I think it is 1. It can equal 2. That is a mystery to me!


“We were able to prove that 1 = 2, after very, very complicated work,” said Radziwiłł.
https://www.quantamagazine.org/a-numerical-mystery-from-the-19th-century-finally-gets-solved-20220815/

I read the above article and have no idea what it was so about. But when I saw you could prove 1 equals 2 I lost my shit.

b=a
ab=a²
a²+ab=a²+a²
a²+ab=2a²
a²+ab-2ab=2a²-2ab
a²-ab=2a²-2ab
1(a²-ab)=2(a²-ab)
1=2


Huh ?

Dave the Atheist Wrote:
-------------------------------------------------------
> b=a
> ab=a²
> a²+ab=a²+a²
> a²+ab=2a²
> a²+ab-2ab=2a²-2ab
> a²-ab=2a²-2ab
> 1(a²-ab)=2(a²-ab)
> 1=2
>
>
> Huh ?

b=a
(a²-ab)= 0

you divided by 0

Bad Atheist!

how about this?

consider 0.9999999 (forever)

let x = 0.999999 (forever)
so 10x = 9.9999999 (forever)

10x - x = 9
9x = 9
x = 1

so 0.9999999 (forever) = 1

Normally, I would offer a

QED (quod erat demonstrandum)

at the end of my algebra, but will pass this one time and finish with a tip of the bowler to our English friends and offer a

BYU (Bob's Your Uncle!)

It's always that last step. Blah blah, therefore the church is true.

IN on math tread !!! ~


why was '6' afraid of '7' ??? ~










because '7' ate '9' ~




wait ~



what is this thred about ? ~

OMG, actual math-like bloviation. Will wonders never cease. :)

Proofs that two obviously unequal numbers are equal usually (always?) bury a divide by zero in a bunch of obscure algebra. The whole point is to be deliberately obscure, which with algebra is pretty easy. I'm not a huge fan of algebra, for precisely that reason. I do not, in general, find algebraic proofs that enlightening. I find them a necessary evil. Geometric proofs can often be intuitive, and enlightening, real "A ha!" moments.

Having now dissed algebra, the second example, of 0.999... = 1.0 is real mathematics. As long as you are willing to accept the Platonic ideal of an infinitely long string of 9s, then 0.999... is indeed equal to 1.0.

We actually see that sort of thing all the time in calculators. 1 / 3 = 0.333... We don't freak out about that.

The same math as above works for one third:
x = 0.333...
10x = 3.333...
10x - x = 3
9x = 3
x = 3 / 9 = 1 / 3

Extra credit problem
Convert 0.142857142857... back to the ratio of two integers. Same idea as above, but you have to multiply by something much larger than 10.

All repeating decimals can be converted to the ratio of two integers.

All repeating rifles eventually need to be reloaded.

The universe is so god-blessed deforgiving and anti-remitting!!

Brother Of Jerry Wrote:
-------------------------------------------------------
> OMG, actual math-like bloviation. Will wonders
> never cease. :)

>
> Extra credit problem
> Convert 0.142857142857... back to the ratio of two
> integers. Same idea as above, but you have to
> multiply by something much larger than 10.
>
> All repeating decimals can be converted to the
> ratio of two integers.

1428570/999999 = 0.142857142857

For the folks who don't dig algebra, I just tell them I will believe 0.9999 (forever) is different from 1 if they can find a number in between them.



I actually use this example in the derivation of the Black-Scholes model for my students when I am showing how Ito's Lemma can be invoked to make annoying things disappear by subtracting an equally annoying thing (in this case delta times the stochastic term) from it.

and voila!

Bob's Your Uncle, all over again!

I am a lot of fun at a party...

Oh, stochastic processes!

There's nothing more erotic than random probability distributions!

But for the laws of random probability I would not have, could not have, married live women!

Probably!

Close. You put an extra 0 in the numerator.

142857 / 999999 = 1 / 7.

OK, not an easy fraction to reduce!

misplaced post



Edited 1 time(s). Last edit at 09/02/2022 11:57PM by Brother Of Jerry.

Brother Of Jerry Wrote:
-------------------------------------------------------
> Close. You put an extra 0 in the numerator.
>
> 142857 / 999999 = 1 / 7.
>
> OK, not an easy fraction to reduce!

