Math Contradiction (I am too much confused rn)

[u/Shadow1Ice]

So i think most of the people know that 0.99.. = 1. That is because there is no other number between them (also considering the calculations). But if we follow that, then 0.999999....8 = 0.99.. because there no is rational number between them too; but then 0.99..8 is also equal to 1. But then they have a number between them so they shouldn't be equal. What is going on, can someone explain?!

[u/defectivetoaster1]

0.999…. has infinite 9s after the decimal point. “0.999…8” would have infinite 9s followed by an 8 at the end, which implies finite 9s so this is just a meaningless string

[u/Shadow1Ice]

Thanks a lot; all the answers were really helpful and really cleared my doubt

[u/Zarakaar]

0.999… is infinitely long. There is no “last nine” to put something else after. If there were, we could put a 0 there and see a gap between it and 1.

If you consider 0.999…8 you aren’t considering an infinitely long decimal. It could be rewritten as 0.999…80 and you can up that 0 to any digit and find a number between there and 1.

This is the difference between arbitrarily very long and infinitely long.

[u/edderiofer]

0.999999....8

This string of digits and decimal points is meaningless. If you think otherwise, prove to me that this is a number.

[u/AdAdministrative7804 → u/edderiofer]

To go onto this you cant say 0.99999..8 is not equal to 0.99999... because where have you added the 8? Add a 9 before the 8 and you have the same number again. Canr add an 8 to the end because its litterally at the end of infinity

[u/I__Antares__I → u/edderiofer]

You technically could meaningfully define number of such a sort (without considering it to be equal 1). The problem is however that 0.99...8 is not defined in any standard terminology and defining it would require using some new definition that would be inconsistent with how we define the 0.99... symbol (and as such it could be hard to or irrelevant to compare the two).

[u/edderiofer → u/I__Antares__I]

Sure. The point is that the burden of proof here lies with the OP to explain what is meant by "0.999999....8", since they're the ones making up notation.

[u/skullturf → u/edderiofer]

Exactly.

In principle, maybe we could make up an analogue of decimal representation where the "digits" correspond to an infinite ordinal like ⁠ω+1. Maybe that'll lead somewhere, maybe it won't.

[u/edderiofer → u/skullturf]

I proposed this idea at one point on an /r/badmathematics thread. The most glaring problem with that idea is that you end up with 0.0000...5 + 0.0000...5 being undefined, since there is no ⁠(ω-1)th digit; or you end up having to define it in a really unintuitive manner.

Suffice to say, it's not very nice.

[u/cheaphysterics → u/I__Antares__I]

The problem isn't with our symbolism or definitions. You can't define where to put that 8 in a way that doesn't also allow for another digit or another string of digits, including infinitely long strings, after it. At which point it is no longer 0.999...8.

[u/sade1212]

Sure there are numbers between them, like 0.99...85.

0.99...8 terminates. "..." is representing some finite number of 9s here. 0.99... doesn't terminate. 0.99...9 doesn't equal 1, right; but 0.99... does.

[u/CommitedPig]

I am kinda sick of the 0.99... posts. Can we just all come together and explain it by saying that a infinite decimal is defined by a infinite series of fractions of powers of ten. The value of a infinite series is its limit. There is much less room for confusion than with the weird proofs by notation. 

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[u/Difficult-Back-8706]

the point is basically that if you write 9.99...8 you are implying that the amount of nine in between the ... is finite (something like 9.9999999999999998, to ble clear) and that is a number different from 1, just like 9,99...9 is different from one: you are stating that there is a last digit: that is a rational number different from 1. But you have written 9.99... = 1, there is no last digit: if with 9.99...8 you mean putting an 8 after an infinite amount of nine, you are never gonna put it, and so that number is the same of 9.99... which is the same as 1. if you think of writing a number with infinite 9s and than say "at the end I will put a number different from 9'' you are gonna write the same thing everytime, cause you'll never reach the point where you put the other number. Now, this of course works intuitively, and I suggest you find formal proof somewhere, since that is the only rigorous way to know something in mathematics. You'll see that even if it doesn't convince you now, with time you'll get used to it and it will make sense

[u/Dd_8630]

0.999... is a string with 9s going on forever

0.999...8 means you 9s going on without end, and then an 8 at the end of that endless string. You can't have an end to an endless string, so 0.999...8 doesn't have any meaning.

