1/10

Not a troll, I believe that's true in math.

Or I'm wrong, pretty stoned right now.

Over 160 replies man. Unoriginal but effective!

: D

Lethe did the job for you.

True

Don't you mean 0.99.../10?

LALALALALA! CAN'T HEAR YOU!

http://www.youtube.com/watch?v=O-Y-ua3WBi4

r u sum kinda SUPER GENIUS?

There is no number between them. Because of the density of real numbers, between any two different numbers there is an infinite amount of different numbers. Hence, 0.(9) = 1.

1/3 + 1/3 + 1/3 = 3/3 = 1

1/3 = 0.333333...
0.333... + 0.333... + 0.333... = 0.999...

0.9999... = 1

But now we have to prove that 1/3=0.333333....

1) Divide 1 by 3
2) Get .333333...
3) ???
4) profit!

But what divide means?

To separate into parts.

doesn't work

It does - 1/3

1/3 is a fraction.

That's called division.

x=0.999...
100x=99.999...

we subtract the first equation from the second

100x-x=99.999...-0.999...
99x=99

x=1

yea I do remember this is what I was taught in school

both are correct

vedic's one doesn't actually do anything, it just reduces problem to proving that 3*0.(3)=1 which is essentially the same problem as the original one. If you can derive that 1/3=0.(3) then you can use same method to derive that 1=0.(9) without doing any intermediate steps involving 1/3.

you're right, it kind of solves itself using the data it's supposed to prove

prly because 0.(9) is just a mathematical construct anyway

just as kg explains it is nearer to 1 than anything else, indistinguishable even so the mathematical language works out

It's a valid proof. The whole 0.999 = 1 thing is just really an issue of the decimal base.

You are essentially solving the same problem twice.

But you still cannot write that 1 == 0.999...
:)

Yes you can, they are the same number.

Thanks for educating. :)

Seriously.

http://en.wikipedia.org/wiki/0.999...

technically 0.99999.... is not a number. its an infinite series.

I don't see how it's not a number. It can be expressed as a sum of an infinite series, but that doesn't stop it being a number.

How to you know that such arithmetic laws apply to infinite strings

Those infinite strings are just shortcut notation that we use for infinite sums like a_0+a_1*(10^-1)+a_2*(10^-2)*a_3*(10^-3)+...
I.e. 0.99999... means 0+9*(10^-1)+9*(10^-2)+9*(10^-3)+...
Those arithmetic laws can be rather trivially proven for such sums.

You cant just throw around plusses and minuses willy nillally!!!!!! The numbers go on fooooooooooooooreeeeeeeeevvvvvvvvvaaaaaaaaaaaaaaaaaaaaaaarrrrrrrrrrrrrrrrrrrr

You can also view in from another angle (which is perhaps more intuitive):

Between any two real numbers there's an infinite amount of other numbers (to construct them just take averages, first of the two staring numbers, then of the first number with the first average etc.).

We take the contraposition of that (i.e. instead of "two different real numbers imply that there is an infinite amount of numbers between them" we take "if there are no different numbers between the two starting numbers, the starting numbers are not different").

We can clearly see that's the case. You cannot construct a number closer to 1 than 0.99... (because at each point you already have a 9 in place, it can't get any closer).

Well you can prove anything you want using that sort complicated stuff. 0.99 gets closer n closer to 1 but it can't ever reach it. it's like ta;lking half of somethin forever it gets closer to 0 but never quite. its called LIMITS

http://en.wikipedia.org/wiki/Limit_%28mathematics%29

1 is the limit of 0.99999999999999999999999

To talk about limits you need a sequence.

If your sequence is a_0=0.9 ; a_1=0.99 ; a_2=0.999 and so forth then indeed as n approaches infinity then a_n approaches 1.

We, however, are talking about the number 0.99.. with an infinite amount of '9's after the period. It is equal to 1.

e:

Well you can prove anything you want using that sort complicated stuff.

That's just a bit silly.

The mistake you are making is thinking that the amounts of 9 is growing (forever) and therefore getting closer to 1. It is not an amount that keeps increasing. The amount of 9's is infinite at all times.

I think your problem is to comprehend infinity.

0.99 gets closer n closer to 1 but it can't ever reach it.

No.
You are implying that 0.999.. is some sort of running process in progress, but it's not a process, it's a constant number, it doesn't change, so you're assuming completely different thing when trying to imagine what 0.999... is.
0.999.... is what we get AFTER the process you thinking about is finished. I.e. it IS the limit, not anything "in progress", it doesn't get anywhere, it's what we get when we are ultimately finished adding all the infinite amount of nines at the end.

You can't finish adding an infinite number of things.

You can't do the process, but you can know what the result of it will be.

or
9*1/9
9*0.1111111111111111

:ASD

Retard alert!

something about this reminds me of Zeno's Paradox.

i like numbers

i like numbers too, want to share your numbers with me?

hahaha

bind "h say nice1"

h

nothing last forever, your lagic is flawed.

bencha nurds

I can bench 2 nurds...







per side

So if I write a conditional statement in c++ like
if (1 == 0.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999)

what will happen?

Actually I tried it:

#include <iostream>
int main()
{
long double a = 0.3333333333333333333333333333333333333333333;
long double x = a + a + a;
std::cout << x << std::endl;
}

Output: 1

clearly hax

Precision isn't that awesome.

If I run this on my sparc machine long double will be quad precision instead -_-

erase a 3 from a.

So it's impossible to divide something by 3 and have equal pieces.

Take pizza, divide to 3 persons, 0.0000...1 pizza still left?

Get back to making edawn rulesets. :@

The difference between 0.(9) and 1 is infinitely small. In other words, there is no difference.

The difference is 0.

or is it 0.000...1?

It is the same number. The difference is 0. The nines never end, so you can't add 0.0...1 more.

When you're adding 0.000...1, what decimal place is the 1 added in an infinite number? If the 0.999... is infinite, there is no final place to put the 1. Any place before the "end" would result in you getting 1.00...999999....

That is not correct. There is no difference at all, not even an infinitely small one.

Infinitely small is the same as no difference at all.

False.

the issue is with how people visualise real numbers (especially those with an infinite decimal expansion). Instead of thinking of them as actual numbers, visualise a number line, with the integers (whole numbers) labeled on it. Every point on this line corresponds to a real number, and two numbers are equal if and only if they lie on the same point.

Now lets try to label this line with decimal representations of numbers.

Its fairly trivial to label numbers with finite decimal expansions, i.e. 0.125, you can just divide the relevant unit section up into 10^n equal parts (where n is the final decimal place) and count along until you reach your number.

We can label rational numbers (fractions) in the same way. If our number is p/q, we split each unit up into q equal gaps and count along p.

But how do you label a point with an infinite decimal representation? It has to be a single point, (i.e. it cant "get closer" to anything, it is only one number) and it has to make sense, i.e. if you truncate it to n decimal places, your infinite number has to be reasonably close to this, and be closer for each decimal place you add.

The way we do this is by taking the limit of the sequence of truncated decimal numbers. This satisfies both properties, and it can be shown that this limit is the only number consistent with these premises.

In the case of 0.333... this sequence is 0.3, 0.33, 0.333, 0.3333, ...

The limit of a sequence is a unique number such that eventually, every term in the sequence is arbitrarily close to this number.

We can rewrite the above sequence as :
3/10, 33/100, 333/1000, ..
or
1/3 - 1/30, 1/3 - 1/300, 1/3 - 1/3000, ..., 1/3 -1/(3*10^n), ....


It is easy to see that each term gets closer and closer to 1/3, and, if you chose n large enough, you can make it as close to 1/3 as you like (i.e. arbitrarily close), so we say the limit of this sequence is 1/3,
This means that 0.333... = 1/3, by our definition, so we can label this point by dividing each unit into 3 equal segments, and counting one along from zero. This point can be labeled 1/3 or 0.333...

Now apply this to 0.999...

the sequence is 0.9, 0.99, 0.999, 0.9999, ....

We can rewrite this as 1-1/10, 1-1/100, 1-1/1000,..., 1-1/10^n, ...
and we can see that if we choose large enough n, it is arbitrarily close to 1, so the limit of the sequence is 1, so 0.999... = 1.

This means the point 1 on the line has two different labels as a decimal expansion, 1, and 0.999... and this is consistent with our definitions. They are the same number written in different ways.

This applies to all other terminating expansions in the same way, e.g. 0.125 = 0.124999.....


Reading it back, this probably doesnt make much sense for someone who hasnt studied maths and already knows this :(
I was trying to explain why this works, since the common proofs arent really satisfying IMO.

This means we can label the point on our line with 0.333..., or 1/3 and 0.333... = 1/3.

OH YEAH!? WHY!?

Because we agreed that notation 0.3333.... represents a number that is limit of sequence 0.3, 0.33, 0.333, 0.3333, ... and that limit is 1/3

what the fuck is wrong with you people

so it really isnt 1 but since the 9 goes forever we are just gonna sit here and say it is 1.

yeah bs.

and that bitch didnt kill her daughter.

it really is 1 but since the 9 goes forever, some not very clever people are just gonna sit here and say it is not 1.

You can't say it's "not very clever people". This isn't exactly intuitive and without some background in mathematics isn't that easy to grasp.

I'm not talking about people who are not familiar with mathematical concepts, I'm talking about some people who violently argue that mathematicians are wrong before even trying to understand the issue.

lol @ this thread and kg. perfect thread for "wtf is this i dont even ....".

no wonder people hate math. idiots making stupid ass explanations.

I'm sorry your nitpicking didn't work out.

if you do 0,333333333333333333... x 3, no matter how many dots are there, you'll never gonna get 1 as a result, so by basic math rules that can't be true

6/2 = 3 reverse 3 x 2 = 6

0.33333.... by definiton of that notation is limit of sequence 0.3, 0.33, 0.333, 0.3333, ..., so it is 1/3. If you multiply 1/3 by 3 you'll get 1.

