Is 0.9999 with the nine repeating equal to 1
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After 1 vote and with 5 points ahead, the winner is...
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repeating: an action done over and over again
equal: same in numerical amount/value
- Algebraic equation proof
- Denominator of 9s proof
- Geometric series
However, there are mathematically coherent ordered algebraic structures, including various alternatives to the real numbers, which are non-Archimedean. Non-standard analysis provides a number system with a full array of infinitesimals (and their inverses).[49] A. H. Lightstone developed a decimal expansion for hyperreal numbers in (0, 1)∗.[50] Lightstone shows how to associate to each number a sequence of digits,
The standard definition of the number 0.999... is the limit of the sequence 0.9, 0.99, 0.999, ... A different definition involves what Terry Tao refers to as ultralimit, i.e., the equivalence class [(0.9, 0.99, 0.999, ...)] of this sequence in the ultrapower construction, which is a number that falls short of 1 by an infinitesimal amount. More generally, the hyperreal number uH=0.999...;...999000..., with last digit 9 at infinite hypernatural rank H, satisfies a strict inequality uH < 1. Accordingly, an alternative interpretation for "zero followed by infinitely many 9s" could be
The intelligibility of the continuum has been found–many times over–to require that the domain of real numbers be enlarged to include infinitesimals. This enlarged domain may be styled the domain of continuum numbers. It will now be evident that .9999... does not equal 1 but falls infinitesimally short of it. I think that .9999... should indeed be admitted as a number ... though not as a real number.
My opponent has failed to address all types of numbers, including non-real numbers. If we look at the Wikipedia page on this math statement, it says
In layman terms, this means if we assume there is a smallest number X where X>0 but there is no number such that X>Y>0, then X is the difference between 1 and 0.999... As all my opponent's formulas rely on the idea that 0.999... is a real number, he fails the idea when applied to hyperreal numbers.
a function when going with X to infinity could be "equivalent" to 0.999... but never reach 1
Rebuttal: 0.999… only tends to 1. It is not equal to 1. We only have an approximation.
Reply: It is true that for the sequence a_n = 0.999, it is the limit as n→∞ that is equal to 1. However, since 0.999… is defined to be that limit, it is defined to equal 1.
Without using this definition for infinitely repeating decimals, there would be many numbers, such as 1/3 = .333..., that we wouldn't be able to write out as decimals since there exists no finite sum of tenths, hundredths, thousandths, etc. that exactly equals 1/3
doesn't consider the possibility that it might not be the same interpretation in every circumstance.
Rebuttal: 0.999... and 1 are not equal because they're not the same decimal. With the exception of trailing 0's, any two decimals that are written differently are different numbers.
Reply: 0.999...=1 is another case when two decimals that are written differently are, in fact, the same number. The fact that there are two different ways to write 1 as a decimal is a result of the role that infinite sums have in defining what non-terminating decimals mean.
The decimal system is just a shorthand for writing a number as a sum of the powers of 10, each scaled by an integer between 0 and 9 inclusive. For example, 0.123 means 1/10 + 2/100 + 3/1000. Similarly, 0.999... means 9/10 + 9/100 + 9/1000 + ... and the value of this infinite sum is equal to 1.
Rebuttal: Infinite sums don't make any sense. It's not possible to add up infinitely many things, so any infinite sum is only an approximate value, not a real value.
Reply: Without using infinite sums, there would be many numbers, such as 1/3 = .333..., that we wouldn't be able to write out as decimals. The definition of such an infinite sum is rigorous, but strange in that the sum is defined to equal the limit approached as many terms are added together, whenever this limit exists.
Not all infinite sums of fractions can be evaluated. For example, 1/2 + 2/3 + 3/4 + 4/5 + ... could not have a real numerical value. But the definition of an infinite sum includes the restriction that an infinite sum only has a well-defined valuation when, as we add up terms of the series, the total sum zooms in towards one, specific value. In the case of 3/10 + 3/100 + 3/1000 + ..., 1/3 is the value being approached. There would be no other way to define 1/3 as a decimal otherwise since 1/3 is not equal to any finite sum of tenths, hundredths, thousandths, etc.
In the case of 9/10 + 9/100 + 9/1000 + ..., 1 is the value being approached as more and more of the terms are added together.
For a more complete explanation of infinite sums, check out the Infinite Sums wiki page.
This argument is valid in that the perspective that we're adding "term by term" is how we are evaluating the limit.
Pro offers three proofs, and soundly supports them. Con does not challenge those, but challenges the resolution with a counter proof. That is a fine tactic as I've said before:
"Con has a duty to attempt to disprove (or cast strong doubt upon) the resolution, be that by providing direct evidence against it, or refuting all the evidence provided by pro."
However, in math it gets tricky, because depending on the type of math employed it may be true or false. This means it is usually true, but occasionally false (as con puts it: "it might not be the same interpretation in every circumstance").
Arguments: Pro
In real numbers, they equal. I think I would leave this tied had material from con's final round come earlier allowing it to be addressed. This is a key thing because it's a math debate, and this is where con finally walks us through some numbers.
Sources: Overwhelmingly Pro
Con's R1 is (save for three lines) a wiki page which is not directly linked, and the cutoff between it and his material is unmarked. He points to this, but has a hard time putting any of it into his own words. This had a detrimental effect on his arguments.
Pro on the other hand used sources to support his case, but still explained his numbers to be not dependant upon the appeal to authority.
Plus, were we to just go with sources instead of making arguments, pro would win anyway from using trustable math websites (a better appeal to authority).
Conduct: Leaning pro...
I am choosing to leave this tied, but con, please be careful in future regarding making your own case rather than a quick copy/paste from a source (had the source not been mentioned, I'd likely make it a 7-point loss).
In future, please be careful to mark quoted material to differentiate it from your own work. An in line quotation usually just calls for double quotation marks.
Large chunks of text, I would suggest the quotation tool. ctrl+bracket to increase the levels within it.
please vote! I really appreciate if you do!
Thanks! Please vote!
Very interesting!
Though I didn't completely read this, personally, I've always agreed since third grade that 10 - 9.9999999999 repeating was equal to 0 and have seen a lot of proofs on it too.
Hey ragnar can you please vote?
Very good first round. I'll have to give this another look later.
Vote
I just realized I already refuted your argument but you keep bringing it back up acting like i did not refute it.
Please Vote
You realize that your round 3 argument was already refuted in my round 3? Please look at the 1st time I cited brilliant.org in the round 3.
you also have not rebutted any of my claims which means you say everything I said is correct.
This guy fails to realize its not n geometric sequence, it is a fractional infinite geometric sequence.
Also 1/10^infinity is so small it is undefinable like the width of a line
please watch this video: https://www.youtube.com/watch?v=--HdatJwbQY
Ghandi series are some bank notes used in India. Can you please answer my question?
infinities get really weird. Look up the Ghandi series and how you can derive 0, 1, 1/2, from 1-1+1-1+1-1+1....
It is 0.1134 not 31134. And that is a calculator upside down which is not the same as right side up.
Can you please define "not same in every circumstance". Also, nobody says 31134 instead of hello. My argument still stands.
wrong. "31134" can mean "hello".
What do you mean "not same in every circumstance"
A number is a number. It is the same value in every equation and every instance of mentioning.
Thanks!