[split] 0.999... equals 1

[edk]

Oh, really?

[Tiberius]

Or:

http://wolframalpha.com/input/?i=0.1%5E%E2%88%9E

[LastPoet]

Mathematica is such an awsome app

[Modular Moog V]

True, you cannot use infinite values in algebra, but we aren't. 0.999... isn't infinite, it is infinitely long. There is a big difference. Technically, all values can be represented as infinitely long:

1 = 1.000...
3.64 = 3.64000...

Etc.

There are various proofs that 0.999... = 1, not all of them are algebraic.

Sometimes there are questions when "infinity" is used. it is a concept, some say a limit. But what about the "set" of infinite limits?

Can anyone demonstrate to us what the difference is between "infinite", and/or "infinitely long"?

---

Are all numbers infinte

True, you cannot use infinite values in algebra, but we aren't. 0.999... isn't infinite, it is infinitely long. There is a big difference. Technically, all values can be represented as infinitely long:

1 = 1.000...
3.64 = 3.64000...

Etc.

There are various proofs that 0.999... = 1, not all of them are algebraic.

Sometimes there are questions when "infinity" is used. it is a concept, some say a limit. But what about the "set" of infinite limits?

Can anyone demonstrate to us what the difference is between "infinite", and/or "infinitely long"?

[IATIA]

True, you cannot use infinite values in algebra, but we aren't. 0.999... isn't infinite, it is infinitely long. There is a big difference. Technically, all values can be represented as infinitely long:

1 = 1.000...
3.64 = 3.64000...

Etc.

There are various proofs that 0.999... = 1, not all of them are algebraic.

Sometimes there are questions when "infinity" is used. it is a concept, some say a limit. But what about the "set" of infinite limits?

Can anyone demonstrate to us what the difference is between "infinite", and/or "infinitely long"?

I have already shown one equation in which 0.999... and 1.000... are not interchangeable. The problem is infinity and we just need a better way to handle infinities.

(Technically, all numbers are infinitely long, but we will stick with tradition and consider any infinitely long number that does not end in zero.)

Take for instance this 'infinite' set -> ( 0.1, 0.01, 0.001, ...) in which only the last number is 'infinitely long' (in the set of transcendentals, all numbers are 'infinitely' long. )

The "last' number would be 0.000...1, but our math at this time does not allow anything tagged to the end of an infinite string (and actually that is a judgement call 1.0 x 10E-infinity) which invalidates the 'last' number in this set, but because the set is infinite the 'last' number is discounted (or "called" zero). This makes the set short one number but again, because of infinity and the way we handle it, infinity-one still equals infinity and the set is still complete without the 'last' number. Without this 'last' number we have nothing to subtract from one to get 0.999...(again, judgement call 1-(1.0 x 10E-infinity)) so it is accepted as 1, but they still are not interchangeable.

0.000... = 0.000...1 = 0.000...pi all equal 0

An infinite string of zeros is not necessary because it does not change any values, however a string of any other number changes the value every time it is repeated.

By definition 0.1x10E-infinity = 0.1x10E-infinity x 10

But if you divide both sides by 0.1x10E-infinity then 1=10 and that is why one cannot use infinities for anything but limits. An infinitely long number is considered valid and when complications arise, just say 0.999...=1 and all is well.

A number can be infinitely long, but not infinite. A set can be either.

[edk]

I have already shown one equation in which 0.999... and 1.000... are not interchangeable. The problem is infinity and we just need a better way to handle infinities.

(Technically, all numbers are infinitely long, but we will stick with tradition and consider any infinitely long number that does not end in zero.)

Take for instance this 'infinite' set -> ( 0.1, 0.01, 0.001, ...) in which only the last number is 'infinitely long' (in the set of transcendentals, all numbers are 'infinitely' long. )

The "last' number would be 0.000...1, ..........

Which equation?

If there are infinite zeroes and then a 1, there is no 1. What would the smallest number larger than 0.0...1 be? Are numbers preceded by an infinite number of zeroes even ordered? Any of these numbers with infinite zeros followed by some other numbers is simultaneously equal to itself, larger than itself and smaller than itself, which doesn't sound much like a number to me.

[IATIA]

Which equation?

here

If there are infinite zeroes and then a 1, there is no 1. What would the smallest number larger than 0.0...1 be? Are numbers preceded by an infinite number of zeroes even ordered? Any of these numbers with infinite zeros followed by some other numbers is simultaneously equal to itself, larger than itself and smaller than itself, which doesn't sound much like a number to me.

That is why infinity is a problem. In scientific notation one can show these infinitely long strings (including the next number up or down), but they are all still considered to be equal to 0.

[1 ≤ |x| < 10] xE-infinity=0

Infinity - x, where x is any number still equals infinity. Any attempt to use infinity in an equation results in invalid answers.

[little_monkey]

Don't know if anyone pointed this out:

1/3=.3333...

Multiply both sides by 3

1=.9999...

Why is math so GODDAM consistent?

[edk]

That is why infinity is a problem. In scientific notation one can show these infinitely long strings (including the next number up or down), but they are all still considered to be equal to 0.

[1 ≤ |x| < 10] xE-infinity=0

Infinity - x, where x is any number still equals infinity. Any attempt to use infinity in an equation results in invalid answers.

In the equation you posted, f(0.999...) is also undefined.

Using literal infinity sometimes results in invalid answers, in the same way using zero sometimes does. In general it is fine as a coefficient but you can't reduce an equation with it because all you prove is that infinity == infinity. By your logic, the square and square root operations are invalid: (-1)^2=1 => sqrt((-1)^2)=1 => -1=1

I think you may have missed my point that the infinite zeroes thing precludes the difference between 0.99... and 1 from being a number (by your definition of 0.99...). This means we can add an irrational number and something that isn't a number at all and come out with a rational number. Ignoring this though, would you agree that if we scale this up by, say, a billion, (10^9)*(0.0...1) + (10^9)*(0.99...) = 10^9? If so, either the infinitesimals are different sizes (a contradiction in terms) or 0.99... is not consistent with the rules of multiplication.

Also: What is 0.99...^2?

But we're not dealing with infinity here, we're dealing with infinite repetition, in this case of a single digit. You clearly understand that repeating decimal representations where the repeating unit has a finite number of digits evaluate to a rational number, so why there is a special case for when all the digits are equal is beyond me. I demonstrated a method for finding the rational number that's equal to a given repeating decimal.


It's clear that 0.900900900900... =900/999; 900.900900... - 0.900900 = 999(0.900...) = 900 => 0.900... = 900/999. It is easy to see how this can be generalised to a sequence of any number of zeroes; 0.900000900000... = 900000/999999, for example. When applied to 0.99..., a special case of the above where the length of the string of zeroes is 0, we see that 0.9... = 9/9. No infinity is required anywhere.

[houseofcantor]

But it's mathematical chicanery. The mind always wants to make numbers real. Here we go: 5.13 x 10^61. The number of the universe; well, age in Planck time. But the difference between finite and infinite is not a number, it is a perspective. One cannot write 9/10^(n) on to every n quark in the universe to add to to one, just ain't done; so whether or not it really sums to one is a metaphysical concern.