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Author Topic:   A question of numbers (one for the maths fans)
kuresu
Member (Idle past 801 days)
Posts: 2544
From: boulder, colorado
Joined: 03-24-2006


Message 151 of 215 (326088)
06-25-2006 11:31 AM
Reply to: Message 150 by riVeRraT
06-25-2006 11:20 AM


Re: Talking about math
actually, Mod's equation makes perfect sense.

This is going to asssume you have no basic algebra skills, just for simplicity.

x=0.999999...
10x=9.999999...
10x-x = 9.999999... - 0.999999...
9x = 9

Therefore x=1

What you do on one side of the equation, you must do to the other side of the equation.

We establish that the variable, x, is equal to .999 . . .
If we multiply .999 . . . by 10, we get 9.999 . . .
If we subtract .999 . . . from 9.999 . . ., we get 9.

On the other side of the equation,
if we multiply x by 10, we get 10x.
Subtract x from 10x, and you will get 9x.

then, 9x = 9.

Now then, all that's left to do is find out what x is.
To do that, we divide the 9 (that's attached to the x) into the other 9, which gives us

x = 9/9, or x = 1.

So, 1 = .999 . . .


This message is a reply to:
 Message 150 by riVeRraT, posted 06-25-2006 11:20 AM riVeRraT has responded

Replies to this message:
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riVeRraT
Member (Idle past 207 days)
Posts: 5746
From: NY USA
Joined: 05-09-2004


Message 152 of 215 (326089)
06-25-2006 11:36 AM
Reply to: Message 148 by SuperNintendo Chalmers
06-25-2006 10:50 AM


Re: ack
Looking at the graph at the bottom of the first link, is always how I had pictured .999... in my mind. That is how I can conceive .999... becoming 1. The resolution of the infinite 9's becomes so small that it becomes 1.

My question is, why does .999... become 1, and .333... never be anything more that .333... ?

Or how about this.

In the equation 2+1-1=2 You can reverse the last 2 calculations, and still get the same answer
2-1+1=2, but here you can't:

∞ + 1 - ∞ = 0

∞ - ∞ + 1 = 1


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riVeRraT
Member (Idle past 207 days)
Posts: 5746
From: NY USA
Joined: 05-09-2004


Message 153 of 215 (326090)
06-25-2006 11:46 AM
Reply to: Message 151 by kuresu
06-25-2006 11:31 AM


Re: Talking about math
Yes, I took algebra, but rarely have to use it. So it is very rusty.
That's about as far as I got, as I had a rough childhood, and had to drop out and start working. So I do appreciate everything, that everyone is saying to me here. It's not like I don't think about mathematical concept's all the time. It is my favorite subject. I use math immensely in my work, and my hobbies, but not at calculus levels. I have been thinking lately of going to college, to get an engineering degree, it's never too late.

I invented parts of trigonometry out of necessity, because I am a master sheet metal layout mechanic. At one time, I needed to figure out what the other sides, and angles of a triangle would be, if I only knew one angle, and one length.

I had invented some formulas, but I can't seem to find them. I later learned that it was similar to sin, and cos.

Sheet metal fabrication is all about triangulation, and after doing for so many years, you gain a different perspective on the world. There is nothing I couldn't break down into a 2D pattern. I could cover you in metal.

Now, about the formula. If that formula works for 10x, shouldn't it work for 2x? also shouldn't the result be similar when we use .333...?

Why does .999... resolve to a 1, and .333... never resolve?


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kuresu
Member (Idle past 801 days)
Posts: 2544
From: boulder, colorado
Joined: 03-24-2006


Message 154 of 215 (326093)
06-25-2006 11:51 AM
Reply to: Message 152 by riVeRraT
06-25-2006 11:36 AM


Re: ack
∞ + 1 - ∞ = 0

I do believe that that should equal 1, not zero, but you might be right because adding one to infinity would give you infinity.

Well, let's take a look at your .333 . . . problem.

x = .333 . . .
10x = 3.333 . . .
10-x = 3.333 . . . - .333 . . .
9x = 3
x = 3/9 = 1/3.
1/3 = .333 . . .

wheras in the other equation, 1 = .999 . . .


All a man's knowledge comes from his experiences

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kuresu
Member (Idle past 801 days)
Posts: 2544
From: boulder, colorado
Joined: 03-24-2006


Message 155 of 215 (326094)
06-25-2006 11:56 AM
Reply to: Message 153 by riVeRraT
06-25-2006 11:46 AM


Re: Talking about math
10x is just easier to use. But I'll go ahead with 2x.

x = .999 . . .
2x = 2
x = 2/2, or 1.

x = .333 . . .
2x = .666 . . .
x = .666 . . ./2, or .333 . . ., or, 1/3.