Zero? as in nothing?

uh you can just ignore it then, and I got the right answer

you never heard of new math?

:)

dagnabbit, this is what happens when you don't check your answers carefully

apologies to Gladys for talking smut

In my limited experience a party doesn't get exciting until the Greeks show up.

Sorry, but I overslept on this one:

"Proofs that two obviously unequal numbers are equal usually (always?) bury a divide by zero in a bunch of obscure algebra. The whole point is to be deliberately obscure, which with algebra is pretty easy. I'm not a huge fan of algebra, for precisely that reason. I do not, in general, find algebraic proofs that enlightening. I find them a necessary evil. Geometric proofs can often be intuitive, and enlightening, real "A ha!" moments.

COMMENT: I don't quite know what to make of this. The algebraic rule that one cannot divide by zero is not arbitrary, or a rule of mere convenience. Moreover, it does not result in mathematical 'obscurity,' quite the contrary, it results in mathematical rigor. Finally, I am not sure what you expect by a proof being "enlightening." In mathematical proofs you are not necessarily expected to have an 'aha' moment at the end, in the sense that the conclusion is intuitively independently 'seen' as true, independent of the rule-guided steps of the proof itself. In any event, I would guess that you are in a small minority of mathematicians who have this negative view of algebra, and I find it quite baffling. A mathematical physicist would no doubt find it bizarre. Simple geometric proofs are indeed fun and can be directly enlightening. But then, we *do* need analytic geometry, which, of course, gets us right back into algebra.
_______________________________________

"Having now dissed algebra, the second example, of 0.999... = 1.0 is real mathematics. As long as you are willing to accept the Platonic ideal of an infinitely long string of 9s, then 0.999... is indeed equal to 1.0."

COMMENT: Consider the series 0+.9+.09+.009. . . . This series clearly tends to *converge* to 1. However, mathematically, constructively considered, it does not equal 1, and never will. The Platonic ideal (capital P) in mathematics is a reification of all of mathematics, without arbitrarily adding 'infinity' or the 'infinitesimal' as a metaphysical convenience without mathematical context. Thus, the infinite set of real numbers is within the Platonic realm. But this does not mean that any given divergent or convergent series is represented in the Platonic realm as being equal to either 'infinity' itself (divergence), or to the number that a series might only *tend* to converge to. If it did the legitimacy of mathematics would be undermined. In short, the Platonic realm is not a blank check to fix any mathematical problem, solve any mathematical mystery, or to fill in the blank with respect to any needed or helpful mathematical generalization or specific result. In physics, this is one reason why the method of renormalization remains to some extent controversial, notwithstanding how well it 'works'.
______________________________________________

We actually see that sort of thing all the time in calculators. 1 / 3 = 0.333... We don't freak out about that.

The same math as above works for one third:
x = 0.333...
10x = 3.333...
10x - x = 3
9x = 3
x = 3 / 9 = 1 / 3

COMMENT: This example is telling. Calculators 'round off' numbers; they do not 'fix' the infinite series by turning irrational numbers into rational numbers or integers. The same is true with the above .9999 . . . series. Eventually, the calculator gives you a 1. But no matter how many 9's there are in the series, there will always be another one between that number and 1, making the rounding off feature a convenient 'illusion,' mathematically speaking.

Now, consider the relation between the radius and circumference of a circle 2(PI)R Since PI is irrational, the ratio between the radius of a circle and its circumference must also be irrational, the circumference must therefore be indefinite, when e.g., R=1? In this instance, we might ask, 'Doesn't the Platonic realm "fix" the problem? by making the infinity of the circumference a definitive 'holistic' reality? Well, yes. Here the definite, holistic, idea of 'infinity' must be invoked in order to get the geometric result that is evident and obvious from experience. But this valid metaphysical assumption is very different from calling upon Plato's realm to make the convenient rounding of numbers also actually real, definitive quantities of our mathematical expressions. .999999999 = 1, is rounding approximation. 2(PI)R is a definitive reality.

I hope all this makes sense. I would welcome your comments explaining why any of the above is mistaken. After all, you are the mathematician here. I am admittedly just a lowly 'pretender' who--for the benefit of RfM--should best remain 'undisturbed.' :)

Mathive aggressive.

