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

.
1 - 9/10 = 1/10
1 - 99/100 = 1/100
1 - 999/1000 = 1/1000
. . . .
1 - 9999.../10^infinity = 1/10^infinity = 0.0

as the number of nines increases the difference from one decreases.

As that goes to infinity the difference approaches zero.

At infinity that difference/quantity is zero.

in decimal
1.0 - 0.99 = 0.01
1.0 - 0.999 = 0.001
1.0 - 0.9999 = 0.0001
etc.
at infinity difference 1.0 - 0.99999..... = 0.000000 .... infinite zeros .... 00001

I did a project in a 7th grade science fair in sherlock holmes style asking what happened to the missing 0.0000....00001, so i can empathize with your question.
time must have made me comfortable with it though. Maybe some study of infinite series helped too.
.

Owww my brain...let's just file math away until 5th period tommorow. I agree with the statement that we cannont grasp infinty...I mean try to imagine a number that is so unimaginaby small that no matter how far you extend it, it never quite reaches 1

[edit on 3/24/2005 by UnMature]

Originally posted by UnMature
Owww my brain...let's just file math away until 5th period tommorow.

I agree with the statement that we cannont grasp infinty...I mean try to imagine a number that is so unimaginaby small that no matter how far you extend it, it never quite reaches 1

Try imagining what happens when you finish counting all the integers, and keep going past them. You end up coutning an infinite number of infinite numbers.

To answer the question in the topic, which I missed, no. No it is not flawed, for all the reasons put forth here. All it means is that things behave oddly, which is the least of our problems.

Sounds like the old half the distance question to me. Ask the engineer and the mathematician the same thing: If the wall is 1 unit of distance away and with each iteration of the process you go .9 of the distance (then .9 of the distance left) to the wall, will you ever reach the wall? The mathematician says, no, you will never reach the wall. The engineer says, you will get close enough for all practical purposes.




[edit on 26-3-2005 by Icarus Rising]

Originally posted by Icarus Rising
Sounds like the old half the distance question to me. Ask the engineer and the mathematician the same thing: If the wall is 1 unit of distance away and with each iteration of the process you go .9 of the distance (then .9 of the distance left) to the wall, will you ever reach the wall? The mathematician says, no, you will never reach the wall. The engineer says, you will get close enough for all practical purposes.

[edit on 26-3-2005 by Icarus Rising]

Your engineer's answer reminds me of this other one on a classical theme:

The optimistic guy will say that this glass is half full, while the pessimistic will consider it half empty. But the engineer will think that the glass is simply twice too big...

Ahh, I love those jokes. So much.

Anyway, kind of like Zeno's Paradox. Basically states that you can't get anywhere, because in order to get somewhere, you have to go half way. Then, before you go the remaining half, you half to go half of that. And so on and so forth.

Calculus solved all those problems, with infinite sums converging at a finite number i.e. the distance.

Ahhh. But a limit is not a destination, and that is the differential.

Or, as my complex variables prof used to say (at 815am, FGS), when you go to look outside, it makes a difference if you stick your head out, and then open the window, or open the window, and then stick your head out.


peace out

Originally posted by Icarus Rising
Or, as my complex variables prof used to say (at 815am, FGS), when you go to look outside, it makes a difference if you stick your head out, and then open the window, or open the window, and then stick your head out.

That's the definition of wise, I believe.

But, if you take the limit as the sum of the numbers approach 0, you'll get a finite number, which in a physical sense such as the one you gave, is the distance.

The half thing reminded me of the half-life concept. When the parent material becomes daughter material basically half turns every certain amount of time. This keeps happening, but it's never completly daughter.material. This is why radioactive material will rot away in their secure vaults for eternity, there's always going to be some amount of material radioactive, whether it is 0.00005 or0.0000000000000000000000000000000000000000000000005.

Originally posted by BluePostman
The half thing reminded me of the half-life concept. When the parent material becomes daughter material basically half turns every certain amount of time. This keeps happening, but it's never completly daughter.material. This is why radioactive material will rot away in their secure vaults for eternity, there's always going to be some amount of material radioactive, whether it is 0.00005 or0.0000000000000000000000000000000000000000000000005.

Sort of. Except with half-lives you do eventually run out, because you run out of atoms. In math, you never run out of decimal places.

I love it when you prove my point for me!

Originally posted by Icarus Rising
I love it when you prove my point for me!

Except you said

for all practical purposes

With both the half-life example and the ladder example, it does, in fact, reach the number completely. The math reference was referring to the fact that if you took a number and halved it, you'd go to infinity.

0.9r is an irrational number, it can't be expressed as a ratio, as three thirds equals 1.

When we talk about 0.3r, we mean 1/3. It repeats because there's the tiniest little bit, that multiplied by 3, the little bits add up to form the little smidgeon that's in between 0.9r and 1. Granted there isn't anything in between it, but still... =\

I still think we should do with it what we did with leap years. That'll fix that lousy decimal system.

Originally posted by Xar Ke Zeth
I still think we should do with it what we did with leap years. That'll fix that lousy decimal system.

Every four years add a day to february?! How does THAT help?

Originally posted by Amorymeltzer

Originally posted by Xar Ke Zeth
I still think we should do with it what we did with leap years. That'll fix that lousy decimal system.

Every four years add a day to february?! How does THAT help?

Erm... How doesn't it help? *Looks around*

I wasn't being entirely serious, but it'd sure put an end to such things if it was applied to these numbers in the same general idea.

I'm going a little off topic here, but all maths is linear. If only there was such thing as Cyclical maths, something might happen. Unfortunately I did a Google search for that exact phrase, and nothing came up.

What, exactly, was your suggestion? You didn't really say, except make a reference.

all maths is linear

How is all maths linear?

This 'problem' can easily be solved, as has been said, using calculus and that fact that infinite sums have finite results.

I got lost in thought, and thought about something like what we do with a leap year, but yeah, I don't really know much from there.

I haven't done calculus, so I'm just going by on the knowledge I currently have. All real numbers can be represented on a line, complex numbers on a complex line, etcetera. I'm not sure if cyclical maths is even possible, it was just a thought I came up with. I'm not that particulary talented in the mathematics field.

Edit: Actually imaginary numbers would be on an imaginary line, and complex numbers would be in a complex plane. =\

[edit on 28/3/05 by Xar Ke Zeth]

Ah, you mean like a clock system maybe? It would 'solve' the problem, I believe, because you can't ever arrive at it.

Clock systems are stupid...

[edit on 3/28/2005 by Amorymeltzer]

Good point, it'd just be neglecting the problem. But maybe that problem only exists with our decimal system?