Is 1 = 0.999... ? Really?

Even if it was not equal to one according to that proof it would still be a rational number since it is not random or non repeatable. Even though
it has an infinite number of decimal places the same as any irrational number. Any number whether rational or irrational can be expressed as a
fraction but only rational ones technically count as fractions. Because neither the numerator or denominator can tend to infinity. Interestingly
though most numbers are actually irrational as Wikipedia correctly says

So 0.999… is technically a “rational number”.

But can’t anyone explain why it cannot be equal to 1?

It can be expressed as the fraction 9/9 :smiley:

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I have to correct myself when I said I did not think 0.999 … is equal to I because they are the same number just expressed differently
The Wiki proof is correct and is accepted by mathematicians. It does appear counter intuitive but that does not actually make it false

The question demonstrates a quirk of human perception and the limitations of the decimal system. It’s similar to an optical illusion but it’s a mental illusion. :smiley:

Emmmm…
… no. :sunglasses:

Well, that is what I was expecting for at least someone to say. That was Wiki’s take on it.

But happens to be wrong. :sunglasses:

If you say so then it must be no. :laughing:

I’ll give you a hint:

0.999… ≡ ∑[from 1 to ∞} of [9 * 10^-n]
That is:
0.9 +
0.09 +
0.009 +
0.0009 +
.
.
.

And for that reason, you know that 0.999… is NOT equal to 1.000…

Just think about what “…” actually means.

Why sum an infinite number of numbers when you only need to sum 9?

1/9+1/9+1/9+1/9+1/9+1/9+1/9+1/9+1/9=1

Represented in decimal form :

.111_ + .111_ … = .999_ and also = 1

That’s it in a nutshell. No infinite series required. KISS

Feels somehow unsatisfying?

Well, that’s life.

But is it really?
… not really.

So you say. :smiley:

What does the “…” really mean?

If there is no exact ‘1’ then there is no exact fraction of one.

therefore, no amount of integers even an infinite amount would ever add up to one, because there would be a numerical entropy caused by difference betwixt the whole and the inexact proportion.

The basic issue with this is that the “…” means that the string of decimal numbers never, never, never gets to an end point. And that means that it never gets to the point wherein it is actually representing the fraction that it was “trying” to get to, its “limit”.

1/3 does NOT actually = 0.333…

Base 10 decimal notation simply cannot be used to represent “1/3”. That is what the “…” is telling you.

No decimal number that ends with “…” ever truly represents its limit. And even though for all practical applications, it might as well be equal, it can be misleading to just ignore that missing final infinitesimal when combining more complex concerns or comparing numbers that are very close, yet infinitesimally different.

1 = 0.999… ← The way I read this is: you have 0.999 and then a little extra. ← That’s what the “…” means. It means 0.999 and more. Why can’t the “more” be 1 - 0.999?

Irrational or rational numbers with infinite places [ either decimal or non decimal ] are accepted just as much as real or whole or other rational numbers even
if they cannot be written in their entirety. The most famous irrational number of all is pi but no one has a problem with it having an infinite number of places
Every irrational number has to be abbreviated because that is the only way to express them. And if you think 0.333 … is not base ten then what is it ? Is there
a rule which says irrational or rational numbers with infinite places cannot be written in base ten ? No because 0.333 … actually uses the notation of base ten

I think that you’re missing the point, as do all of those claiming that it is merely an issue of notation. I would rather explain this to a mathematician, but…

The “…” notation, specifically means that there is no end to be obtained. That means that it never, ever gets up to being exactly 1.0.

And to prove the point:
The number “0.999…” is formed of the following infinite series:

90% of 1 +
90% of the remaining 10% +
90% of the remaining 1% +
90% of the remaining .1% +
90% of the remaining .01% +
90% of the remaining .001% +
.
.
.
Note that always and forever, only 90% of the remaining amount up to 1.0 is ever added to the sum. The very definition of the number “0.999…” forbids the inclusion of the very last infinitesimal amount that would allow it to get up to 1.0. It is always and forever only taking 90% and must always and forever leave 10% of whatever was left, thus never, never, never reaching 1.0.

It is forbidden to ever equal 1.0 by its very definition.

What ever formula was used to create the 0.999… does not resolve to 1.0, but always just an infinitesimal amount less.

0.9r leaves 0.1r, and both are [potential] infinities.

Surely that means 0.9r is infinitely less than 1.

its one of the reasons ‘why’ reality isn’t made of lego [why qm and relativity exists].