They were using a proposed improvement to the cubic large sieve, and discovered that using that "Improvement" allowed them to show that 1=2. They rightly concluded that the proposed improvement was incorrect. They then used the original unimproved version of the cubic large sieve and were able to demonstrate what they were trying to prove (Patterson's conjecture - that the sum of cubic Gauss sums should converge on N^5/6).

From the article:

"Dunn and Radziwiłł, like Heath-Brown before them, found the cubic large sieve indispensable for their proof. But as they used the formula that Heath-Brown had written down in his 2000 paper — the one he believed to be the best possible sieve, a conjecture that the number theory community had come to believe was true — they realized something wasn’t right. “We were able to prove that 1 = 2, after very, very complicated work,” said Radziwiłł.

At that point, Radziwiłł was sure the mistake was theirs. “I was kind of convinced that we basically have an error in our proof.” Dunn convinced him otherwise. The cubic large sieve, contrary to expectations, could not be improved on.

Armed with the rightness of the cubic large sieve, Dunn and Radziwiłł recalibrated their approach to Patterson’s conjecture. This time, they succeeded.

“I think that was the main reason why nobody did this, because this [Heath-Brown] conjecture was misleading everybody,” said Radziwiłł. “I think if I told Heath-Brown that his conjecture is wrong, then he probably would figure out how to do it.”

brb ~



using google ~


in b 4 ~



copy pasta ~

So that's how it turned out?

I had thought the mathematicians would have realized that all their studies were in vain and then quit their university positions and got jobs flipping burgers.

Pizza delivery. Better tips.

Not mysterious, but the most Mormon number is the square root of -1 (because it’s imaginary).

Good one. What is the square root of iniquity?

Back in an earlier incarnation, I was a math major. The professor in my "Foundations of Mathematics" course offered the following proof that there are no "uninteresting" numbers:

Assume (for purposes of argument) that the set of uninteresting numbers is not empty; then the set must have a smallest element: that is, there must be a smallest uninteresting number. But being the smallest uninteresting number automatically makes this number interesting. Thus the smallest uninteresting number is both uninteresting and interesting, which is a contradiction. Therefore our original assumption that the set of uninteresting numbers is not empty is false, and all numbers are interesting.

And in closing, here's a possibly related limerick from George Gamow:

There was a young fellow from Trinity,
Who took the square root of infinity.
But the number of digits,
Gave him the fidgets;
He dropped Math and took up Divinity.

Are you telling us that nature abhors an empty set?

The empty set is a pivotal concept in theoretical mathematics. Actually it works very hard: the empty set is a subset of every other set, including itself. That can't be easy.

I would switch to theatrical mathematics!

There is actually a "most irrational" number. Google it.

The ancient Greeks didn't consider 1 a number at all, but rather the generator of all the other numbers. And they didn't have a zero, which really hobbled their ability to do algebra-like stuff.

Speaking of 1, a prime number is an integer that can only be evenly divided (that is, divided with a remainder of zero) by itself and 1. Question: is 1 a prime number?

"That depends on what the definition of 'is' is."

--WJC, feeling your pain while smiling with evident sincerity

>Question: is 1 a prime number?

Answer: No.

Yes, but that is because there's an unspoken addendum to the usual formulation. Thus

A prime is a number that is only divisible by itself and one (and those two factors must be different integers.)

See the Fundamental Theorem of Arithmetic

https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic

and

https://en.wikipedia.org/wiki/Prime_number#Primality_of_one

"Primality of one"
.
.
.

"If the definition of a prime number were changed to call 1 a prime, many statements involving prime numbers would need to be reworded in a more awkward way. For example, the fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1, because every number would have multiple factorizations with different numbers of copies of 1."

Also

https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell1/cald5.pdf

"The real problem with one being a prime only became apparent in the 18th century when
mathematicians expanded their purview to wider notions of integers, such as the Gaussian
integers and general number fields. In that context, they were forced to address the notion
of units, to separate the concepts of irreducibility and primality, etc. The generalization of
prime to unique factorization domains clarified the role of unity and now informs the way
we define primality in the ordinary positive integers"

LOL. Exactly the right answer, though much more detailed than I expected. I was aware of the awkwardness of having to change a lot of theorems about primes to say "for all primes greater than 1, it is true that..." instead of simply being able to say "for all primes".