1/3 is a third of something, one third of whatever. that's why it's presented as 1/3. a third part.

decimals is something different and can't work that way.

0.333333333333 is not 1/3.

the 3s go to infinity. so that part, that difference, becomes smaller and smaller, to infinity. but it *never* ceases to exist, it just becomes small beyond your point of comprehension. no matter how small something is, it exists. and as long as that difference exists, no matter how small it is, it can't be 1, cause there's something left in each of the 0,3333333s part.

This is gonna be fun.

decimals is something different and can't work that way.

decimals are numbers, just as 1/3. One number can be represented in many ways. For example, 0.5 can be represented as 1/2 or, for example, in binary system it's 0.1. Those are only different symbols that mean the same number.
Just as well, you can represent 1/3 as 0.333..., it's still the same number, because by definition of decimal representation of real numbers, by symbols 0.333... we represent the number that in fractional representation is written by symbols 1/3.

the 3s go to infinity. so that part, that difference, becomes smaller and smaller

When you think of 3s going to infinity, you are thinking NOT of the number 0.3333..., you are thinking of the sequence 0.3, 0.33, 0.333, 0.3333, 0.33333, ...
That sequence is NOT the number. It's a sequence. Sequence and number are completely different things altogether. To get a number from our sequence, we use operation of taking a limit, and limit is by definition a point to which we can get as close as we can with our sequence.
So when you say:

that difference, becomes smaller and smaller, to infinity. but it *never* ceases to exist, it just becomes small beyond your point of comprehension.

In mathematics terms your words mean that at some point the elements in aforementioned sequence will become as close as we want to 1/3, and that means (by definition of what limit is) that 1/3 is limit of that sequence. And that limit is the number that we in mathematics agreed to notate by our infinite list of decimal digits, not anything else. Just because we defined our decimal notation that way.

1/3 is not a number, 1/3 is another way to write down 1:3, the same as 4/9 is not a number, it's a fraction, something you NEED to calculate in order to GET a number. 10/2 IS five, when you calculate 10/2. 1/3 is not possible to find, cause there is no number that you multiply with 3 to get the result of 1.

the decimals can go as far to the right as you want them to.

become as close as we want

don't know what this means, noone forbids you to say 3 x 0,333... is 1, anyway. for me it's not, it's the same as saying 4,498 = 4,50 cause it's easier to calculate later on.

it's a fraction, something you NEED to calculate in order to GET a number

If we calculate 1/3, what number we will get?

1/3 is not possible to find, cause there is no number that you multiply with 3 to get the result of 1.

Is it possible to find 1/2?

If we calculate 1/3, what number we will get?

my point exactly.

Is it possible to find 1/2?

yes. that's 0,50 and it stops there.

If we calculate 1/3, what number we will get?

0.333...

... is not a part of mathematics

An ellipsis is also often used in mathematics to mean "and so forth". (c) wiki

ahhh (c) wiki terribly sorry lol.

1) math needs to go both ways.
2) if it doesn't go both ways, it's not correct.
3) if 1/3 = 0,33333..., it means that 0,33333... x 3 must give 1. though it doesn't cause 0,33333... is not a number, it's nothing.


4) the human need to define an answer even in places where it can't be defined is well known, the brain works that way, but i find it hilarious that a certain guy ^^ has linked a certain youtube video a couple of days ago which could perfectly be applied on this very example. he'll know.


anyway what's the difference between 1/3 and 1/6? do you also use dots in a "let's say" way?

1) math needs to go both ways.

What about taking the absolute value of something? How does that go both ways? What's the inverse function of abs(x)?

3) if 1/3 = 0,33333..., it means that 0,33333... x 3 must give 1. though it doesn't cause 0,33333... is not a number, it's nothing.

0.33...*3=1

anyway what's the difference between 1/3 and 1/6?

Exactly 1/6.

do you also use dots in a "let's say" way?

In mathematics the ellipsis is used only in the way I mentioned afaik.

ellipsis is just another way of saying "this goes on forever, let's just wrap it up." and doesn't make it mathematically correct.

No, it means and so forth. I.e. that something follows a pattern established in earlier examples.

0.33...*3=1

take a piece of paper and feel free to prove me how that's right.

0.33...*3=1

take a piece of paper and feel free to prove me how that's right.

Apply what kg said earlier.

x = 0.333...3...
10x = 3.333...3...

We substract the first equation from the second:

10x - x = 3.333...3... - 0.333...3...
9x = 3
x = 3/9
x = 1/3

We multiply both sides of the equation by 3:

3 * x = 3 * 1/3
3 * x = 1

QED.

i'm not interested in dots, that looks like a child's been playing with a pencil...

Can you find anything wrong in the demonstration I just presented you with? If not, kindly stop talking.

yeah, the dots, they are funny.

this is not the wheel of fortune.

so please stop boring me with that crap.

At this point you're really either trolling or a complete idiot, and in both cases you end up being an idiot.

if that means you're finally gonna shut the fuck up, thank god.

You asked for a demonstration and you got it. Either show me what's wrong in it or stop talking.

i'll try.

so you divide the number 1 into 3 parts.

1st part) 0,333333333333333333333333333333333333333
2nd part) 0,333333333333333333333333333333333333333
3rd part) 0,333333333333333333333333333333333333333

(we don't have infinite space at our disposition, obviously.)

so here's the thing. no matter how many 3s do you put in each and every line (and it goes to infinity), it's still a digit 3 in the end of every line, it doesn't change. it's not 2, it's not 4, it's just digit 3 repeating itself. infinity or not, it's still a line of 3's. 3 lines ending in 3's will always produce a combined result of a 9, and you kinda need a 10 in total to get a result of 1. no matter how long it is, you will always miss that part. so it's irrelevant how long it goes.

Congratulations, you just failed to prove my demonstration wrong. Look at my demonstration and show me which step I got wrong.

You really do seem unable to understand the meaning of infinity.

erm, you draw dots, which are another way of saying "this can't be calculated, let's just give it a name and wrap it up".

i just logically explained how your theory is crap. and it ends there, hopefully with you understanding it.

just go back to comment 120, to infinity, if needed.

Dots are a way of visually representing the rational number in question. One can also write 0.(3). In fact you used the dots yourself when you asked for a demonstration, something I just provided you with and that you are unable to refute mathematically. Comment #120 is you displaying your lack of understanding of what infinity means.

so let's see.

0.33333....3..... x 3 is 1

well, i'm gonna say that 0.33333....3..... x 3 equals 0.99999999....9.....

that must be true.

Yes, that is true. 1 and 0.999...9... are the same number.

(you still haven't found anything wrong in my demonstration btw)

nope, they are obviously not the same number.

please don't try to explain infinity btw, it makes you look stupid.

Yes, they are. You can even go take a look at kg's demonstration earlier in the thread. Feel free to find anything wrong with what he said.

are you trying to prove that it became "accepted" or to prove that it's logically correct?

It is mathematically correct.

#120

ah yeah, or if you're implying that math doesn't need to make sense, we can wrap it up.

Stop referring to your comment #120 which does not in any shape or form prove that it is mathematically incorrect to assert that 0.999... is 1. Address kg's demonstration if you're going to attempt to prove it wrong mathematically.

#120 is a simple logical way to show how 3 line of 3's can't result in a total of 1. flawless.

No, comment #120 is you displaying your inability to grasp infinity. In fact, you even wrote "3 lines ending in 3's", when those lines would precisely not end because they're infinite.

If you're going to assert that 0.999... = 1 is mathematically incorrect, you need to show that something is mathematically wrong in the demonstration that kg provided earlier in this thread. I'm all ears.

that's not even a demonstration, that's just a couple of =='s missing a / inbetween.

about that infinity thingy, if it confused you.

to infinity = without an end, the 3's just repeat themselves without changing, without, stopping, ever. right?

How is this not a demonstration? Which step is, according to you, wrong?

Nothing confused me in what you said, I was just pointing an indication of your inability to fully grasp the concept of infinity. Your post is completely irrelevant anyway, since you're trying to argue that 0.9999... != 1, but all you do in your post is explain that if you do 0.333... + 0.333... + 0.333..., you get 0.999..., and I'm not disputing that. You indeed do get 0.999..., the point is that 0.999... = 1.

again the dots. for fucks sake.


look, i'm not gonna prevent you from accepting stuff that makes no sense. if some math freak spent months and months and moths of his life battling with all those 3's before his mind exploded and screamed FUCK IT I SAY IT'S 1, that's fine. go ahead.

i just don't accept stuff that makes no sense.

good night.

It makes perfect sense if you understand basic maths. Again, feel free to point out anything wrong in the demonstrations when you come back.

edit: and the dots are a way of representing an infinite number of 3's instead of writing

"0,333333333333333333333333333333333333333
(we don't have infinite space at our disposition, obviously.)"

like a complete moron, so I don't know wtf your problem is with that.

I don't intuitively understand that 0.99..=1, but when taking into account the density of real numbers, it becomes much easier to grasp.

Yes it is. Have you seriously never heard of rational numbers? Rational numbers are also all real numbers. This is basic stuff that you learn in school...
1/3 and 0.3333333....3..... both represent the same rational number.

yes. that's 0,50 and it stops there.

But let's say we're talking to a Martian with 3 fingers who uses base 3 for writing numbers instead of base 10 that we use.
We ask him: what is 1/3? And he answers: hey it is 0.10, and it stops there.
Then we ask him: what is 1/2? And he will say: well, it is 0.1111111.... and OMG there are infinite ones, I cannot compute that number!
So how come that for one person number 1/2 exists and 1/3 does not, but for another person with different cultural background, it's completely opposite: number 1/3 is perfectly legitimate and has finite representation, but 1/2 on the other hand has infinite digits and it's not a number at all according to you?
We generally want numbers to be objective and independent of whoever uses them, because in that case we can be assured that if you get some result (i.e. compute some value), then that result holds for everyone else, everyone can use it then, not just you, so one person can enrich knowledge of the whole humanity, and therefore only with objective numbers it's possible to make progress in science. In your concept of "numbers", some people will have "numbers" different from other people. It's not very useful to have "numbers" like yours, don't you agree?

when the martian appears with his own way, i'll be happy to say he's right.

