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Topic

# 0.999 recurring is not equal to 1

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Science
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Round 1
Pro
This debate is derived from a previous debate on the same topic.
For the benefit of the audience, I will elucidate most of the possible mathematical terms I am going to use.
Since CON has the mantle of proofing 0.9999 recurring equal to 1,
CON must explain why as to 0.999 recurring must be equal to 1,

Definition of EQUAL:being the same in quantity, size, degree, or value.

CON must sufficiently explain all the phenomenon and questions PRO is going to put up,

1.If 0.9999 recurring is equal to 1 then it necessarily means an infinity of numbers are equal to 1. PRO will elucidate,
0.999 is number which is included in the list since 0.999 recurring essentially goes on to infinity, as is 0.9999 as is 0.99999 as 0.999999 but these are not the only numbers in that short space since between 0.999 and 0.9999 there are further an infinity of numbers. 0.9991, 0.99912 etc the list goes on, thus is essence if this statement is true in essence an infinity of infinity of numbers are equal to 1.

2. What are the ramifications of this statement on mathematics as we know,
a) how will point functions be defined? A point function is a function that is exists only on a particular point, that is, it's domain is of a single point, if 1 is that point,then it means in essence an infinity of infinity of points satisfy the condition and domain and the range of the function would change. Thus functions and their calculations would become impossible to compute.

b) How will limit, differentiability and continuity of functions change? some functions have the tendency to hover between positive infinity and negative infinity thus limits, L'Hospital's Rule, Differentiation by 1 st principle are some tools used to evaluate them. The entire sub-domain of Mathematics will collapse if we hold this true.

c) How will set theory work?  If is defined that the set has only natural numbers or whole numbers, or an integer ,then how will the students calculate the number of points in a given set because there will be an infinite number of points in each set. If set theory does not work the ability of modern computers to do vector calculations also falters because be it R or python we enter numbers in set as an input. If vectors does not work , solid mechanics, fluid mechanics, and a lot of other domains of fields will be distorted.

d) what about binary systems and other computational fields dependent on the binary system, since the computer only reads 0s and 1s . what does 0.9999 recurring means for the binary system.

e) What about quadratic and other polynomials of more than 2 degree , how will the roots of the polynomials be calculated? will they be real or imaginary ?

CON must sufficiently demonstrate the effects of all the above fields PRO have not even picked up co-ordinate geometry.

SUMMARY: Even if the statements become technically intense for a reader who essentially does not have a background in Maths, PRO has sufficiently demonstrated a few fields that basically govern everything we do in the word that will collapse if this statement is rendered true.

PRO did not feel the need to cite any website as all the topics covered till higher secondary in most of the countries around the world so readers should not have a hard time making sense of PRO's arguments .

PRO looks forward to CON's reply.

Con
I will just copy paste myself from another debate on this (my opponent wanted this based on that debate, he challenged me and was happy to have this debate).

Key:
* multiplied by
/ divided by
r = recurring
others are obvious

Third Proof
0.9r / 3 = 0.3r = x
x * 3 = 3x
3= 3 * 0.3r = 1

The 10x - proof

x = 0.9r
10x = 9.9r
10x - = 9x = 9.9r - 0.9r
9.9r - 0.9r = 9
9x/9 = 9/9
x = 9/9 = 1

The 0.9r never has a last 9 to reach because it is inherently endless, the idea being it is endlessly approaching a destination value of 1.0

What I argue is that it's actually equal to 1.0

A common objection is that, while 0.999... "gets arbitrarily close" to 1, it is never actually equal to 1. But what is meant by "gets arbitrarily close"? It's not like the number is moving at all; it is what it is, and it just sits there, looking at you. It doesn't "come" or "go" or "move" or "get close" to anything.
On the other hand, the terms of the associated sequence, 0.9, 0.99, 0.999, 0.9999, ..., do "get arbitrarily close" to 1, in the sense that, for each term in the progression, the difference between that term and 1 gets smaller and smaller. No matter how small you want that difference to be, I can find a term where the difference is even smaller.

Pro is saying that a value of 0.0r with a '1' at the end separates 0.9r and 1 and would separate 0 from the difference between them. The problem with this is how infinite series work, meaning that the '1 at the end' is not even a value capable of being considered mathematically.

Let me put it differently, if something is infinitely approaching a value, this implies that it never stops that approach until it's reached the value. If you deny that it's approaching the value, you also deny that the 0.0r ever can have a 1 at the end of it. In other words, both ways around Con is defeated.

infinite

1extending indefinitely ENDLESSinfinite space
2immeasurably or inconceivably great or extensive

If we deem the numbers to be 'approaching', then we concede that they never ever stop that approach and can't ever stop until the 0.999... hits 1 (theoretically).

Alternatively, if we deem the number static and refuse to perceive 'movement towards' then we also need to understand that the number 0.0recurring with a 1 at the end is impossible to write as a number or mathematically express at all, defeating your static value as being plausible. The infinite 0's cannot ever stop being infinite zeros to then have a 1 at the end, they are infinitely close to 0.

It is not an infinite of numbers that are equal to 1, instead 0.9recurring is an illusory value that infinitely and endlessly approaches 1 such that at the point where you go 'ah there's that amount of 9's, it's impossible to stop ever even once until you either approach one or 'die trying' since it will go on forever.

If you argue it's a static value, that's completely fine. The static value is infinitely close to 1, meaning that the infinitesmally small distance between 0.999... and 1.000... is itself incapable of being anything other than infinitely close to 0 itself.
Round 2
Pro
I will concede the debate not because of any of CON's arguments but because of this limit proof , which is very rudimentary.
I was wrong to make a huge fuss out of it, might have even borderline insulted CON, I apologise and cede the debate.
Con
In the final Round of this debate, I will prove myself wrong (as I am playing devil's advocate in this debate and want to show you the flaws in my own arguments). However, if Pro stops conceding, I am willing to fight to the bitter end.

The choice is his.
Round 3
Pro
I told you In personal message I will concede and apologise, I am a man of my words. Show the flaws though I am curious.
Con
Forfeited