I hadn't thought of the fact that including 1 as a prime would also screw up unique factorization, since 1 could be included as a factor an arbitrary number of times.

I am pretty sure this is the only time in the entire history of RFM that "unique factorization domain" has appeared in an RFM thread. For those not familiar, a UFD has the property that when you factor a number, no matter how you do it, you will come up with the exact same list of prime numbers.

That is a property so basic that most people are (a) well aware of it, and (b) don't even think about it. It is also the property that allows us to recombine those primes in whatever order we want, and get the same result. Example: an egg carton (2 x 6) and a plastic blister pack of mini cupcakes (3 x 4) both constitute a dozen. Everybody, including those people who keep claiming they can't do math, know this. They intuitively understand the properties of a Unique Factorization Domain. (a dozen = 12 = 2x2x3). Ancient Sumerians thought having lots of primes in their number bases was pretty handy, and they were right. A gross = 144 = 2x2x2x2x3x3, Sixty (as in minutes, seconds) = 12x5 = 2x2x3x5. 360 (degrees in a circle) = 12x5x6 = 2x2x2x3x3x5)

Somewhat surprisingly, it is possible to create mathematical structures where unique factorization does not apply. You can factor some values into more than 1 list of primes. That is so foreign to our everyday understanding of arithmetic, that mathematicians have gotten burned by coming up with theorems that actually turned out to be incorrect because they had assumed that factorization was unique in the system they were investigating, and it in fact was not. It is in a way a much fancier version of proving that 1 = 2, which started this thread.

Incidentally, a UFD is not to be confused with a North Dakota/Minnesota Uff da. https://en.wikipedia.org/wiki/Uff_da

That is fascinating. I have always assumed that numbers always reduce to unique factorizations.

I'd ask for an example but might find it incomprehensible. . .

Separate OT thread some day. Examples I have seen in books do tend to be complicated, numbers consisting of an integer plus a sqrt term, and they even confuse me. I did come up with one on my own that is simple and I was rather pleased with it. I saw it years later in a book by Paul Nahin, and he listed a number with two factorizations. I had found an even earlier example in the series that I assume he must have skipped over.

BTW Nahin does outstanding general interest math books, though they all but require calculus to really follow them, and complex analysis wouldn't hurt. He's an electrical engineer, and is not in the least shy about putting actual math in his books.



Edited 2 time(s). Last edit at 09/03/2022 03:14PM by Brother Of Jerry.

Well, I read math books for fun sometimes so I look forward to your next O/T discourse.

There's also "Goldbach's conjecture," one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.

The conjecture has been shown to hold for all integers less than 4-times-10-to-the-18th, but it remains unproven despite considerable effort.

Prove or disprove it, and fame and fortune await you.

Cheers.

https://en.wikipedia.org/wiki/Goldbach%27s_conjecture

And I just remembered one of my grad school physical chemistry profs demonstrating that all odd numbers are prime, as follows:

1 is prime;
3 is prime;
5 is prime;
7 is prime
9 is--well, that's probably just experimental or operator error;
11 is prime;
13 is prime;
that ought to be enough to prove it--what more do you want?

misplaced



Edited 1 time(s). Last edit at 09/03/2022 01:27PM by Brother Of Jerry.

RfM is full of math nerds. *LOL*

Don't worry, ghawd loves trains, too.

e

Avocado's number

Like I say to my wife when watching certain Olympic sports that go way outside the classic running and jumping: "But would it save you from being eaten by a bear?"

BE THE BEAR !!!

Some Olympic sports would put the bear to sleep, so "yes."

Most definitely "yes."

Henry B asked some questions way up thread that I didn't see until today, thanks to Elder Berry's snarky but very funny bit of math/language humor, "mathive aggressive". That's a real groaner, and I wish I had thought of it.