So, does number that is result of calculation of 1/3 exist or not? If it does, then what number it is?

nope, in our math, it doesn't. but luckily for us, we don't need infinite number of decimals for anything, so i don't see why it bothers you so much, a couple is more than enough.

So what is number in your sense?

i just said it doesn't exist. there is no "in my sense". in this mathematical system it's not possible to find a number that you can multiply with 3 and get 1 as a result. you can play with dots etc, as a way to get around it, but it still doesn't exist.

i can live with that.

No, I'm asking what do you mean by a "number" in general. What do you call a number?

in this example, it's a final product of a mathematical equation that goes no further.

12124 x (25 + 12 +15) - 1252 is indeed a number, in the end.

so if you take 1/3, and multiply it 3 times, you'll obviously get 3 thirds, and that is 1. case closed.

but 1/3, as a math operation, means you're dividing 1 into 3 parts, and our way of doing math finds no final result to that.

so it's all good if you ignore the part where 1/3 means you divide that "1" into "3" parts.

How do you know that something is a final point?
Why 0.5 is the final point, but 1/2 is not?
Why 0.(3) cannot be the final point of some "mathematical equation"?

you could have just said "our math is maybe not the only math and nothing is certain" in the very beginning instead of pulling that now.

anyway

It's not really "our math", it's your math, because in the math that is generally used, numbers have pretty much nothing to do with what you described as numbers.

that's not necessarily math, it's logic. or common sense, if you wish.

I cannot see it as common sense to differentiate between 1/2 and 0.5, for example. Or, for example, that for some people there is number that equals to 1/2, while for other people there isn't. Where is sense in that?

1/2 is a fraction, and means 1 divided in 2 parts.

0.5 doesn't have that task. it just is.

Why can't I say that 0.5 means 0*10^0+5*10^(-1)=5/10=1/2 and 1/2 "just is" instead and doesn't have any task of being digits multiplied by powers of ten?

But using a different base (and they're all equivalent) you get that 1/3 "just is" and 1/2 doesn't exist.

ok lets put it this way, in "your math", with no infinite decimals we can only write fractions with denominators whose only prime divisors are 2 and 5.
In "our math", with a logical definition* of an infinite decimal, we can write any rational number (fraction).

Guess which one we use.

* Your issue seems to be that we have to "define" what the dots mean, and we are "defining" 0.999... = 1, and it doesnt ultimately make sense.

But this isnt so, this is the only logical definition for an infinite decimal (i.e. any other definition wont make sense unless it gives the same answer), in the same way we have to define 1+1 = 2, and 1*x = x. There isnt any maths you can do to derive it, it is just obvious, so we define it to be so. You obviously dont know enough maths to understand why, or even understand the definition, and noone is going to explain it to you properly and rigourously on ESR. I had an attempt to squeeze the related maths into one post above, but none of it was explained properly, or it would be far too long.

This has gone from a quest for knowledge, and looking to be proved wrong, to assuming you are right, and refusing to believe anyone who knows what they are talking about. If you want to go any further, read a first year undergraduate mathematical analysis textbook (heres one that doesnt assume too much prior knowledge), then you will understand the definition we are using for our infinite decimal completely, and why it makes sense.

YES IT DOES. Did you not read my post? 1/3 and 0.3333333....3..... both represent the same rational number. Look it up on wikipedia or elsewhere if you've never heard of rational numbers.

please fuck off with dots. there's a reason why i'm ignoring you.

thanks.

Do you know what a rational number is?

edit: here is how merriam-webster defines a rational number: "a number that can be expressed as an integer or the quotient of an integer divided by a nonzero integer".

Is 1 a integer? Yes. Is 3 a nonzero integer? Yes. Is 1/3 the quotient of an integer divided by a nonzero integer? Yes.

it doesn't matter how you call it, christ, the names are irrelevant.

1/3 x 3 = 1. that's clear.

cause 1/3 stands by itself, as an item.

but further in, 1/3 is a math operation, and it has no final result, cause it fails to satisfy a simple rule of math.

What simple rule of math?

a x b = c > > > > c / b = a

1/3 * 3 = 1 >> 1/3=1/3

??

This is basic math. 1/3 and 0.333...3... are two expressions of the same rational number. A rational number can have its decimal expansion either terminate after a finite number of digits or begin to repeat indefinitely the same finite sequence of digits. In this case there is one digit repeating, "3".

awesome.

why is the digit 3 repeating itself?

I'm not sure you understand what you're asking. If you want to read more about repeating decimals, read this.

Quite simply because base 10 has no room for any other number - it's a result of the decimal system, and will happen in any base with different numbers.

Try to think of it a different way: distance. Consider the numbers 0.9999... and 1. What is the distance between 0.9999... and 1? What would you have to add to 0.9999... for it to get to 1? There is no answer, as the 9 will repeat forever. Even if you add a 1 to the hundred millionth millionth millionth decimal place, there will always be more 9's that continue after it, and you will end up with 1.0000...9999.... - clearly passing your mark of 1. Ergo, the distance between 0.9999 and 1 is 0, and they occupy the exact same place.

ahh stop /threading :/ Lethe was doing well :D

Never taken calculus, eh?

very nicely put.

become as close as we want

we define 0.999... as the limit of 0.9 ; 0.99; 0.999 ; ...

that's what as close to 1 as we want means. the number 0.999..=1


also, a fraction is also a number. doesn't matter if you write it as 10/2 or 5.

alright, so is 7 x 6 a number?

Of course.

which one?

each one that belongs to the equivalence class of the relation "=" (if I got the terminology in english correct)

i.e. 42, 43-1, 7*6 are the same number

yeah, in the end, they are.

so 7x6 is 42, cause 42/6 = 7

that's how it goes.

This relation "=" is between numbers. To even write 42=7*6 you have to agree that 7*6 is a number.

1+1=2

HEY GUYS AM I COOL NOW LIKE U

you idiots

math doesn't exist.

It's all a lie

1+3+3=7.

I think the real issue we are all avoiding here is 'does objective morality exist?'

Of course.. not.

:ASD

counting up

someone introduce the limit thing ,I forgot how to use, to lethe. He obviously didn't learn about that in school, and ofc he's not gonna understand something when he's ignorant on the matter. doesn't seem to keep repeating 0.33... thing as he is confused about it's math meaning is not the same when we use it in writing.

should have something to do with this iirc:
http://en.wikipedia.org/wiki/Limit_of_a_sequence

At least three different people already pointed this out, if he wants new knowledge I'm sure he will check it out.

okay. was tired and only got to the end of their first discussion. I thought i could add some unknown information, guess i was too slow :)

Simple answer: math is a flawed system. Yes, it's true, the difference between .9 and 1 = infinity, but, only within the system of mathematics (a man-made structure). Science is a more effective implementation of math, in that it measures the realities around us, rather than the hypotheticals allowed within a man-made construct.

Difference is infinity => difference is unbounded => 1 - 0.5 less than 1 - (0.9...) => 0.9... less than 0.5

WHAT HAVE YOU DONE

a math thread on esr without czm?
this cannot be

This is a noob topic. Czm is a pro.

Gotta love Lethe's endurance. :D

vedic's explanation was the best

It's solving the same problem twice. If you can say that 0.33...=1/3, you can use the same algorithm to establish that 0.99...=1

He likely means this: http://www.esreality.com/?a=post&id=2113040#pid2113040

lol

math is potat

just think of it like a curve where 1 is a straight line the curve gets closer but it never touches it thats liek 0.99(9's foreveer)

we say 1=.9999999... because its so close it dont matter but its just rounding cus hte difference is so small

the limit of 0.999999999999999.......... is 1 but the the number itsself IS NOT =1

The number 0,999... IS equal to 1, as it has been repeatedly demonstrated in this thread (and repeatedly explained to you).

the limit of 0.999999999999999.......... is 1

You are confusing this number with sequence "0.9, 0.99, 0.999, 0.9999, ....", which indeed has limit 1, but sequence (infinite number of numbers following each other) is not a fucking number (just one point on the number line, even if humans sometimes give it strange alias using idea of infinite string of symbols).

The representation of real numbers by decimal fractions is inherently constructed in such fashion that there happen to be two ways to arrive at number 1 (and at other whole numbers) with decimal representation.

When you write number as "1234.56789", it is actually a short way of writing "1234.56789000000000...", and what it really means is that the whole part "1234" selects the segment (of length 1) of real number line where your number is supposed to be, and the fractional part ".5678900000...." is the "address" of your particular number on that segment.

The nuance is, that consecutive segments are overlapping in one point, e.g. [0, 1] overlaps with [1, 2] at number 1, so you can either write address of number 1 in [0, 1] segment, or you can choose to write its address in [1, 2] segment, and that's why you get two ways of writing the same number.

So, either you can start with segment from 1 to 2 by writing whole part as "1" and then continue by excluding everything that is bigger than 1 by choosing all further decimal positions to be zero (1.00000.....). Or you can start with segment from 0 to 1 by writing whole part as "0" and then exclude everything that is less than 1 from [0, 1] by choosing all further decimal positions to be 9 (0.9999....). In both cases you address exactly the same number 1 from the segment you started with.

If only we had some sort of concept of two things being relationally equivalent. Maybe we could us such an idea to identify different representations of the same underlying object. This whole subject seems cauched in mystery :(

We have such concept!
You are already talking about it, so it must exist in your brain!
It's very simple really. Imagine a city that looks like this:

C O O L L E R
O
O
L
L
E
R

Then for people of left-to-right street, house depicted with letter C is "at the left end of the street", but for people of top-to-bottom street, house C is "at the top end of the street". There it is - two representations of the same underlying object.

