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jsilvela

jsilvela

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Let me explain it better.

1 - 0.9 = 0.1
1 - 0.99 = 0.01
1 - 0.9999 = 0.0001

1 - 0.9bar = 0.0bar (A never ending sequence of nines results in a never ending sequence of 0's)

So since 0.0bar clearly equals 0 then 1 must = 0.9bar.


EDIT: @BDWoody 's works also
So, agree with the gist of it, but there's an implicit use of limits.
In your sequence of 0.1, 0.01, 0.001 .... you will never get to 0.0bar. You can never get rid of that pesky "1", even if it is farther and farther to the right.

If you want to explain 0.9bar = 1 this way to someone who doesn't know limits, there's some hand-waving involved.
In contrast, BDWoody's proof requires no hand-waving.

So, interestingly, in analysis books you learn the construction of the real numbers with Dedekind cuts.
Dedekind cuts contain an infinity of numbers, but are easier to reason about than "a never-ending sequence of decimals".
The gist is to define a real number as the set of all rational numbers greater than it.
Having done that, you need to define all the arithmetic operations, and prove that indeed this set of numbers works as a number.

E.g. you define the square root of 2 as, the set of rational numbers such that their square is greater than 2.

So, you could think of defining 0 as the set of all rational numbers greater than 0.
 

antcollinet

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So, agree with the gist of it, but there's an implicit use of limits.
If you are talking limits you are misunderstanding the concept of infinity.

0.9bar has an infinite sequence of 9s after the point. It doesn't end.

So similarly the 1 - 0.9bar has an infinite sequence of 0s. There is no limit at which you have to append a 1 to the end.


I accept that @BDWoody 's works without needing to grasp infinity.
 
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jsilvela

jsilvela

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If you are talking limits you are misunderstanding the concept of infinity.

0.9bar has an infinite sequence of 9s after the point. It doesn't end.

So similarly the 1 - 0.9bar has an infinite sequence of 0s. There is no limit at which you have to append a 1 to the end.
I'm misunderstanding infinity, but you're not, I take it.

Making notions of limit formal took a large effort in the mathematics of the 19th century.
You can't really apply intuition and common sense.

Limits are somewhat finicky and frustrating, so there has been some effort in formalizing so-called infinitesimals.
Which are more abstract to construct.
 

antcollinet

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I'm misunderstanding infinity, but you're not, I take it.
I was referring to the person referred to in your post who "doesn't understand limits". My wording using "you" didn't make this clear. Sorry.
 

tomtoo

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If i would be a 1 i would be upset. Couse i would say what ever you do you .9period iam always a little more. And i will this be forever. ;)
 

holla

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If you are talking limits you are misunderstanding the concept of infinity.

0.9bar has an infinite sequence of 9s after the point. It doesn't end.

So similarly the 1 - 0.9bar has an infinite sequence of 0s. There is no limit at which you have to append a 1 to the end.


I accept that @BDWoody 's works without needing to grasp infinity.
I think this is just a clash of meanings for the word "limit". In mathematics a "limit" is a technical term that does not mean that something has stopped.

The statement that 0.9999.... = 1 is really the same thing as the formal statement that the limit of the sequence

0.9
0.99
0.999
0.9999
...

is 1.

What it means is: think of any value you like, no matter how small. Eventually, the sequence of numbers above gets within that distance of 1, and stays within that range forever.

For example if you think of the value 0.05, then after two steps, the sequence is within 0.05 of 1 and every later element of the sequence is also in that range.

By the way, the BDWoody proof that 0.999... = 1 is very nice from an intuitive point of view but takes a bit of work to make into precise mathematics. To define what 0.999... means, you need the notion of limit as I tried to describe above. Then to show that you are allowed to say

10 x 0.999... = 9.9999...

and other such operations -- which seem obvious from an intuitive point of view -- takes another pile of work. It does all work out in the end because everything in the argument is appropriately well-behaved: the limits we need do exist, and the operations we are performing are all continuous, so they interact as you would expect them to with the sequences and their limits.

And that is what first semester undergraduate mathematics students do all day. Well, until algebra class starts.
 
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jsilvela

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By the way, the BDWoody proof that 0.999... = 1 is very nice from an intuitive point of view but takes a bit of work to make into precise mathematics. To define what 0.999... means, you need the notion of limit as I tried to describe above. Then to show that you are allowed to say
exactly right.

I learned formal calculus with my beloved Spivak, and then have read graduate level analysis with parts of Rudin. Love Stein & Shakarchi.

But there's the book by Omar Hijab Introduction to Calculus and Classical Analysis, which uses non-standard analysis and infinitesimals, and (not having read it), I have to wonder if he has a point, if it would be an easier path to learn without all the epsilon-delta arguments.
 

boswell

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limits need something to approach a value, often 0 or ∞ (its reciprocal? = big philosophical argument there ) or any other number
the number 0.99999... (infinitely recurring) needs to be regarded as a series ie a sum of terms
0.99999... = 0.9 + 0.09 + 0.009 + 0.0009 + ....(infinitely recurring)
as the number of terms approaches infinity the value of the series approaches 1

a more formal presentation of the BDWoody( and all others) proof is as follows
let x = 0.99999...
x = 0.9 + 0.09 + 0.009 + 0.0009 + ....(infinitely recurring)
10 x = 9 + 0.9 + 0.09 + 0.009 ... (infinitely recurring) the conceptual leap is that as the series has an infinite number of terms both the decimal parts of x and 10 x are exactly equal so when we subtract one from the other on LHS and then RHS you get
9x = 9 ( the exactly equal decimal parts subtract to 0) the rest is
x = 1
 
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jsilvela

jsilvela

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I'm trying to find physics results that are impressive and accessible. Can't think of any.