Anyway, here's Henry's post and my responses:

Henry Bemis Wrote:
-------------------------------------------------------
> Sorry, but I overslept on this one:
>
> "Proofs that two obviously unequal numbers are
> equal usually (always?) bury a divide by zero in a
> bunch of obscure algebra. The whole point is to be
> deliberately obscure, which with algebra is pretty
> easy. I'm not a huge fan of algebra, for precisely
> that reason. I do not, in general, find algebraic
> proofs that enlightening. I find them a necessary
> evil. Geometric proofs can often be intuitive, and
> enlightening, real "A ha!" moments.
>
> COMMENT: I don't quite know what to make of this.
> The algebraic rule that one cannot divide by zero
> is not arbitrary, or a rule of mere convenience.
> Moreover, it does not result in mathematical
> 'obscurity,' quite the contrary, it results in
> mathematical rigor. Finally, I am not sure what
> you expect by a proof being "enlightening." In
> mathematical proofs you are not necessarily
> expected to have an 'aha' moment at the end, in
> the sense that the conclusion is intuitively
> independently 'seen' as true, independent of the
> rule-guided steps of the proof itself. In any
> event, I would guess that you are in a small
> minority of mathematicians who have this negative
> view of algebra, and I find it quite baffling. A
> mathematical physicist would no doubt find it
> bizarre. Simple geometric proofs are indeed fun
> and can be directly enlightening. But then, we
> *do* need analytic geometry, which, of course,
> gets us right back into algebra.
> _______________________________________

This is a case of miscommunication. I didn't mean that the division by zero is somehow an obscure rule. Indeed, I think it is one of the few rules of algebra that a great many people can quote, even if they are not entirely sure why it is so, or when it matters. Ya can't divide by zero.

[Until you take calculus, that is, when you learn that limit sin(x)/x as x approaches zero is 1. And then you cover L'Hospital's Rule, and really go down the zero and infinity rabbit hole. For anyone who took calculus and hated it, the phrase "limit as x approaches zero" is enough to trigger nightmares.]

What was obscure in this particular proof is that a²-ab happens to be equal to 0 because a = b. Unless someone was playing pretty close attention, or had seen tricks like this before, it is very likely they would not recognize that a²-ab is equal to zero, and is not just some random value-unspecified bit of algebra. That's what I was objecting to.

I see memes on FB now and then to the effect that "Only people with an IQ above 140 can correctly solve this problem:" followed by a deliberately confusing arithmetic statement that is designed to trick you into messing up the order of evaluation of the operations. Some of them depend on using two different symbols for division ( ÷ and / ) or exponentiation, which is used so rarely by most people that they are not aware that compound exponentiation differs from the standard rules for add, sub, multiplication, div.

It is the math equivalent of writing a sentence with a triple negative and two subordinate clauses, and seeing if the reader can correctly untangle the mess.

They are bad English, and bad mathematics. If an expression is complicated enough that there is a fair chance it will be misevaluated, it should either be parenthesized to make the evaluation order obvious, or broken up into several statements.

As for algebraic proofs being unenlightening, I remember in algebra II and even in calc class they would cover math induction. An example is prove the formula for the sun of the first n squared numbers is ... wait for it...

n(n+1)(2n+1)/6

You add the next square term (n+1)² and beat on it for awhile with algebra until you beat it into the form of (n+1)((n+1)+1)(2(n+1)+1))/6, which shows that if the formula was true for n, it is also true for n+1. If you are not familiar with math induction, take my word for it. :)

OK, fine. I could do the algebra. What I wanted to know was where in bloody hell did the original formula come from? It's not like you can just pull something like that out of the air. The math induction proof of its correctness tells you not a thing about where the original formula came from.

That's what I found very frustrating with algebraic proofs.


One more tidbit on algebra. When I took Algebra II, I hated the chapter on factoring polynomials. There are a few simple forms you just learn to recognize. For quadratics, which are far and away the most common form you run into in intro to physics, the quadratic formula will spit out the factors for you. Why on earth are we messing with all this "factoring" nonsense? That was my reaction to factoring. That quarter in Algebra II in HS was the one and only D I ever got at any level of school in any subject. I really didn't like factoring.

Years later, I bought Mathematics from the Birth of Numbers, by Jan Gullberg. He is a Swedish surgeon whose son went into engineering, so he created a math "cheat sheet" for his son, a quick survey of a great number of topics in math that might be of use to an engineer.