Finally someone puts it a way we can all understand

http://www.wolframalpha.com/input/?i=Sum%5B9%...nity%7D%5D

Actually....
if the 9's go on "forever" (that being infinite) then we can put an equal sign between:
0.999999999.... = 1-1/infinity.
By definition 1/infinity = 0 (at least by most definitions... some philosophers claim different though)

So 0.9999999.....(to infinity) = 1 - 0 = 1.

Yeah, we missed your expert opinion

No, we can't put an equal sign between what you wrote, because infinity is not a number. So all that follows next is bullshit.

Indeed

I really hate your view on math... it is so limiting and suffocating.

This is the type of math people get thaught in highschool and end up hating the subject.

You probably also would say that "X / 0 = BS". But that taking the limit of 1 / X can approach infinity when X approaches 0.

Although linguistical it is correct this is so fundamentally fucked up that this limited view on math limits science as a whole.

Trust me when I say that the only reason that X / 0 = undefined is because people can't imagine infinity and computers can't (yet) deal with it.

This whole thread appeared exactly because of this loose imprecise understanding of things.

You cannot resolve a loosely formulated argument by introducing another equally loose formulated argument, because it is not possible to say which of them is more true than the other. OP might read your explanation, then come and say to you: "but hey, 0.9999... = 1-1/infinity, and 1/infinity is infinitely small number, so 0.9999... is DIFFERENT from 1 by infinitely small amount!". So you didn't explain anything and didn't resolve the problem, but just introduced more confusion since both your and OP's arguments are seemingly correct at the level they are formulated and contradict each other. Who is right? You only can tell if you explain what exactly you mean by saying this stuff, without allowing any vague interpretation.

In this case, the issue of 1 having two decimal representations comes from small nuance in the precise construction of real numbers model as decimal fractions, and you cannot really explain this nuance without getting down to "low level" of precise definitions. There cannot be a "hand waving" elementary explanation for something that is caused by low-level definition "glitch".

it's not loose and it's not imprecise.

You think that an infinitely large number "must be a certain really large number" which is the cause of your confusion.

As X / "any number" = "something small". Therefor X / "A really large number" = "something insanely small".



As you point out in your post: "OP might read your explanation, then come and say to you: "but hey, 0.9999... = 1-1/infinity, and 1/infinity is infinitely small number, so 0.9999... is DIFFERENT from 1 by infinitely small amount!" "
That is true. Because an "infinitely small amount" = "nothing". A finitely small amount = something.

1 - nothing = 1.

The "glitch" you speak of is not in my post... it is in yours. You need to have special laws for infinity which don't apply to 0. I don't need any special laws for that as the behaviour of inifinity is dominated by the (in math somewhat difficult to understand) concept of 0.

The glitch has nothing to do with infinity at all. The fundamental reason why 0.99...=1.000 is because segments [0, 1] and [1, 2] have common point 1. Infinities don't take any part in explanation of this issue, and even simply mentioning them is completely misunderstanding the point and confusing the issue further.

When you talk about infinity, you always really talk either about some limit or about some set. In our case there are no limits involved, 0.999... is just a single number, so there is nothing to apply the notion of infinity to.

Then you missread the question as "The nine's go FOREVER" in the origional post....
Which is also in my post. Which introduces (as I point out in my 1st post) an infinite series of 9's in which case it is the same as
1- 1/infinity.

You could write it as a series of equations:

(9/10^1) + (9/10^2) + (9/10^3)......... + (9/10^inifinity) ......(and beyond! ;) )

You can also chose to write it as
1-(1/10^n) but then obviously n has to be infinite... which you cannot comprehend.... So you can solve by taking the limit of 1-(1/10^n) as n goes to infinity.

Let me beat you to your next question btw...
1.1111111111... = 10/9
2.2222222222... = 20/9
3.3333333333... = 30/9
............
10 = 90 / 9.

0.99999999.... = 9 / (9+(1/10^infinity) )

Well, you see that your explanation begins making more sense when instead of talking about infinity as about some number (which it isn't), with which we can do anything we want algebraically and magically and unexplainably get results which we want, you make yourself more precise and say that by writing infinity symbol you are talking about infinite sum, and that it's value equals some limit.

Still, your argument only proves that 0.9999... is indeed one of ways to write 1, but it is not the actual reason why 1 has two different decimal representations, and you cannot tell from it that every rational number also has two of them, but irrational numbers have only one.

what precisely do you mean by 'an infinitely large number'?

Any integer with infinitely many digits.

I don't understand. I asked you what you meant by "infinitely large number". You respond with "any integer with infinitely many digits". Three minutes later you say that you know such an integer does not exist.

?

well if I'd ask you to write down any "infinitely large number" would you be able to do it?

You can try using the following steps:
1) Write down any digit and go to step 2.
2) Precede/succeed that digit by step 1.

No I wouldn't be able to do it. I asked you what you meant by such a concept since you brought it up.

well people who have no clue about math either only write down numbers and operators to "do math"... or they think only in operators and their utility.

each of those views requires a different way of explaining infinity. P4r4 thinks that infinity must be "some really large number" (aka it 'must be bound' somewhere / have some upper limit) because he cannot see how to write it.
But the thing is you can't write it as a number (which is why there is the infinity symbol...)

P4r4 thinks that infinity must be "some really large number" (aka it 'must be bound' somewhere / have some upper limit) because he cannot see how to write it.

Lol wut?

You try to counter my point with OP saying: "but hey, 0.9999... = 1-1/infinity, and 1/infinity is infinitely small number, so 0.9999... is DIFFERENT from 1 by infinitely small amount!".

That means that somewhere infinity must be bound. Aka it must correspond to 'a real number'.
Because if the infinitely small amount difference is 0 that means you actually agree with me. If it != 0 then infinity must be a number.

That means that somewhere infinity must be bound. Aka it must correspond to 'a real number'.

I really hate your view on math... it is so limiting and suffocating.

This is the type of math people get thaught in highschool and end up hating the subject.

Although linguistical it is correct this is so fundamentally fucked up that this limited view on math limits science as a whole.

Hey it's your view.... not mine.

I just write down what you say and then take logical conclusions from there.

You say that 1/infinity is an infinitely small number which is different from 0 (If it is equal to 0 you agree with me).

You really believe that sarcastic example of how your loose mathematics can be used to contradict you is "my view"?
You see how you cannot use things like "1-1/infinity" without actually falling down to that "limiting and suffocating" precise description of what this equation really means in terms of limits and infinite series?

Then tell me where I'm contradicted?

You say that x/inf != 0 (or x*0 as I would prefer).
I reply by saying that if x/inf != 0 then infinity must be finite (which = YOU being wrong or well you missinterpreting me)....
To which you then reply that because in YOUR view "inf must be finite" I am wrong by saying that it is not?

Are you having some trouble reading p4r4's posts or something? He said the exact opposite, namely that infinity is not a number, and he's perfectly right.

btw I do know that there are no integers with infinitely many digits.

But when someone can only view math as "working with numbers" (Which is what I suspect p4r4 is suffering from). Then this is the best explanation I have for them.

x/0 is undefined because otherwise it would break mathematics

it doesn't break anything?

Assume x>0.
Let x/0=a for some a>0.
Then x=0*a.
But 0*a=0.
So x=0. But x>0. Da fuck?!

That happens only because you say that x/0 must be some a > 0.

When we just take infinity as being the opposite of 0 there is no problem at all.

Unless ofcourse you want to multiply infinity with 0 which would need to be 1 by definition.
(just like 0 * any real number = 0).

But still this addition doesn't "break" any current math.

edit: there are some points which get 'phillosophical' after this stage though. But the addition of x/0 = inf. doesn't mess up any math.

What would it mean for x/0 to be defined without an 'a' such that x/0=a? If you claim x/0 is a legitimate number I am allowed to ask you to show it to me (i.e on the number line).

There's no philosophy here, just maths

Why is math restricted to a number line?

I didn't say that, you've totally avoided my question. Saying there exists a number 'a' such that x/0=a is what it means for x/0 to be defined. From your other posts it seems like you are saying a=inf

yep...
to be more precise (if you want all math to keep sane) a = x*inf.

I didn't say that, you've totally avoided my question. Saying there exists a number 'a' such that x/0=a is what it means for x/0 to be defined. From your other posts it seems like you are saying a=inf.

Btw my wifi is going crazy so some of my replies aren't going where they should be.

a small addition btw to the above post (I've ommitted an important detail as I wasn't paying attention).

You solve it by doing it in a similar fashion as Euler solved his imaginairy numbers.

Except we don't need to add anything really...
just that 1 = 0*inf.

(opposed to i^2 = -1)
The concept of Euler's "imaginairy unit" applies though. Call it a "Weird infinity unit" if you like.

so if a number divided by itself = 1, and infinity/0 = 1, is infinity equal to 0?

hey wp.
My first mistake of the thread :D

I'll fix the origional post because inf/0 should be inf^2 ofcourse :D

The Weird infinity unit should be 0*inf :D (not inf/0).

btw in my defense it was almost 5 in the morning ;)

This whole thing is so naive. We've departed the standard number system by adding inf. Let me ask you then: what is 0*inf equal to? I've taken two elements from your new system and multiplied them together so you should be able to tell me. You've already said its not 0.

1

(btw it's 2 posts up... "Except we don't need to add anything really...
just that 1 = 0*inf.")

what 2*0*inf equals to?

2

..... we can keep doing this till infinity man ;)

But isn't 2*0 equal to 0?

I smell a trap. Run Weird!!!!

Assume x>1.
Then x/0=inf.
So x=0*inf.
0*inf=1.
x=1. But x>1. Your system is inconsistent.

Here's another:
Assume x>1.
x/0=x*(1/0)=x*inf.
But x/0=inf.
The structure of your system seems to change based on what question you are answering. This isn't a coincidence.

unfortunately for you I've already countered that problem in a different post as well.

What I said right here (in this part of the tread) is that when you add x/0 = inf it doesn't destroy any already valid math.
When you want to "completely switch" to a new form of math it is fine with me but then x = x*(0*inf).