.333 . . .does resolve, as a fraction. if you're looking to get it to resolve as a decimal, it will keep going for infinity.


All a man's knowledge comes from his experiences

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Replies to this message:
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riVeRraT
Member (Idle past 207 days)
Posts: 5746
From: NY USA
Joined: 05-09-2004


Message 156 of 215 (326095)
06-25-2006 12:01 PM
Reply to: Message 136 by Modulous
06-25-2006 9:55 AM


In 0.999 we don't lose the 9/1000. All that happens is that the 1000ths column becomes 0 because the digits are shifted up one column (in base 10). when there are an infinite number of 9s we don't put a zero at the end, but the essential operation (shifting the digits up a column) can still happen. This is part and parcel of the number system.

In wikipedia, what I am saying is considered wrong, but worthy of mention.

quote:
Some argue that, in the second step of the equation given above, 10x is 9.999...0 and not 9.999... but this is not the case: the right-hand side does not terminate (it is recurring) and so there is no end to which a zero can be appended.

http://en.wikipedia.org/wiki/Recurring_decimal

If we can't add a 0 to the end, then 2*.999... = 1.999... ?
And if so used in your equation, x still equals 1?

Even though 2*.999 = 1.998
You don't find a problem with this?

Shouldn't infinity and time be directly related to one another?
Could it be, that at the moment you multiply an infinite number by a finite number, it becomes finite?

I was thinking this all along, but hesitant to say this.


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 Message 136 by Modulous, posted 06-25-2006 9:55 AM Modulous has responded

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riVeRraT
Member (Idle past 207 days)
Posts: 5746
From: NY USA
Joined: 05-09-2004


Message 157 of 215 (326098)
06-25-2006 12:04 PM
Reply to: Message 155 by kuresu
06-25-2006 11:56 AM


Re: Talking about math
x = .999 . . .
2x = 2
x = 2/2, or 1.

2x=2?

That assumes .999... is 1.

That is wrong anyway.

I think it should look like this:

x = .999...
2x = 1.999...
1x = 1

Again, we are ignoring the last number, because there is no last number. That is what I find to be the problem.


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kuresu
Member (Idle past 801 days)
Posts: 2544
From: boulder, colorado
Joined: 03-24-2006


Message 158 of 215 (326099)
06-25-2006 12:14 PM
Reply to: Message 157 by riVeRraT
06-25-2006 12:04 PM


Re: Talking about math
THe original equation, where you multiply by 10 proves that .999 . . . is 1.

.999 . . . * 2 does not equal 1.999 . . .. It would make less sense if this were so.

Now, .999 * 2 is 1.998, because you are dealing with a non-recurring decimal.

x = .999...
2x = 1.999...
1x = 1

and you still end up with 1 = .999 . . . in that equation.
if x = .999 . . ., and if 1.999 . . . divided by 2 is 1, then x = 1 and .999 . . ..


All a man's knowledge comes from his experiences

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lfen
Member (Idle past 2965 days)
Posts: 2189
From: Oregon
Joined: 06-24-2004


Message 159 of 215 (326101)
06-25-2006 12:28 PM
Reply to: Message 134 by riVeRraT
06-25-2006 9:41 AM


You cannot prove infinity. I am 40 years old, and for the first half of my life, I believed infinity could exist. For the second half, I started to doubt it. There is a possibility that infinity does not exist.

It sounds like you are confusing concept with the universe. Infinity is a mathematical concept, just as a point and a line are concepts. A line is defined as having two dimensions. It has no width. You won't find actual lines out in the world that have no width. They don't exist. As far as I know at this point, well, at least for the known universe matter is assummed to be finite. There is a very large number of electrons but not an infinite number.

But we are talking about mathematics. Do you accept that between 0 and 100 there are an infinite number of numbers (you can call them points as well) and that between .99999 and 1 there are also and infinite number of numbers? Infinite is not a number it means that no matter how far you count, or how many times you divide the line you can still keep counting, still keep dividing, there is no end to it. Now in the universe there may be a limit to how small you can make a division but we are talking mathematics here, pure concept.

lfen

ABE: I'm sure they prove infinity in number theory every day. Some one here can do this better than I but the proof is along the lines of pick some number as large as you want, call it n. You claim there is no larger number. But by adding 1 to it we have n+1>1. If you say that is the largest well we make another number by adding 1 to that.

