

Hope I was of Help.

Joe

 

edit:

This guy is only 14, he's only been doing algebra for about year or two, so his knowledge is limited.

 

he's also entirely correct

[Dieter]

KingJoe confirmed my suspicions, and he seems legit to me, so I bow down to thee!

[EEVILMURRAY]

However, the ... after every number suggests that the 9's go on into infinity. Then we get the problem of infinity. The problem is, you cannot enter an infinate number into an equation and get an exact answer, you can only make an estimate. Take pi for example. This is used to find the area of a circle. To use the full number, you wouldn't get the calculation done, because the decimal places go on for infinity. Scientist have tried to see how far they can go before it finally ends, unfortuneately it's a random pattern, so goes on forever. They've mangaed to get it to several million decimal places, and it just keeps going. So to make things easier we round the number up to 3.14. This way, a calculation can be done easily. However, you will never get the exact number you require. But to get it perfect does not concern us, as it's a waste of our time. To round something up or down, simplifies it to the point we need. The only people who need to be precise in these cases are people like NASA. They needed to get the calculations for the mirror in the Hubble space telescope accurate to millionths of a milimetre in order for it to work. We only need to calculate as far as we can see. So to get it more accurate than say several hundred atoms either way would be wasteful, as there's not much an atoms difference is going to make in comparrison to what our own eyes see. OK this my be confusing, but basically, you only need as many decimal points as necessary to your bounderies. If you're measuring the area of a circlular section of say a clock face, you aren't going to be more accurate than a milimetre at most, because it isn't necessary. So we come back to the statement of 1 = 0.99999999... No this is incorrect! 0.99999999... going on into infinity never ever reaches the exact value of 1. It get's extremely close however. So close, that it isn't any cause of our own concern. Therefore at this stage, you can round it up to the nearest whole number "1" and say that's it's 1. In actual fact it isn't, but you can say it's pretty damb close. It's so close in terms of human thinking, that it really doesn't matter. We don't need to know how many billions of 9's there are after the decimal point as that's close enough to 1 for us to just call it 1 for simplicity. It's still not 1, but it's really damb close. The connection cannot be proven with an equation, simple as that. X will always equal the same number. Otherwise, it isn't an equation. Equations are all about equality. if this guy put that crap down in an exam, he'd get a big fat 0 marks, because an equation must be equal on both sides, so the value of X cannot change. That is impossible! This guy is only 14, he's only been doing algebra for about year or two, so his knowledge is limited.

 

[Shorty]

I can't believe I'm saying this on a primarily nintendo based internet forum, but you guys are a bunch of geeks.

[Dieter]

I can't believe I'm saying this on a primarily nintendo based internet forum, but you guys are a bunch of geeks.

 

Because maths is so much geekier than gaming, right? :P

 

What do you think a console does?

[#1cubeplayer]

Argh, I'll be damned if I argue over this, but I just want to say that 0.9999rec technically isn't equal to 1 or anything else other than itself. If you say it is, you have rounded. Simple as that, no?

 

Now to do some real mathematics homework. *sigh*

 

EDIT: I didn't read the last page so I didn't know I was going against someone with a degree. I still believe that 0.9rec isn't 1 unless you round.

[zatoichi]

going by that theory, what is the exact measurement of a quarter of 1?

[BGS]

err, did you really ask that?

0.25

...duh.

[Supergrunch]

I've realised the problem with this thread. The people who are mathematically orientated know that 0.999rec = 1, and provide proofs to show this. The people who aren't mathematically orientated don't believe this is true, but won't accept mathematical proofs to the contary because they're not mathematically orientated.

 

It be a vicious circle I tell thee!

[kopo]

I don't have a degree in math but it is assumed by many scientists and mathemathicians (and me) that 0.999... =1.

The problem people have is the concept of infinity.

It never ends. Think of it as a HUGE number, so huge in fact, that it's impossible for sums and substractions to change it. For example infinity minus 99999 billionzillionkwadrillion is still infinity. I also believe that +infinity - infinity is impossible, because both are so big. I'm not sure of this, i simply forgot.

 

As for the the proof that 0.999... = 1 see above somewhere

[zatoichi]

so how can a quarter be 0.25, when a third is 0.333rec?

 

4 quarters at 0.25 = 1

3 thirds at 0.333rec = 0.999rec

 

And so why isn't 0.333rec 0.34? In which case 1.02 = 1 as well?

[kopo]

I don't see how you can conclude if 0.333rec = 0.34 that 1.02 = 1.

[zatoichi]

multiply the 0.34 by 3, as you would the 0.333rec. If they're the same number, that is.

[Dieter]

multiply the 0.34 by 3, as you would the 0.333rec. If they're the same number, that is.

