The whole 0.999...=1 argument (1 Viewer)

[z600]

A friend just showed me a post where people debate over whether 0.999...=1.
Here are some proves i found on various forums

1/3 = 3.3333333333
2/3 = 6.666666666666
3/3= 9.99999999

x = 0.999...
multiply both sides by 10

10x = 9.999...
subtract the first equation from the second

9x = 9
so
x = 1


You guys agree with this?

 

[Frigid]

what the hell are you on about?

1/3 =/= 3.3333333333
2/3 =/= 6.666666666666
3/3 = 1 =/= 9.99999999

 

[Affinity]

you got some obvious errors there, besides that can you tell me why 0.3333333.... = 1/3?

 

[z600]

I merely paste them off the net, its not my prove for the solution, look up 0.99..=1 in google.

 

[dom001]

1/3 =/= 3.333333, nor does it equal 0.333333.

0.33333 is merely a rounded representation of the rational number 1/3.

Yes, if you multiply that rounded figure by 2, you obtain 2/3 = 0.66666.

However, as you probably know, 0.666...6667 is a better approximation for 2/3.

It is for this same reason that the 'proof' of 0.9999 = 1 is absurd.

You cannot round a fraction, perform several operations on it, then expect to bring it to a different number.

So, 0.9999 =/= 1.

 

[Bank$]

Think of it like this 1/3 = 0.33333(rep)
u get this mumber [0.33333(rep)] if u divide 1 by 3 or 10 by 30, 20 by 60 and so on.

so therefore 1/3 = 0.33333(rep)

L.H.S (left hand side)
=1/3
1/3 * 3 = 1

R.H.S
=0.33333(rep)
0.33333(rep) * 3 = 0.99999(rep)

since L.H.S = R.H.S
therefore

1 = 0.999999(rep)


note: rep = repeater

 

[Trebla]

It's the whole concept of the limiting sum.
0.9999..... = 0.9 + 0.09 + 0.009 + ................
= 0.9/(1 - 0.1)
= 1
0.9999..... approaches 1 as the number of decimal places increases. It's almost like the behaviour of an asymptote in graphical terms. It's not merely an approximation.
Think about it. If 0.999999999.......... has infinite decimal places then it equals one because that is its limiting sum. The reason why many argue otherwise is because we do not see infinity as a definitive number. In theory it does equal one.
If you still don't understand then an appropriate analogy is the asymptote of the simple rectangular hyperbola y = 1/x. As x --> infinity, y --> zero. Therefore in theory, at the point of x = infinity (which is basically undefined) the ordinate would equal zero. The same applies for 0.999999......

 

[Trebla]

Haha, I didn't see you were in Year 10 going onto Year 11 lol...so you might not understand what I said above....

 

[Optophobia]

How does

3/3= 9.99999999

?

 

[aussiechica7]

ya i've thought about this. I think this was the kind of thing he was trying 2 say:

1/9 = 0.1111111111111111111111111111111111111111...

9/9 = 9 x 0.1111111111111111111111111111111111111111... = 0.9999999999999999999999999999999999999...

However 9/9 = 1.

Therefore 0.9999999999999999999999999999999999999... = 1.

 

[Slidey]

No it isn't, Brogan.

The x=0.33..., 10x=3.3..., take them away proof is taught as part of the 2unit course.

And yes, z600, you're right; 0.9... does equal one. Or at least, it has a limiting sum of one. It's all to do with base 10 representations of numbers:

0.9... is really 9/10 + 9/100 + 9/1000 + ... to infinity. You will later realise this to be a geometric series to infinity and you can sum it via the formula S = a/(1-r), here a = 9/10, r = 1/10, so we get S = 9/10/9/10 = 1.

Remember however that this is a limiting sum to infinity. Some would argue that a limit is not the same as the actual value. Whatever.

 

[pLuvia]

No it isn't, Brogan.

I think he's addressing the fact that z600 typed that 1/3=3.33333.. etc

 

[Affinity]

Oh no brogan's trolling this forum too now

 

[Forbidden.]

I don't feel like arguing about basic arithmetics when it can be solved in the classroom in several seconds.

 

[Optophobia]

2 / 3 = 0.666666667

 

[Templar]

2 / 3 = 0.666666667

Not quite. 2/3 does not have a terminating decimal representation.

 

[xclo]

this reminds me of a picture my friend showed me

 

[BrotherBread]

Think of it like this 1/3 = 0.33333(rep)
u get this mumber [0.33333(rep)] if u divide 1 by 3 or 10 by 30, 20 by 60 and so on.

so therefore 1/3 = 0.33333(rep)

L.H.S (left hand side)
=1/3
1/3 * 3 = 1

R.H.S
=0.33333(rep)
0.33333(rep) * 3 = 0.99999(rep)

since L.H.S = R.H.S
therefore

1 = 0.999999(rep)


note: rep = repeater

Sorry but that is not really correct. The way I see it is that 1/3 =/= .33333333.... to start with. 1/3 is merely a representation of .33333... . A symbol in society may represent a company it doesn't mean that the symbol itself is that company. By the above logic that you have displayed coupled with my POV (note it is just a POV I'm sure someone far more qualified, or even someone less, will dispute what I say, which is fine) shows that regardless of the manipulation of the values 3/3 =/= .99999...

 

[toadstooltown]

BrotherBread, while it isn't the most elegant proof it is correct. 1/3 is a naturally occuring number (no duh) but our representation it as a bas ten decimal (and many numbers of course) leads it to be a repeating decimal of 0.333333....etc.

There are several proofs, some more tangible than other but all are correct. 0.99999... = 1 isn't that too hard to comprehend, for the first time in maths you can kinda argue that it is 'close enough', in layman's terms.

 

[darkliight]

Re BrotherBread, both "1/3" and "0.333..." are just sets of symbols. Those sets of symbols both respresent the same real numbers though. In the same way, the sets of symbols 1/2, 0.5, 8/16 etc represent the same real number too.

Like everyone else said, 1 = 0.999... = 3/3 etc. Nothing mysterious.