Edited by lfen, : after thought added


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nwr
Member
Posts: 5586
From: Geneva, Illinois
Joined: 08-08-2005


Message 160 of 215 (326102)
06-25-2006 12:29 PM
Reply to: Message 152 by riVeRraT
06-25-2006 11:36 AM


Re: ack
∞ + 1 - ∞ = 0

∞ - ∞ + 1 = 1


Neither of those actually make much sense. You cannot treat infinity as if it were an ordinary number.

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riVeRraT
Member (Idle past 207 days)
Posts: 5746
From: NY USA
Joined: 05-09-2004


Message 161 of 215 (326103)
06-25-2006 12:30 PM
Reply to: Message 155 by kuresu
06-25-2006 11:56 AM


Re: Talking about math
Ok, try it with .5x

x=3
.5x=1.5
-.5x= -1.5

That works.

x=.999...
.5x=.4999 (5?)... or maybe just .4999... let's use .4999...
-.5x = -.499
x=1.0020040080160320641282565130261
But we are ignoring the 5, can we do that?

x=.333...
.5x=.166...
-0.5x=.166

x=?

Isn't this proof you can't ignore the last number?

Shouldn't the 10x in mod's equation be
9.999...0 - .999... be 8.999...1

Edited by riVeRraT, : mistake


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riVeRraT
Member (Idle past 207 days)
Posts: 5746
From: NY USA
Joined: 05-09-2004


Message 162 of 215 (326104)
06-25-2006 12:33 PM
Reply to: Message 159 by lfen
06-25-2006 12:28 PM


just as a point and a line are concepts.

Concepts exist in time.

Do you accept that between 0 and 100 there are an infinite number of numbers

I do not accept it, or not accept it. I think about the possibilities.

Infinite is not a number it means that no matter how far you count, or how many times you divide the line you can still keep counting, still keep dividing, there is no end to it. Now in the universe there may be a limit to how small you can make a division but we are talking mathematics here, pure concept.

A concept that has no end. So if the last number of a infinite numver has to change, then we have a problem houston.


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riVeRraT
Member (Idle past 207 days)
Posts: 5746
From: NY USA
Joined: 05-09-2004


Message 163 of 215 (326106)
06-25-2006 12:37 PM
Reply to: Message 158 by kuresu
06-25-2006 12:14 PM


Re: Talking about math
and you still end up with 1 = .999 . . . in that equation.

Yes, I know, I was pointing out your mistake.

2*.999... = 1.999...8
There is no last number, so we drop the 8 or never put it there.

Just like 10*.999... = 9.999...0
We drop the zero, or we never put it there, because there is no end.

This is opposite of what I am trying to point out. But considered the correct way.


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lfen
Member (Idle past 2965 days)
Posts: 2189
From: Oregon
Joined: 06-24-2004


Message 164 of 215 (326107)
06-25-2006 12:43 PM
Reply to: Message 162 by riVeRraT
06-25-2006 12:33 PM


So if the last number of a infinite numver has to change, then we have a problem houston.

Two errors here. First there is no such thing as an infinite number. There are infinite sets, infinite series, etc. But infinity is not a number.

Two the last number of an infinite number doesn't make sense. If you are referring to the last digit of a decimal representation of pi for example there is no last digit. There is not a last digit to .999... If there were a last digit it wouldn't be an infinitely repeating decimal, it wouldn't be an infinite series, it would be finite, that is to say countable.

So there is no last digit, so the last digit, which never existed in the first place, can't change. There is no problem. The calculus works. You can cross a room.

lfen


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NosyNed
Member
Posts: 8863
From: Canada
Joined: 04-04-2003
Member Rating: 4.6


Message 165 of 215 (326108)
06-25-2006 12:44 PM
Reply to: Message 153 by riVeRraT
06-25-2006 11:46 AM


Resolving
Why does .999... resolve to a 1, and .333... never resolve?

They both resolve.

.9 is not equal to 1.
.99 is not equal to 1 but it is closer.
.999 is not equal to 1 but it is closer yet.
.9999 is not equal to 1 but it is still closer.
.999999999999999999 is not equal to 1 but it is very close indeed.

.9... (the ... means an endless row of 9's. We need to be VERY careful with our notation) IS equal to 1 the difference is now zero.

.3 is not equal to 1/3
.33 is not equal to 1/3 but is closer.
.333 is not equal fo 1/3 but it is pretty close.

.3... is equal to 1/3

You may think there is a diffence because the first case is a whole number (1) and the second is a fraction (1/3). That doesn't make a difference to what we are talking about.


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