It was so detailed and so good that he got the thing published. It is over a thousand pages of dense material, so it was basically the cheat sheet from hell. It is much like wikipedia - it gives you enough information that you will at least know where to look for additional information.

He has a section on factoring polynomials that gives 5 guidelines to factor the most common polynomials that you are likely to run into. He then says this about factoring:

"More intricate problems of factoring are only rarely encountered in practice. Yet many curricula call for seemingly endless drills of factoring. Instead of nurturing an interest in mathematics, such "education" - bordering on harassment - could block the path to more exciting mathematical challenges."

I am vindicated!!


>
> "Having now dissed algebra, the second example, of
> 0.999... = 1.0 is real mathematics. As long as you
> are willing to accept the Platonic ideal of an
> infinitely long string of 9s, then 0.999... is
> indeed equal to 1.0."
>
> COMMENT: Consider the series 0+.9+.09+.009. . . .
> This series clearly tends to *converge* to 1.
> However, mathematically, constructively
> considered, it does not equal 1, and never will.
> The Platonic ideal (capital P) in mathematics is a
> reification of all of mathematics, without
> arbitrarily adding 'infinity' or the
> 'infinitesimal' as a metaphysical convenience
> without mathematical context. Thus, the infinite
> set of real numbers is within the Platonic realm.
> But this does not mean that any given divergent or
> convergent series is represented in the Platonic
> realm as being equal to either 'infinity' itself
> (divergence), or to the number that a series might
> only *tend* to converge to. If it did the
> legitimacy of mathematics would be undermined. In
> short, the Platonic realm is not a blank check to
> fix any mathematical problem, solve any
> mathematical mystery, or to fill in the blank with
> respect to any needed or helpful mathematical
> generalization or specific result. In physics,
> this is one reason why the method of
> renormalization remains to some extent
> controversial, notwithstanding how well it
> 'works'.
> ______________________________________________
>
> We actually see that sort of thing all the time in
> calculators. 1 / 3 = 0.333... We don't freak out
> about that.
>
> The same math as above works for one third:
> x = 0.333...
> 10x = 3.333...
> 10x - x = 3
> 9x = 3
> x = 3 / 9 = 1 / 3
>
> COMMENT: This example is telling. Calculators
> 'round off' numbers; they do not 'fix' the
> infinite series by turning irrational numbers into
> rational numbers or integers. The same is true
> with the above .9999 . . . series. Eventually,
> the calculator gives you a 1. But no matter how
> many 9's there are in the series, there will
> always be another one between that number and 1,
> making the rounding off feature a convenient
> 'illusion,' mathematically speaking.
>
> Now, consider the relation between the radius and
> circumference of a circle 2(PI)R Since PI is
> irrational, the ratio between the radius of a
> circle and its circumference must also be
> irrational, the circumference must therefore be
> indefinite, when e.g., R=1? In this instance, we
> might ask, 'Doesn't the Platonic realm "fix" the
> problem? by making the infinity of the
> circumference a definitive 'holistic' reality?
> Well, yes. Here the definite, holistic, idea of
> 'infinity' must be invoked in order to get the
> geometric result that is evident and obvious from
> experience. But this valid metaphysical assumption
> is very different from calling upon Plato's realm
> to make the convenient rounding of numbers also
> actually real, definitive quantities of our
> mathematical expressions. .999999999 = 1, is
> rounding approximation. 2(PI)R is a definitive
> reality.
>
> I hope all this makes sense. I would welcome your
> comments explaining why any of the above is
> mistaken. After all, you are the mathematician
> here. I am admittedly just a lowly 'pretender'
> who--for the benefit of RfM--should best remain
> 'undisturbed.' :)

There are actually a fair number of mathematicians here, though I am one of the few that prattles on about it.

Now, 0.999... = 1

You state that constructively, this is not true. Yes, in any actual application, you do not have an infinite number of 9s. You must quit at some point and decide that is close enough for whatever you are doing. That's why I said you have to be willing to accept an infinitely long string of 9s, which is the Platonic ideal, not constructible in the real world.