That meaning that your equations would have to be:
Then x/0=x*inf.
So x=x*0*inf.
x*0*inf=x.
x=x.

I just showed allowing x/0=inf let's us show that x=1 AND x>1. You can't just conclude it doesn't break standard math when I have shown that it does by demonstration

in standard math x/0 = error (or undefined...).
Standard math also doesn't contain x*inf so how can it break something that isn't in?

I don't know what's going on anymore. You are the one who claims adding x/0=inf won't break standard maths.
Your conception of inf, with 1=inf*0 allows us to prove that any number equals any other number - sounds pretty broken to me.

"When you add x/0=inf doesn't destroy any already valid math"

simple... the math you use to show me being wrong is 'currently invalid' math as x/0 is not valid. Also you are hammering on my first post while I've already said (quite a few times) that if you want to keep it sane then x/0 = x*inf.

If you add x/0 = inf then it just 'adds to the concept'.
Everyone can understand it when you say 0.9999999... = 1-1/inf without equating all the math rules.

That you guys want to know all details of this kinda shocked me.... Probably this is because esr is obsessed with trying to find anything where I am even remotely inconsistent.

Your first question (the starting post of the topic) is answered easily if you can imagine 1/inf.
But if you would want to have fully consistently working math from there it becomes harder to explain.

And why that is, is simple. In 'normal math' 0 = 0+0. If you add infinity to that concept then you have that 0 is actually 'a unit'. This would cause some properties which are (in normal math) to become false. Although, with the right rules, all current math remains valid.

As noted below the major concequence of 'rewriting everything to fit with infinity' is that 0 != 0 + 0. But if you want it is possible to write a fully consistent mathematical model (note that some things will have to change) in order to allow this.

If you guys are really really interrested I could make a post about it... but it'll take me a bit of time (and seeing how I have stuff to do right now it'll be somewhere near the end of next week I think)

You: it is possible to affix the concept of inf to the standard number system without breaking it.

Me: if you do that, we can show that 1=2 (this is true if x/0=x*inf and 0*inf=1 like you said). This is not true in the standard system.

Conclusion: we cannot add this concept to the standard system without breaking it.

Well then I would say: "you are wrong in saying that 1=2 because of the given rules." (as I've already shown)

If you think that reply makes any kind of sense you should be very worried

I am worried for you :(

btw read fau's posts as well ;)

2/2 = 1 so 2 = 1 * 2
inf/0 = 1 so inf = 0 * 1
inf/0 = 1 so 0 = inf * 1

kind of strange.

0/3=0..

)))

would anybody, please, think about the children?

1 child or 0.99999 child?

There are some nicely proven theorems stating e.g. that between any two distinct real numbers a and b (such that a < b) you can always find another real number c, such that a < c <b. If you can't do it then a=b.

Now assuming that 0.(9) is distinct from 1, you therefore assume that 0.(9) < 1. Both 0.(9) and 1 are real numbers. Now dear quasi mathematicians on esreality: let a=0.(9), b=1. Can you please find a real number c that satisfies the theorem above in this case? :)))

To avoid confusion: 0.(9) is just a different notation for 0.999...

/thread

very well QuasiMathNumber = a + x (b-a) where 1 > x > 0.

There you go ;)

clever, quite clever. No numerical value (apart from a) can be attached to this QuasiMathNumber though :)

ehmmm... no numerical value can be a? (as no number is infinite in size)

(I'm a hard person to beat at math m8... many professors have tried ;) )

"I'm a hard person to beat at math m8... many professors have tried ;)"

lmao

good that it makes some people laugh :)

I love how people of ESR goes mental and starts a math seminar over an insignificant post.

Well this is ... weird.

that's why I like it :D

You are sort of describing the hyper reals, but even in that system you can't just write 1/0=inf because it leads to contradictions as I have shown.

edit: this was meant to be a reply to the post below

adding for lolz

The stuff about infinitely small (infinitesimal) numbers isn't crazy nor new. This is how Newton and Leibniz thouht about the real numbers. The problem was that it was loosely formulated and led to contradictions.

In XVIII/XIX century mathematical analysis created by Newton and Leibniz was formalised and formulated without infinitesimals by Cauchy and many others.
In XIX century the modern (one taught in school nowadays) definition of field of real numbers was created by Cantor, Dedekind and others.

Later in 60s Abraham Robinson returned to old concepts of infinitesimals (they are rather intuitive) and created his nonstandard analysis based on modern model theory (as saturated model of arithmetic iirc). It's correct but has it's own problems and isn't widely used.
In this model 0,(9) and 1 are two different numbers. (Not all numbers have decimal representation though)

In response to Weird's "ideas":
You can define any abstract objects in math. Real numbers is just a set so you can throw in various stuff like apples and giraffes. The point is that real numbers besides being a set of objects have their mathematical structure (they are ordered field - read on wiki) if your extended numbers aren't ordered field anymore then they aren't very interesting or usefull from the mathematician's point of view.
For example complex numbers extend field of real numbers (they aren't ordered though) and indeed are very usefull.

Hi there.

Aren't your last 2 lines contradicting eachother?
"if my extended numberes aren't ordered field anymore then they aren't very interesting or usefull from the mathematician's point of view.
For example complex numbers extend field of real numbers (they aren't ordered though) and indeed are very usefull. "

Also I don't really see as to why it would not be ordered as it seems that the >= even allows me to call it ordered.

(0*inf = 1 x/0=x*inf inf/inf = inf^2 x/inf = x*0
The only problem you run in to is the same one Euler ran in to... which is that 16 + x/0 = 16 + x*inf.)

Yes they do. I should say "probably". Ofc such partial extensions (they extend only part of original structure - in this case field structure but not order) can have many different properties which make them useful (eg complex numbers are algebraicaly closed contrary to real numbers).

Your extension of real numbers by inf isn't a field. One of the field's charasteristics is that multiplication should distribute over addition. In your extension (0+0)*inf = 0*inf = 1 but (0+0)*inf = 0*inf + 0*inf = 1+1 = 2. It's not paradox nor contradiction - it's just your extended * operation doesn't distribute over addition anymore.

Not being a field isn't the worst thing that can happen to you - there are far more pathological objects in math. I just don't see it's purpose. You can infinitely create new mathematical objects but only the ones that help you to answer an interesting question are considered important.

Ehhhh,
That multiplication distributes over addition is not a property of an ordered field right? (I don't think it is)


You do however understand math nicely :D
You are also correct on this being not clearly defined.

I would actually like your example to be 2 as then addition makes more sense from a numerical point of view.
But this is the point where it becomes more philosophical than true math.
I see 1/inf as an infinite series of 0 decimals (aka 0.0000......). But then the infinitely last element of the series (that which we never reach) would be a 1.

So then 0 + 0 = 2/inf.
The idea is that we have a "unit" (aka similar to euler's imaginairy unit)

Multiplicative distributivity isn't a property of a field?? Come on man are you trolling?

It just occurred to me how ironic this post is since Im the one who made this travesty of a thread
: D

according to the definition on wikipedia the distributivity is not part of the field.
The ordered field only refers to each singular element having properties related to another singular element.
That (x+y) * z = x*z + y*z doesn't seem to be a condition for the field to be ordered (it is a general mathematical 'truth' though).

It's a condition for the field to be a field...

ahhhh..... ok.

Math people with their terms.... they make things needlessly complex :D

2nd mistake of the thread noted ;)

That's the thing, it's exactly complex as it needs to be to make it rigorous. These things need to be well defined to allow us to coherantly reason with them - relying on our intuition leads to errors and contradictions (like with your x/0=inf idea)

That multiplication distributes over addition is not a property of an ordered field right? (I don't think it is)

It is.

Mathematics and philosophy overlap in some places. For example mathematical logic is part of mathematics and philosophy.
Fields, orders and their extensions are far from philosophy though. They are basic mathematical objects well studied and understood.

Instead of your "0,(0) last element 0" you should use 1-0,(9). This is why I said that not all hyperreal numbers have decimal representation.

You can read more on infinitesimals on wikipedia. I didn't read it but looks like a good overview.
I know how disappointing it is to read about something you came up with. Unfortunatelly it's inevitable in science as ancient as mathematics.

Working with such field-like structures is very tricky. Most basic mathematical theorems you take as granted are not true there.

PS Afaik in all continental Europe comma is used to separate decimal part.

I'll read that wikipedia page tomorrow or smth (have stuff to do :() but it seems (skipped through it in 30 secs) as if he fails to find a solution for 0 because the notion of an infinitely long number is beyond our imagination.

So he prolly gets stuck at the same philosophical question which I try to 'dodge' by saying that 0 can be written as an infinite amount of 0's (decimals) followed by a 1.

What do you mean by "solution to 0"? And what do you mean by infinitely long number?

Beyond our imagination? You are so XVIII century... :D

I'll answer in a few hours (really gotta do smth now :x)

I mean exactly what I say with infinitely long number.
1/3 is a fracture which leads to an infinitely long number which is 0.33333.............333..... etc (to infinity).

Then 3*1/3 = 0.9999999999999999999999999999999.......9999 right?
Then my infinitely long number for 0 = 1 - (3*(1/3)).
Which I abreviate to 0. (tada math is correct)
However this leads to a paradox... because if 1 is "purely 1" (aka 1.0000000) then it can never be equal to 0.999999..... So when you substract 1 with 3*1/3 you end up with a number which is infinite in size (0.00........) but ends with a 1 after you wrote down the infini'th (invented a new word yee) indexed 0 digit (which ofcourse you cannot actually write down but you can imagine it).

Welcome to the 21st century of imagination :D


Now as for what I mean with solution for 0 (didn't write solution to 0).
Is that nowhere in the wiki is there any mention of the number I reference to above. Not having this number as 0 presents us with a problem. Because then no matter how you twist or turn it the "unity measure" (the 1/inf) always has to have 'a value'.

And sooner or later this will lead to contradictions and falsities etc. You need to have an infinite index.


