Does 0.99999... = 1?

[MalneyKerman]

Try subtracting 0.(9) from 1.

1.0000000000000....

0.9999999999999....

0.0000000000000....

Although subtraction is traditionally done from right to left, we can't do that here, because there is no right end.  So we will start from the left.  In doing this, we have to create all the borrows before doing any subtraction.

1 - 0.9 = 0.1.  But there yet remains 0.0(9) to subtract.  So 0.1 - 0.0(9) = ?  We perform a similar step: 0.1 - 0.09 = 0.01.  But there yet remains 0.00(9) to subtract.  This process continues forever.  If 0.(9) were finite, there would be a 1 in the least significant digit of the difference.  But 0.(9) has no least significant digit.  There is therefore no 1 in the least significant digit of the difference, and the difference is 0.  So, 1 - 0.(9) = 0, which implies 1 = 0.(9), Q.E.D.

[kerbiloid]

Spoiler

 

[pmborg]

 

Question if, x = 0.999(9)

is x < 1 or x > 1 or x = 1 ?

Supposing: x = 0.99(9)

and having this equation, to solve in x: 10x - x = 9

Solving: 9x = 9
   
                 9
         x = --- = 1
                 9

So x = 1

 

[Leganeski]

1 hour ago, AlamoVampire said:

It's not possible to add up infinitely many things

 

Why not? There's a perfectly rigorous way to do so, which is what is used in infinite decimal expansions.

1 hour ago, AlamoVampire said:

With the exception of trailing 0's, any two decimals that are written differently are different numbers.

Why? Infinite decimal expansions represent the limit of converging sequences of finite decimal expansions, but different sequences can converge to the same value.

The number 1 already has many different representations, such as 1/1, 3 - 2, and 5⁰. There's no reason why there can't be another one.

58 minutes ago, Ryaja said:

This boils down to the value of infinitesimals

In particular: are there any nonzero infinitesimals? In the hyperreal numbers, there certainly are, but when we're talking about numbers without further context, we mean a real number, not a hyperreal. In particular, decimal expansions such as 0.999... always represent real numbers.

One very important step in constructing the real numbers is taking all the things that would have been infinitesimals and setting them equal to zero.

In the hyperrational construction, the infinitesimal hyperreals are explicitly unified with zero. In the Cauchy construction, all the sequences approaching zero are set equal to zero.

[Ryaja]

this seems to have the same thing as calculating the area under a curve in calculus. What I think this is is that 0.999... is infinitely close to 1 but not quite, I purpose a new concept: Uninfinite(or disinfinite or maybe ininfinite, or superfinity?), an infinitely small number that is greater than 0 but smaller than all other positive decimals and numbers.

Edited December 19, 2022 by Ryaja

[Leganeski]

54 minutes ago, Ryaja said:

an infinitely small number that is greater than 0 but smaller than all other positive decimals and numbers.

This sounds like an infinitesimal. As you mention, infinitesimals can be useful as a way to think about calculus, and both Newton and Leibniz used them in their original formulations of calculus. (Around the 1860s or so, mathematicians figured out how to do calculus without infinitesimals by using limits instead.)

The thing is that infinitesimals are not standard real numbers, and therefore do not have decimal representations at all. (If 0.999... were less than 1, then its square root would be even closer to but still not quite 1, and how would you represent that?)

[Ryaja]

1 hour ago, Leganeski said:

This sounds like an infinitesimal. As you mention, infinitesimals can be useful as a way to think about calculus, and both Newton and Leibniz used them in their original formulations of calculus. (Around the 1860s or so, mathematicians figured out how to do calculus without infinitesimals by using limits instead.)

The thing is that infinitesimals are not standard real numbers, and therefore do not have decimal representations at all. (If 0.999... were less than 1, then its square root would be even closer to but still not quite 1, and how would you represent that?)

I'm specifically talking about an infinitesimal smaller than all the other infinitesimals infinity smaller yet not zero, the tallest possible value.

But ya that is basically what I was thinking of, and I knew what an infinitesimal was before.

Edited December 19, 2022 by Ryaja

[kerbiloid]

6 hours ago, Leganeski said:

infinitesimal

A perfect term to describe a deadline status.

"...It's just an infinitesimal of time till we get ready."
"...You have an infinitesimal of time to do that."
".. We need a infinitesimal of money for that."

And in other cases to describe something very small.

Edited December 20, 2022 by kerbiloid