That's OK. A numeric value for the square root of two is not constructible in the real world either, but we work with it all the time, and we can physically construct the square root of 2 - it is simply the diagonal of a unit square. It freaked out the Greeks that they could construct a line of length sqrt 2, but they couldn't write it down as an exact number. Their feeling that this made no sense is literally how the term irrational as most people think of it (as in "this makes no sense") came to be.

Calculus is completely built on processes that theoretically take an infinite number of steps - that whole "limit as x approaches zero" thing. However close x gets to zero, you can always add another step and get a little bit closer still. Yes, the infinite number of steps is a theoretical construct, but the results of calculus are very exact, real, and reliable.

Same with 0.9999... = 1. The key is that the number of 9s is infinite. As someone earlier in the thread said, if they are not the same, give me one number that is larger than 0.99999.... and smaller than 1. If you can't do it, they are the same number, even if they look radically different.


Infinity is hard to work with. The ancient Greeks just threw their hands up and avoided it, pretending that it wasn't there. When Newton/Leibniz invented calculus, it was another couple hundred years before a rigorous definition of limits showed that what they actually did with calculus was legitimate, and what few parts of what they did was not in fact legitimate. A general theory of the infinite came in the late 19th century with the work of Georg Cantor. I took a course in set theory, and read David Foster Wallace's [yes, he of Infinite Jest, a decidedly non-mathematical novel] book on infinity. The topic still makes my head hurt.

A couple book recommendations.
The Square Root of 2: A Dialogue Concerning a Number and a Sequence, by David Flannery. It is written as a dialogue between a professor and a young person, say in grade 9 or thereabout. A year of HS algebra is enough to get through it, though that by no means implies the math is trivial. You end up doing some amazing things with fractions.

BTW, here is the main sequence the title mentions:
1/1, 3/2, 7/5, 17/12, 40/29 ...

Except for the very first fraction, every new fraction is derived from the numbers in the immediately preceding fraction. See if you can figure out what process you use to generate the next fraction.

And this series of fractions approximate the value of the sqrt 2, each new fraction being slightly closer than the previous one. I find that surprising and fascinating.

The other book is "The Irrationals: A Story of the Numbers You Can't Count On", by Julian Havil. This book requires calculus, Lots of calculus. I have not gone through it in detail, but what I have plowed through was fascinating. It covers ground I had no idea even existed. Havil also wrote a book on the number Gamma which is equally tough sledding, but wonderful material.


And if you made it this far, congratulations. I now return you to our usual ruminations about Rusty and his Magic Pen, and how many temples will be announced next month. ;)

Learning by rote as a form of "harassment" rather than true education:

I submit that it would be wrong, as the word "wrong" is normally comprehended, to deny amongst people generally familiar with Foucault's Discipline and Punish, that your observation is not without merit.

Thanks for this. Here are a few responsive comments:

“OK, fine. I could do the algebra. What I wanted to know was where in bloody hell did the original formula come from? It's not like you can just pull something like that out of the air. The math induction proof of its correctness tells you not a thing about where the original formula came from.”

COMMENT: Well, this is indeed a philosophical problem, but it should be noted that the problem lies at the root of number theory and the foundations of mathematics. Mathematics represents an axiomatic system(s), and the principle of induction follows from such axioms. The rigor of mathematics is dependent upon such axioms within the context of whatever level of mathematical system one is dealing with. It is well known that mathematics cannot stand on its own within a self-contained purely logical structure.

So, if you have heartburn about mathematical induction, it seems to me you should also have heartburn about mathematical foundations generally, and not just certain inductive proofs.
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“Years later, I bought Mathematics from the Birth of Numbers, by Jan Gullberg. He is a Swedish surgeon whose son went into engineering, so he created a math "cheat sheet" for his son, a quick survey of a great number of topics in math that might be of use to an engineer.”

COMMENT: Interesting you should mention this book. When I was an undergraduate philosophy major in the 70s interested in philosophy of science, I became painfully aware that I needed a better background in mathematics and physics and embarked on a personal study regimen to meet that need. I began to look for something in the mathematical literature that was as comprehensive and user friendly as Feynman’s Lectures on Physics. I eventually found a three-volume set by Russian mathematicians, A.D. Aleksandrov, A.N. Kolmogorov, and M.A. Lavrent’ev, called, *Mathematics:It Content, Methods and Meaning.* (1969). Several years later, I too became aware of the Gullberg volume and went through that book as well. These two resources, along with numerous “philosophy of mathematics” books and essays represent my limited knowledge of the subject.