What the infinith index allows me to do is create *any* finite number of integers such that round(integer/infinity) = 0. And you need this in order to keep the math correct.
Now ofcourse you don't have to "round" anything it is just there to make the concept less abstract.
I would much prefer it (aka you can keep doing calculus with infinity when) x/infinity = x*0.

because if 1 is "purely 1" (aka 1.0000000) then it can never be equal to 0.999999

1,000...0... = 0,999...9...

I agree with you.

However this leads to a paradox... because if 1 is "purely 1" (aka 1.0000000) then it can never be equal to 0.999999.....

It's not uncommon in mathematics that there are many instances or representations of the same object. Take rational numbers as an example. 1/2 = 2/4 = 3/6 = ... There are infinitely many representations of 1/2 yest it's still the same, "pure" 1/2.
This is done by defining equivalence relation and treating equivalent elements as the same (not exactly, I'm simplifying). In classical construction of decimal real numbers one defines them as sequences of numbers 0-9 where sequences ending with ...,n,9,9,9,9,9,... are equivalent to ...n+1,0,0,0,0,... Therefore 0,(9) = 1 and there are no problems you mentioned. The answer to op's question is given by definition.

This definition was created in such way because it's creator wanted real numbers to have some specific properties (in this case - dense order). These properties are here from the times of ancient greece when real numbers were strictly connected to line and plane. When asked for real number you imagine decimal representation. Ancient greek would imagine two line segments and their length relation and there would be no question about 0,(9) :-)

The "operation" of dividing isn't defined by itself. If you want to find a number x/y you simply search for number z satysfying equality z*y=x. This is the essence of dividing and if you define it in any significantly different way then this will be just another random operation. x/0 in field is undefined no matter if x is 5, infinity or giraffe because by field's definition z*0=0 != x. What if x=0?

As I wrote previously - formally extending ordered field of real numbers by infinite and infinitely small numbers has already been don by Robinson without loosing ordered field structure and making classical analysis much more intuitive! 1/inf where inf is a number larger than any classical number is called an infinitesimal and it's "infinitely close to 0" but it's not 0 (and can't be without breaking the structure because if x/y = 0 then y*0 = x). Your concept of "infinite index" is actually similar to how we represent non-standard numbers. We decompose them as x = z + y where z is standard real number closest to x and y is non-standard part (your 1-0,(9) or 1 with infinite index). You could as well write (z,y) and y would be "infinite index". It's just the matter of notation.

No offence but this is more like XVIII century imagination. Have a look at this fine XVIII century book containing many more problems similar to yours :-). Fortunatelly all of them were adressed in XIX century by formalisation of mathematics. Modern mathematics has many remarkable and astonishing results on the border of mathematics and philosophy to offer for wide audience. You seem to be truly interested so I'd advice you read some popular science books. I used to read classics myself but I'm going to ask about some good titles. The most interesting results for non-mathematician (for philosopher?) are imho in set theory (XIX/XX century) and mathematical logic (XX century). Theorethical informatics as a branch of mathematics is quite interesting too.

You seem to assume that I'm not good at math.... Or that I am on the philosophical side of science. I've had several years of math on a university level (computer science is pretty math heavy).

You are right that I don't know all of the historical backgrounds in math as I don't care about it.

I also don't care about terms... As I find that people who do not understand math come up with terms to hide their incompetence (I'm not talking about you but in general).
That 1 = 0.99999..... by definition is just some chap not understanding it so he defined what is not neccessary to define.

It's the concepts that matter.... The only problem there is in math is that you need notation to explain the ideas you have to others.

The concept of how I solve this 1 = 0.9999 is SO easy to understand.
But you need to understand what an infinite series is and, as I pointed out in my origional post, hardly anyone understands that. ("Trust me when I say that the only reason that X / 0 = undefined is because people can't imagine infinity and computers can't (yet) deal with it.").
I even have problems understanding how it can be that it doesn't lead to any problems.... but it works because the series of 0's is infinite.

You say that even if 1/infinity is infinitely small then it's still larger than 0.... I propose (or well I tell you) that it is 0. Now the current notations of the math system don't allow for me to say so. And I know that.
But if you want an alternate notation then I'd have to define a new math system (which is fine by me... would take me a bit of time though).

So tell me is it the concept of the idea which is flawed or is it the limiting view on math (limited by notation rather than imagination)?

I don't think that you are bad at math. You think of math in a creative way which is uncommon and this is the only reason why I responded to you. I didn't think that you had any mathematical background because your reasoning has many flaws I'm trying to explain.

I also don't care about terms... As I find that people who do not understand math come up with terms to hide their incompetence (I'm not talking about you but in general).

Strongly formalized mathematics emerged in XIX century. You can read the text I linked to see why. The basic axioms and definitions are there for a good reason - to easily describe mathematical objects used by mathematicans for centuries or millenias. Not altering them because this would mean that all theorems need to be rechecked and prooved again. You can freely create new mathematical objects if you need to but formal approach is recommended to avoid confusion.

That 1 = 0.99999..... by definition is just some chap not understanding it so he defined what is not neccessary to define.

I appraise your independence and critical thinking but it's on the edge of ignorance. I wrote a paragraph explaining the reasons in my last post. Please refer to it.

If you wan't to somehow "solve" or rathe prove that 1 = 0,(9) then you have to start from some other preconceptions anyway.

In standard analysis there aren't any infinite numbers. I was talking specifically about Robinson's non-standard analysis.
Ofc you can say that 1/inf = 0.
If 1/infinity is 0 then infinity * 0 = 1. We have two possibilities now.
1. 1 = 0 and real numbers is trivial field (contains only one element)
2. Your numbers aren't field.
Notice that this reasoning uses purely algebraical properties and doesn't depend on number's representation or notation.

I fail to understand what's actually your idea? I thought it's extending real numbers with infinite and infinitesimal numbers. If that's the case then it has already been done without breaking any important properties by Robinson.

Here is what I don't understand about your view.

1. 1 = 0 and real numbers is trivial field (contains only one element)
2. Your numbers aren't field.

on 1) In no way can I imagine how 1 = 0.... Because all individual additions of 0's do not even come finitely close to becomming 1. So 1 != 0 + 0 + 0 .... an insanely high number
However
1 = 0*inf.

On 2) Well I don't care if they are field? The concepts fit perfectly. What do I care what term people want to put on it? Sure I could go into detail but I'd first have to read up on what his ideas are (exactly).

-------
The only (and I mean only) problem there is that you need to have gradations of 0. Or to put it more in terms you like.... gradations of vanishing entities.

1) It's because I'm talking in algebraic language (this is why I said that it's purely algebraical reasoning and doesn't use number's representation).
1 is just a symbol of multiplicative neutral element (s.t. 1*x = x = x*1 for each x). Perhaps I should note it as e*. In real numbers it's indeed 1.
0 is a symbol of additive neutral element (0+x = x for each x).
Now two symbols can point at one number (think of pointers in programming or 1/2 and 2/4). In case when multiplicative and additive neutral elements are the same (e* = e+) e*/inf = e+ is true and you can easily show that whole field consists of only one element.

The point with being an ordered field is that this concept has been literally extracted or abstracted out of the rational numbers (ordered field that is Dedekind complete has been abstracted from real numbers). The real numbers that had been used for few millenias and emerge from the concept of points on line.
Now If you extend real numbers and your extension preserves the original structure then most of the theorems and preconceptions are still true for your extension!
If it doesn't... well you basically create a new branch of mathematics and you need to build it up from the beginning (basically nothing works: there are no prime numbers, division doesn't exist so you can't define derivative like we do etc.). You could just use the fact old real numbers are subset of your new numbers and just treat numbers with inf in a special way but this is what we do now - have special treatment for symbol of infinity! I don't see how is it any advancement over using old numbers and concept of Cauchy limit.

On the other hand Robinson did it in a compatible way with old numbers and translated a large chunk of mathematics to his new language.

I use the number representation to show the most intuitive algabraic concept of infinity.

Read my posts a few more times... and at least make an effort to understand.
I've already answered the questions you pose in this post.

I'll reanswer this one missconception in your above post in yet another way though just because it's fun.
You say "0+x = x" which is obviously true in the 'current system'.
I say that 0+x = 0+x (last time I said this was "The only (and I mean only) problem there is that you need to have gradations of 0. Or to put it more in terms you like.... gradations of vanishing entities.").

We need to have the 0's because otherwise if when we multiply with infinity you get or lose numbers (kinda like integration/differentiation).
(0+x)*inf = 1+x*inf.

But although we cannot simply remove these 0's nothing changes to any result which we already have. Because the 0 is always infinitely small. It is a gradation of infinitely small but never the less it is infinitely small.
We only need the gradation of 0 when we multiply it by infinity.

To me it seems as if you're stuck in the same web of terms that block all imagination you have. Which is why so many people think math is boring. The education of math is seriously failing :(

So does 0+x = 0+0+x ?

no because:
(0+x) * inf = 1 + x*inf.
(0+0+x) * inf = 2+x*inf.

But that doesn't matter for the previous results in math because each 0 is infinitely small.
Therefor you could say that (x* "my 0") -> "your 0" note that it doesn't go the other way.

I use the number representation to show the most intuitive algabraic concept of infinity.

There is nothing algebraic about infinity. From algebraical point of view it's as good as any other field's element. (ok not exactly, but can you tell me what is "algebraic concept of infinity"?)
Mathematicians have sorted out infinity and infinities pretty well. I recommend some popular science books on set theory and Cantor's results.

I've already answered the questions you pose in this post.

There aren't any questions in my post.

You say "0+x = x" which is obviously true in the 'current system'.
I say that 0+x = 0+x

Which is obviously true in all systems not matter what "0","+" and "x" mean as long as "=" is true equality.

What follows is incomprehensible to me. I don't know the context - what do you mean by 0's (zero's or zeros?)

Can you explain me your idea in one precise post?