But, learning that Eratosthenes measured the Earth's circumference, back before the christian era,
and used a completely simple and logical approach; that blew my mind.


BTW to think that flat-earthism is still with us today :facepalm:
 

antcollinet

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I think this is just a clash of meanings for the word "limit". In mathematics a "limit" is a technical term that does not mean that something has stopped.

The statement that 0.9999.... = 1 is really the same thing as the formal statement that the limit of the sequence

0.9
0.99
0.999
0.9999
...

is 1.

What it means is: think of any value you like, no matter how small. Eventually, the sequence of numbers above gets within that distance of 1, and stays within that range forever.

For example if you think of the value 0.05, then after two steps, the sequence is within 0.05 of 1 and every later element of the sequence is also in that range.

By the way, the BDWoody proof that 0.999... = 1 is very nice from an intuitive point of view but takes a bit of work to make into precise mathematics. To define what 0.999... means, you need the notion of limit as I tried to describe above. Then to show that you are allowed to say

10 x 0.999... = 9.9999...

and other such operations -- which seem obvious from an intuitive point of view -- takes another pile of work. It does all work out in the end because everything in the argument is appropriately well-behaved: the limits we need do exist, and the operations we are performing are all continuous, so they interact as you would expect them to with the sequences and their limits.

And that is what first semester undergraduate mathematics students do all day. Well, until algebra class starts.
And then, as distinct from a mathematicians approach - you can have the engineers approach. Which looks somthing like:

How close does it have to be so it doesn't matter?

So for pretty much all mechanical engineering, you're not going to need more than 6 significant digits, and even that is probably massive overkill.
So 0.999999 is pretty much equivalent to 1 for practical purposes.

Audio electronics - lets say you need to allow for 32 bits - or around 200dB - 10 significant digits.
0.9999999999 is - for practical purposes indistinguishable from 1.

What is the limit? :p How many digits of 0.9bar is really all that will ever be needed for it to make no difference. As a stupid extreme, how about measuring the estimated size of the universe..... in units of the smallest atom - helium.

So - measuring 93billion light years - 8.8×10^26 m, in units of the dimension of helium - stated as 31pm - 3.1e-11

8.8x10e26 / 3.1e-11 = 2.8e37

So to measure the size of the universe in units of the size of a helium atom you need less than 38 significant digits, and even in this ludicrous extreme example, 0.9bar with only 38 9's after the point is indistinguishable to 1.


Just from an engineers point of view, of course. :cool:
 

boswell

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To my eyes with a ruler 1.0 mm looks the same as 0.9 mm, my micrometer might get to tell between 1.00 and 0.99.
I have always wanted my own set of J blocks.

I would highly recommend this book, it describes the developments in reducing tolerances in the making of things.
The paperback is cheap and readily available on Amazon and well worth the read

Exactly.PNG
 

pseudoid

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There are some impressive results in math / comp. sci. that don't require much knowledge to get to.
I was thinking of sharing them in a recent thread on AI, but that was not really the place.
Interesting article in the Journal today about what type of workers get hit the hardest due to generative AI' reach.
["The Robots Have Finally Come for My Job" by Greg Ip]
I guess such AI is considered to be a "white collar crime" more than the marginalization of the blue-collar jobs of the previous industrial, automation and technological game changers.
Besides journalism guaranteed a spot on the hit list, most vulnerable occupations included mathematicians, web-designers and interpreters.
These four professional types are well-paid, and college-educated and are predicted to get hit right in their human capital! Ouch!
University of Pennsylvania researchers also found that dishwashers, m/c mechanics, and short-order cooks were deemed to have no AI exposure, which would threaten their livelihood.
 

BDWoody

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9x = 9 ( the exactly equal decimal parts subtract to 0) the rest is
x = 1

The last time I had to bust out this proof was when my daughter's middle school math teacher needed some remedial work, after trying to embarrass her in class. He apologized the next day to his credit, after he got my handwritten proof and a few thoughts.

the conceptual leap is that as the series has an infinite number of terms both the decimal parts of x and 10 x are exactly equal

Exactly. That's the part that when I first got it kinda blew my mind, and a lot of math later kinda still does.
 

pseudoid

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...Examples of countably infinite sets: the even numbers; the odd numbers; fractions i.e. rational numbers...
Some numberphile at youtube has 'the proof' for :
ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12
From <
> 8.7million views, posted 9 years ago (8minutes)
Their derivation sounds believable, if you ignore some of the transpositions that ensue.
4 years later, some other mathologer at youtube has a compelling rebuttal:
Numberphile v. Math: the truth about 1+2+3+… -1/12 - Convergent and Divergent sums
From <
> 2.6million view, posted 5 years ago (42minutes)
Hope this is not OT.:facepalm:
 

DonH56

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Infinity, blah, I still have some papers and a book on aleph-null vs. aleph-one around someplace. I am definitely not going to look for it.

1 + 1 = 3 for very large values of 1.
 
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jsilvela

jsilvela

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Some numberphile at youtube has 'the proof' for :

Their derivation sounds believable, if you ignore some of the transpositions that ensue.
4 years later, some other mathologer at youtube has a compelling rebuttal:

Hope this is not OT.:facepalm:
I've heard that some divergent series of the sort turn up in quantum physics.
Sounds crazy but I don't know enough :)
 
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