 

kopo wasn't commenting on the 0.33rec (.333rec is incorrect seeing as you have to put the recurring bit twice before the 'rec' iirc :p ) being equal to .34, seeing as it is in fact so.

 

1 = 1.02 is however completely and utterly wrong

[zatoichi]

well then if it is in fact so, then why is 1.02 = 1 wrong? If 0.333rec and 0.34 are the same number, that's like saying 0.34 is a third of 1, which is wrong. It's a third of 1.02.

[Haver]

I wish KingJoe had taught me maths

[Supergrunch]

0.333rec and 0.34 aren't the same number, someone's got a bit confused. The only numbers that we're saying are the same are 0.999rec and 1. I can see why you thought 0.34, although your logic is slightly twisted. 0.999rec (or anything with .999rec on the end) is a special case, as the number infinitely progresses towards 1, and thus (brain-twisting bit) is 1.

[zatoichi]

Wait. You're right. 0.333rec would infinitely progress to 0.3333.....4.

[Supergrunch]

No, it doesn't. I can't exactly explain why because I'm not good enough at maths, but I'm sure someone else here can.

 

It's to do with the fact the the number is 3 rather than 9. 9 is only 1 away from 10, making it much closer to the rounded number, and when the 9's reccur, then it is what was the rounded number, which is no longer a rounded number.

 

I believe (although I may be wrong) that 0.33999rec is equivalent to 0.34.

[The3rdChildren]

Math... Make... Third Children... Brain... Die.

[Zakatu]

Hope I was of Help.

Joe

 

Yes, you were. It was a really helpfull explanation.

 

BUT, even though i'd like to think i'm mathematically orientated i don't really understand it. The sticking point is when you said that because there is no "last" 9 then there is no difference between that number and 1 itself.

 

Its that bit i don't really understand, what you seem to be saying is anything that gets closer and closer for infinite will be the same as what it is approaching.

 

This goes against what i've allways been taught..., that even if something goes on for infinite, it will still never get there. I'm trying to think of an analogy, but if you were constantly accelerating then you would eventually reach the speed of light, right? but according to physics this is impossible. So your saying that if you kept on accelerating for INFINITE you would be going the speed of light?

 

Maybee its a poor analogy.

[Supergrunch]

Actually, according to physics, you can reach the speed of light, but not surpass it.

 

Anyway, are you trying to think of something like the graph of root x- the gradient gets closer and closer to 0, but never actually reaches it?

 

This is a different case. If you don't that proof, then look at some of the earlier ones, such as the ones with thirds, or x=0.999rec etc.

[BGS]

Wait. You're right. 0.333rec would infinitely progress to 0.3333.....4.

 

Argh Zatoichi, please, stop. You don't have that good an understanding of maths so stop saying people are wrong with your flawed mathematics.

 

0.333... will NEVER "progress" to or equal "0.3333......4"

 

BUT.

 

If it were 0.333999... then THAT would equal 0.334

 

It's the nines at the end that are key.

 

 

EDIT: Stop thinking that the 0.999... reaches or 'progresses' to 1 or whatever the number is that has the 0.999... at the end. It doesn't. It just EQUALS that number. the 0.999... goes on INFINITELY. It's an actual infinite rather than a potential one.

[zatoichi]

BGS - 'progress' wasn't my word, I was just using terminology that had been previously used to describe what was going on by somebody else to put what I was saying into context, so if you want to flame someone for their choice of words, flame that person (although you shouldn't because only dickheads flame people.)

 

Also, when did I ever say anyone was wrong? I was just adding to the debate. If you can't handle that, then why do you return to the thread? You could have just as easily said, like others did, 'no, that's not right, because ..... and this is the right answer', but instead you come in mouthing off like a Hyeena with nothing useful to say that hasn't already been said before. Jeez.

 

I was also on course for an A at A-level maths, before I left for other things. Even if wasn't though, why should anybody have to be mathematically orientated to participate in a thread? This forum's for everyone mate, not just elitist bastards.

 

Supergrunch explained it all pretty well. All you did was make me laugh at you.

 

 

 

 

 

 

 

By the way.... 0.999rec = 0.999rec

1 = 1

 

[Zakatu]

Actually, according to physics, you can reach the speed of light, but not surpass it.

 

Anyway, are you trying to think of something like the graph of root x- the gradient gets closer and closer to 0, but never actually reaches it?

 

This is a different case. If you don't that proof, then look at some of the earlier ones, such as the ones with thirds, or x=0.999rec etc.

 

sorry, but i thought it was only light that could travel at the speed of light.

 

*gets very muddled*.

 

I also just don't get why if something is 0.99999... for infinite, WHY that makes it equal to 1? to me thats an approximation. Sure, the difference between it and 1 is nigh on nothing, but there IS a difference. Therefore it is not one.