Also I hope that you already understand that you don't "solve" anything. Real numbers are well defined and there is no infinity there. There are hyperreal numbers with infinity and your "Weird numbers" - they "exist" independently.

you redefine 0 as a unit.
0 = 1/infinity.

This leads to the somewhat "strange but extremely intuitive notion" that:
1 = 0 * infinity.

(added note: you seem to think that this falsifies any math which is already done. But it can't because at the moment x/0 = undefined. So all things that could become false are at the moment undefined.)

The only concequence of this then is that you need to 'count' the 0 symbols when you want to use infinity in a sane way.
x/0 = x*inf.
x/inf = x*0.

Because it is a unit you can't simply remove the 0 as it represents a value when you multiply by infinity.
However the value it represents when you do not multiply by infinity is infinitely small (aka 'in regular math' it won't have different properties than the 0 you are used to, just a slightly different notation as you need to keep track of the x (in x/0 = x*inf)).

(added note: you seem to think that this falsifies any math which is already done. But it can't because at the moment x/0 = undefined. So all things that could become false are at the moment undefined.)

Oh my god. IN YOUR SYSTEM, x/0 is defined. We can show that IN YOUR SYSTEM it is possible prove things that aren't true in the standard system. THEREFORE your system is NOT the same as the standard system with some extra concept for infinity. It is structurally different.

Oh my god... you have no imagination.

Which useful finding then would you be able to prove to be true in the standard system that are false in mine?
And I don't mean just notation wise. As it is obvious that the notation in 'my system' is slightly differen.

So 0+0 = 0 (in normal math) won't work in mine but anything even remotely relevant will.

2/0=2*inf
2=0*2*inf
Now 2*0=0
So 2=0*inf.
0*inf=1.
Therefore 2=1. This is all completely in your proposed system, using the definitions you have supplied. I'll go ahead and assume that you don't need a proof that 2 is not equal 1 in the normal number system.

What about this: x+0 = x+0+(x-x) = x+0+0, contrary to what you said? So what went wrong? If standard maths is not affected (which is what you claim) then this series of steps is totally correct. If you say these steps are not valid, you admit your system is not related to the standard one.

Where it goes wrong is you saying:
"2*0=0"
That should be:
"2*0 = 2*0".

As I said a few times you can't eliminate the amount of zero's.
You have 2 zero's on the left of the = you need to have 2 zero's on the right of the =.

You just mess up because you use the 2 systems through eachother as if they are completely interchangable at any random given point.

Do you think that having 0 multiplied by another number NOT be equal to 0 is not a significant difference from the standard system? Do you think that this wont affect any established results?

What about my second set of questions in the post above? Are we not allowed to have (x-x) at all? If I add this to anything I change the number of 0's! Do you think this will fit just fine with standard maths?!

it is a somewhat significant difference because it fixes a lot of issues.

And no it will not affect any established results.

When you substract 5 from 5 you just cross them off against eachother. I don't need any symbol for 0* (zero-star) as I call it below.
But that is a philosophical point.

Like I said I don't really care if you add 0* into the mix.... But then you'd still have x/0* = undefined.
As well as x - x = 0* (also 0 - 0 = 0*).
and then the notation becomes crappy (in my eyes).
I would much rather say that x - x = <blank space>

Yes or no: x+(a-a) = x+0?
If yes: x = x+0
If no: a-a != 0 which implies a != a. lol.

no.
x+(a-a) = x.

Like I said m8 I would much rather say that x - x = <blank space>.

But if you want you can say it's 0* where 0* has the same fubar properties as 0 has in 'normal math'.

the 0* is useless though (imo) as it serves the same purpose as not writing it at all.

"This leads to the somewhat "strange but extremely intuitive notion" that:
1 = 0 * infinity. "

Extremely intuitive doesn't mean its true. When you multiply two expressions where one tends to zero and one to infinity, it can have different results depeding on those expressions.

It's nice to stretch your imagination and try to develop new theories, but the point is that if we take your idea as an axiom, then we can derive lots of things that implied by this axiom. In this case, it leads to contradictions like the one above. Good thinking exercise though :)

It doesn't lead to any contradiction?

It leads to a question as to "where does infinity start/stop" but theres no contradiction in it.
As you multiply the expressions then as long as expr1 != 0 and expr2 != inf you will get the same value as you get in regular math.

When expr1 = 0 then you should write 0*expr2 though rather than just 0. And similarly for expr2 = inf leads to expr1*inf.

expr1 obviously remains finite untill it reaches 0 and expr2 obviously remains finite untill it reaches infinity.

Look I tried to understand what is the purpose of your "alternative" numbers and to show you how you can precisely and correctly achieve that. (tbh I still think your goal is just typical and intuitional replacing Cauchy limit in calculus with infinitesimals).

You disagree so now I'm going to just listen. And don't hide behind "this is more philosophy than math" Seriously these are very basic structures and there is nothing mysterious about them.

defining 0*inf = 1 doesn't falsify anything but you need to reprove all theorems wich use real numbers. Most of them won't work unless you include special cases for inf. Try proving anything on them - for example that (fg)' = f'g + fg'.

well I do hate limits and like infinitesimals. So you're somewhat right there... but my goal is just to have sane math.

Small sidestep for a few seconds:
What the philosophical question is is "do you need anything to represent absolutely nothing".
Aka even in normal math nobody writes the equation: " = 0" because philosophically it doesn't make sense.
end of sidestep1.
start of sidestep2:
"more philosophical than math" doesn't mean that there is anything mysterious about the structures it just means "does it makes sense to include this".

The 0* in my math... or the 0 in the current math represent exactly the same thing. But the 0* doesn't have any property we need to use if we already have (my) 0. As 0* is the same as a "blank space".
You don't need the 0* for anything other than to say that there is absolutely nothing there. So why write it down?
end of sidestep2

I don't need to reprove any theorems because "what they represent doesn't change".

When you add an infinitely small amount to something that something doesn't change. But I do keep track of it in the math because when you make that something infinitely larger it suddenly matters.

Say I have 1 apple... and I add 1 infinitely small part of another apple. I'd have (in my math) (1+0)apple = 1 apple + 0 apple = 1apple + (1/inf)*apple.
Then what can I measure? 1 apple.
What happens if I take 1 (measured) apple from the real world and make it infinite? Well it could be an infinite apple with many other apples around it. It could also be 1 infinite apple.

Now you are prolly laughing saying how idiotic that is.... but this is what is remarkably similar to the strange behaviour of nature witnessed in physics. Because as we approach 0 Kelvin suddenly most of our formula's break down.

Replying to your "sidestep1"
Yes, semantically denoting nothing is very important. I'll give you three examples.

1. Numeric systems with positional 0 (like in number 100, positional 0 != idea of number 0) were major breakthrough all over the world. Ancient Babylonian probably had the concept of positional 0 but they didn't have any sign for it in the early times. Therefore numbers they wrote depended strictly on the context and can't be deciphered nowadays. Mayas had very interesting positional system with base of 20.
2. Oldancientgreeklanguagedidn'thaveanysigntodenotewhereawordendsandwherenewonebegins(space) :-)
3. Imagine your mother tonque without word meaning "nothing". Yes, mathematical language is a language like any other (slightly more specialised, precise and well defined though).

Other than that in algebraic context 0 isn't "nothing" it's element like any other and has properties you can't deduce from the fact that it's "nothing" (0*x = 0, 0+x = x)

Ok so assuming that your definitions are not self-contradictory (I didn't check) then you don't need to reprove theorems for old numbers and old 0* "replaced" with new 0. You had the same before because as a substructure they aren't any different than old one. Theorems don't work for any expression containing inf so this gives you nothing. You just have your strange inf and you can't do anything with it unless you prove some theorems, but I'm telling you that this is wrong way to introduce infinite numbers and you wont be able to do anything useful (like defining sensible derivation) with it.

Computers have a very interesting position system with the base of 2 ;)

But anyway I don't really care about the usefullness... I think it can work nicely for some of the problems that science as a whole is going to run into soon (or is running into alread).

As like I said... a lot of physics fail when we get (close) to the absolute minimal temperature. This could very well be because of the lack of infinitesimals in current math. (I don't know though... as nobody knows what causes those 'abnormalities')

The thing is though... that (at least to me) the explained 0 is a much much more intuitive meaning than the notion of "pure nothing" which is in the current math.

sidestep1: I still truelly do not see why I should ever write "pure nothing" (from a philosophical stance).

Why don't you use your system to define differentiation like he says? Surely your system can handle calculus? Can you differentiate f(x) = x^2 from first principles for example?

Theres nothing different from normal calculus m8.
We just give a 'new' meaning to 0 which then forces us to keep track of "how many 0's there are".
Just like we need to keep track of how many infinites there are.

The only thing to take care about is that you don't confuse which 0 you are using (your 0 or my 0).
If you're confused then wherever you fail to see how the math works out then just write whereever normal math leads to a 0 any symbol which would represent what I called 0* (aka your 0).

aka 0 - 0 = 0*. (or whatever symbol you feel comfortable with).

So also if you would like to differentiate any number then just write 0*.
f(x) = x -> f'(x) = 0*
f(x) = x^y -> f'(x) = y*x^(y-1)

Then you will whine and bitch about f(inf) and f(0) and I'll re-explain myself again... and you'll still not understand.

It's not leading anywhere untill you grasp the concept.

You don't differentiate numbers...
If there is nothing different from standard calculus, differentiate x^2 from first principle using the standard definitions.

Plz at least read the post

I read it, you didn't differentiate x^2 from first principles. The reason I ask is because if you try to do it using the rules you've outlined, things do not work as they do in standard calculus (far from it).

Dude your like a kid learning how to piss into a toilet. I'm not gonna hold your dick while you ask me retarded questions which are not even needed to be answered.
You don't understand the concept and make no effort to do so.
You can think I'm dodging your question but my patience with your ignorance just ran out.

"questions which are not even needed to be answered."

This shows a profound ignorance of how maths works. I guess we'll leave it here since now you're just throwing tantrums

But anyway I don't really care about the usefullness... I think it can work nicely for some of the problems that science as a whole is going to run into soon (or is running into alread).