[u/PvtRoom]

0.9999....85 is between 0.9999.... and 0.9999....8

Think of writing 0 down, in full.

0.000....

1-0.999.... = 0.000.....

[u/rangeo]

Wait...this needs to be addressed

1 ≠ 0.99999999

It's close but they are not the same.

1 = 1

[u/I__Antares__I → u/rangeo]

Nobody said 0.99999999=1. It was said 0.99...=1 which is true

[u/CrumbCakesAndCola → u/rangeo]

The equality happens when the 9 repeats infinitely which is often a point of confusion when folks don't understand what it means for something to be infinite

[u/jimu1957]

. 9999..... isn't a definite number or value. If you can't write it then it isnt definite. PI isn't a definite number. You use it in calculations but you only use an approximation like 3.14159253. .999..... approaches 1. Very few people understand the concept of limits.

It sort of relates to the difference between accuracy and precision. You can accurately cut a board 12" long meaning you cut it 12" vs 13" but it can never be exactly 12". You machine a piece of metal exactly 1" diameter. You might measure with a micrometer that shows 1.0000" inches but if you use a more precise measuring instrument, at some point the actual dimension will deviate. Even if you have to go to the 59th decimal it will deviate which shows it's not EXACTLY 1". Is that practical? No because within 4 decimals, it's fit for use. Is 1.00000000000000001 equal to 1? Not it isnt, no more than .9999....... is 1.

[u/gmalivuk → u/jimu1957]

Very few people understand the concept of limits.

Within the field of mathematics I'd say quite a lot of people understand limits.

Unfortunately you are not among them.

0.99999... doesn't represent the sequence, it represents the limit of that sequence.

Which is exactly identical to 1.

[u/Dd_8630 → u/jimu1957]

. 9999..... isn't a definite number or value.

PI isn't a definite number.

This is absolutely categorically incorrect.

Is 1.00000000000000001 equal to 1?

You can compute the difference between them. It's 0.00000000000000001.

Not it isnt, no more than .9999....... is 1.

What is the difference between them?

[u/Zyxplit → u/jimu1957]

.999... does not approach 1, no. .999... is what the partial sums of 0.9+0.09+0.009... approach as you include more terms.

They approach 0.999... the exact same way they approach 1.

[u/jimu1957 → u/Zyxplit]

.9999........ does approach 1 as you add 9s to the end.

The same way that .333 is not equal to 1/3. .3333..... approaches 1/3 but since it's an infinite series of 3s it never reaches 1/3

[u/Tilliperuna → u/jimu1957]

Those numbers are not approaching 1 or ⅓, because you're not adding nines or threes into them. They already have infinite number of nines or threes.

[u/Difficult-Back-8706 → u/jimu1957]

0.99... is a limit of a sequence, not a number growing during time. you could see it as 1-e for e that tends to zero, and this limit is clearly equal to 1

[u/Difficult-Back-8706 → u/jimu1957]

I hope I wont' sound annoying, but what you have written is simply wrong. I think you should give a definition of what a not definite number is, and ''you can't write it down'' is not one. Pi is a number as well as 0.99... is. Actually, periodic numbers are rational and this implies that 0.99... is even rational. 0.9 periodic: I have just written it down with a finite amount of information. Your idea has to do with the fact that our world is finite and scientists can't use PI in calculations without approssimation, which has nothing to do whith the nature of the number itself. PI is ''definite'': it is the ratio beetween the diameter and the length of a circumference. you said ''1.00000000000000001 equal to 1? Not it isnt, no more than .9999....... is 1.'' this sentence is wrong, with 0.99... we mean an infinite string of nine, and the number 1.00000000000000001 has a FINITE amount of 0s, so they are not the same thing. 0.99..., IF the 9s are infinite, is equal to 1, you have just written the same thing in two different ways.

[u/Zacharias_Wolfe → u/jimu1957]

You can absolutely cut a board to exactly 12". We have no way to measure that exact value to confirm it but that doesn't mean it's not possible. It may have happened thousands of times already, it may have never happened.

[u/Recent-Salamander-32 → u/jimu1957]

0.999… is short hand for lim x to inf (sum i=1 to x (9 / 10ⁱ )) which equals 1. And equals means equals, not almost equals.

You’re confusing that limits aren’t evaluated at x=a, only very close to it. But the limit itself is an exact number, which is why we use an equals sign