As like I said... a lot of physics fail when we get (close) to the absolute minimal temperature. This could very well be because of the lack of infinitesimals in current math. (I don't know though... as nobody knows what causes those 'abnormalities')

Any reasons to believe so?

notion of "pure nothing" which is in the current math.

You are wrong.

sidestep1: I still truelly do not see why I should ever write "pure nothing" (from a philosophical stance).

To be able to write such sentences?

I have some ideas but they are not yet full proof theories...

Where am I wrong?

If 2 things cancel eachother out.... Why give that anything other than nothing?

The most important thing is that it is intuitive, simple and elegant.

You are wrong by saying that 0 is defined as "pure nothing" in the current math.

There is nothing simple and elegant about your structure. You break previous structure and add no value. Look at complex numbers or hyperreal numbers - these are simple and elegant extensions.

Intuitive? Look at the amount of confusion in this thread.

I don't want to discourage you but you would be better off listening to others and trying to understand their reasons from time to time.

So 0 has a property which you can explain as 'something'?
That it has a symbol?

From mathematician's point of view 0 is as much something as any other number.

Yes - it's often interpreted as "nothing" for example when counting or in probability but in other applications it isn't. For example in geometry 0 can correspond to point on line just like any other number.

0 is also an index..... Zzzzzzz

btw if you want to have it completely compatible then we can just add a definition of something which you can 'strike out' as you do with 0's now:
0star = 0-0.
(0* = 0-0 (where the * is not a multiplication symbol))

Then you can remove the 0* and introduce the 0* whenever you would like. And math will still be the same... Even that x/0* is undefined. (Which is something I dislike)

And how do you define - operation? Because normally it's defined by e+ (0) like this: x = -y if and only if x+y = e+

none of the operators change.

But there is a philosophical question there. Which is partially also at the basics of why the current 0 sucks.

i would much rather have "nothing" as the symbol for (the current) 0.
So (in normal math) 5 - 5 = ... (apparently the layout messes up when I only put spaces rather than '...' ) (which means 5 - 5 = "nothing"). Which you could represent by 0* if you really want a symbol for it (5 - 5 = 0*).
But if theres nothing there I don't see why anyone would want to write it down.

So to really freak you out 0 - 0 = 0*.
x * 0* = undefined. (which I dislike)

Ok so you extend - "operator" from old numbers. But you use 0-0 to define 0* and you haven't defined - operator on new numbers yet. You need to define x-y where x,y are new 0 or inf first.

That 1 = 0.99999..... by definition is just some chap not understanding it so he defined what is not neccessary to define.

It is because the simplest way of making a positional numeral system has a little ambiguity, it has nothing to do with understanding of any math concept, it has no deep reasons, it doesn't touch the question of understanding infitnity, etc, it's a trivial mundane thing that was just simpler to leave as a glitch than to invent stuff that deals with it by inventing more complex procedure of deciding which string of symbols is attached to which number. You can address all numbers from 0 to 1 with a number that starts with symbol "0", and you can address all numbers from 1 to 2 with a number that starts with symbol "1", so it turns out that you can address 1 if you start either with 0 or 1.

you're funny.... that I have to "invent something complex" while all I do is make math more intuitive and easier.

There is nothing at all complex about what I'm saying. It just takes a little imagination...

What you try to do is start with string of digits and try to assign meaning to them and want to make it so that 0.999... means not the same as 1. It is perfectly valid thing to try to do, and it is doable, but you should be aware that "numbers" are things that we distinguish from anything else by how they behave (i.e. by axioms that elements of set of real numbers have to conform to), not by how they look. So string of digits isn't a number by default, and the object that you end up with when you do what you want to do doesn't necessarily behave like numbers or has anything to do with numbers at all. So the moment you forget the numbers and start with strings of digits like "0.9999...", whatever it is you do next becomes completely independent and unrelated to original question of whether the number represented by "0.999..." is the same as number "1.000...".

But if you instead start with actual numbers as just a set of "things" following certain axioms (which is from history and experience an extremely common and useful object, since pretty much everything in mathematics involves numbers in some way), then you don't have much freedom to decide what to do with their decimal fraction representations, and if you follow the simplest, generally accepted way to assign string of digits to numbers, chosen so that all decimal calculations are as easy as possible, then you necessarily have that 0.999... equals to 1.000..., because it just isn't worth it to remove this glitch by making addition or multiplication of numbers insanely complex.
It is also possible to invent a way of assigning strings of digits to numbers in 1:1 way so that different strings always represent different numbers (since those sets have same cardinality), but then it will become very non-intuitive how to work with such strings and do arithmetic operations with them.

I actually say that 1 = 0.99999 because it is 1-1/inf = 1 - 0 which is finitely represented by 1.

Now when I write it like this you could ask the questions that I've already answered above (multiple times). "How can you use inf like that" "it will fail when you do blablabla".

Well I've already shown that if you redefine 0 and then make the smallest of small adjustments to how 0 is used and represented, it will not lead to any problems.

Although there actually is 1 problem which you haven't seen yet which just leads to another definition.

When you redifine anything, it means that you don't anymore want to deal with numbers as they are and arrive at an object that has nothing to do with numbers.
Either you have numbers without infinity or you have 2*inf=inf => either 2=1 or inf=0, or even if you can do anything about it, the object that has different axions than numbers, is not numbers anymore, and it WILL inevitably behave differently in certain cases.
Basically, you invent your own "Weird numbers" and prove something about it. That is by no means prohibited and you are or course free to invent any construction you want and research its properties, but relations between "Weird numbers" that contain infinity as one of its elements have nothing to do with relations between real numbers that don't contain infinity, so the proof that "weird number 0.999..." equals "weird number 1" cannot be used to prove that real number 0.999... equals real number 1 (at least without first proving something about relation between real numbers and weird numbers).

dear god....
You all are so stuck up on definitions that you cannot even imagine stuff which I thought about as a 10yr old.

Have your math teachers all been so focussed on rules that they never showed what math is about?

I cannot imagine what it must be like to be so constricted by rules. Math isn't about rules... it's about concepts. The concept I introduce is in no way violating any other concept.
You guys are just too focussed on 'how you write it down'.
Then I show you how you can write it down.... which you obviously fail to understand because you cannot understand the concept.
And you then reach conclusion that it's false by applying 'how you are used to writing down other concepts'.

Definitions and rules are what actually differentiates one concept from another.
You are offering your own concept that doesn't violate anything, but it's different from the concept of real numbers. What you say it's not false, it is simply about your own mathematical object, not about things that we all know as numbers.
I am not uncapable of imagining what you came up with. In fact, it is a fairly common idea to try and build various structures that naturally incorporate infinity. But unfortunately, the useful structure of "numbers" that comes up in mathematical applications is not the one that has infinity in it, and structures like yours with infinity just don't emerge that often and commonly studied mathematical objects don't usually behave like you want them to.

I never said it was the same... I said that all the things which are true in normal math are also true in this system.

That you guys freak out and ask me to rewrite every equation known to man is just retarded.

I did use it (in my origional post) without explaining every detail though... because it seems to me that the concept is too obvious to need any explanation.

When you take two baskets of apples and throw them all into one common basket, they add like natural numbers, and even if you invent your own weird natural numbers with your special addition law, apples won't care about this at all. This is kinda physical example, but the same thing happens in mathematics: if some quantity is represented by a real number and it follows all real number rules, then even if you invent your own weird numbers, that quantity won't care about it at all and won't know anything about your special infinity number and rules to work with it.
The concept is obvious, it may be nice, it may be super cool, but the quantities in mathematics usually behave like real numbers (as defined by appropriate axioms), not like your concept that you invented.

Sure you go ahaid and think that... I'm not gonna explain how you cannot distinguish between your apples and mine. I already did that.

You actually haven't resolved any kind of "2=1" paradoxes that were pointed out to you.

there are no '2=1' paradoxes because there are no '2=1' instances unless you fuck up notation.
I pointed out where you guys fuck up notation --> no more paradox.

You introduce not just notation, you introduce a different element 0* with properties that cannot be present within real numbers.

strangely enough the 0* I introduce is the real number 0 which you are familiar with.
I just don't have the need to introduce 0* because of my 0 and my view that if "it's nothing then it should be represented by nothing".

But I guess you don't understand it no matter how often I repeat myself.

This is what you say about your 0*:

Where it goes wrong is you saying:
"2*0=0"
That should be:
"2*0 = 2*0".

But with real number 0, 2*0=0. So clearly your 0* is not the real number 0.
With real numbers, as they appear in mathematics, there is absolutely no history of how much zeroes there are. So whatever your 0* is, it doesn't depict actual behaviour of most mathematical quantities.

clearly you haven't read how I defined 0* yet you think you know all about it... story of this thread innit.... people not reading me but then asking all sorts of irrelevant questions.

I said 0* (readable as zero-star, which has nothing to do with any multiplication as you assume) equals:
x-x = 0*
so also
0 - 0 = 0*
Which means that
(x-0)-x = -0
and
(x-0)-(x-0) = 0*

Now before you go whine at "OMFG BUT THAT'S JUST HOW OUR MATH WORKS" I only put it in because you guys can't understand the concept and to stop the retarded barrage of irrelevant questions.

Then you get the next retarded question "BUT WHAT IS 0 / 0*" which is undefinable which shows you how retarded the current math system actually is. As it contains an element for which an operation is undefined (how retarded is that?)

Is 2 multiplied by 0* equal to 0* or not?
Is 0* multiplied by infinity equal to 1 or not?

both have to be undefined for it to remain true.

But the whole notion of it (imo) is silly... but if you guys want it in then it can be in :s.... unintuitive stuff

What "it"?
Is 2 multiplied by 0* undefined or not?

ah will you look at that.... I actually misread a post of yours w00t ;)

2 * 0* = 0*
0* * inf = 0*

soz (I thought you asked about / ) :x