In any event, it is quite clear from the Gullberg volume that there is a practical orientation—rather than philosophical one, the latter of which he has essentially no concern (e.g. no problem with mathematical induction as I recall). His statement, as you quoted (as repeated below), represents that orientation:

"More intricate problems of factoring are only rarely encountered in practice. Yet many curricula call for seemingly endless drills of factoring. Instead of nurturing an interest in mathematics, such "education" - bordering on harassment - could block the path to more exciting mathematical challenges."

Note, however, that your annoyance with factoring is manifestly different from your annoyance with algebra. With algebra, your problem appears to be entirely philosophical, and not at all practical, since mathematical induction obviously works!
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“Now, 0.999... = 1”

“You state that constructively, this is not true. Yes, in any actual application, you do not have an infinite number of 9s. You must quit at some point and decide that is close enough for whatever you are doing. That's why I said you have to be willing to accept an infinitely long string of 9s, which is the Platonic ideal, not constructible in the real world.”

COMMENT: Yes, but notice again that this is a practical move, and not a move grounded in mathematical logic or proof, since, of course, .999999 . . . simply does not equal 1 unless (1) you accept infinity as a platonic reality; and (2) you draw an arbitrary line where infinity is allowed take over the finite and magically converge to 1, or diverge to whatever number you need at the time. In short, whenever you invoke “Platonism” you find yourself knee-deep in mathematical philosophy, and the issues of realism, nominalism, ontology, intuition, and thus mind, become front and center. (The mathematical positions of logicism, intuitionism, formalism, and structuralism have exhaustively considered such issues in all their varieties.) And just as with your concerns about mathematical induction, your philosophical concerns about “Platonism” should kick in.
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“That's OK. A numeric value for the square root of two is not constructible in the real world either, but we work with it all the time, and we can physically construct the square root of 2 - it is simply the diagonal of a unit square. It freaked out the Greeks that they could construct a line of length sqrt 2, but they couldn't write it down as an exact number. Their feeling that this made no sense is literally how the term irrational as most people think of it (as in "this makes no sense") came to be.”

COMMENT: Correct. But did the eventual utility of irrational numbers dissolve the Greek’s philosophical concerns? Of course, one can ask the same question as to negative numbers, imaginary numbers, complex numbers, transfinite numbers, etc. Historically, mathematical utility came to dominate and override ontological concerns, with intuitive axioms added in support of such utility, and the unstated assumption that if it words so beautifully within a system, it must in some sense be 'true.'
_____________________________________________________

“Calculus is completely built on processes that theoretically take an infinite number of steps - that whole "limit as x approaches zero" thing. However close x gets to zero, you can always add another step and get a little bit closer still. Yes, the infinite number of steps is a theoretical construct, but the results of calculus are very exact, real, and reliable.”

COMMENT: Yes, but what does that tell you about Platonism—remember this was the term you introduced, and the term that sparked my interest in the first place. Is there a distinction to be made between the infinitesimal limit as applied to calculus, and the rounding of .999999 to 1? To my mind, the former seems much deeper, i.e. more 'platonic friendly;' whereas the latter seems more a matter of convenience. (I admit that this is my own intuition.) One reason why the former may be more appealing is that the infinitesimal calculus as applied to quantum physics might suggest that there is a discrete limit in nature at the Planck scale before infinity is reached (and physics breaks down). Anyway, that basically is the point I was trying to make.
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“Same with 0.9999... = 1. The key is that the number of 9s is infinite. As someone earlier in the thread said, if they are not the same, give me one number that is larger than 0.99999.... and smaller than 1. If you can't do it, they are the same number, even if they look radically different.”

COMMENT: The key is that the number of 9999s *never* equals 1. Here is a number larger than 0.99999 and smaller than 1: 0.999999. What am I missing here? Obviously, the difference eventually becomes trivial, but it would seem to me that in "pure" mathematics, where "pure” logical rigor is called for, the distinction needs to be acknowledged and remembered.

Finally, thanks for the book recommendations. Allow me to recommend a couple of my own: Paul Benacerraf and Hillary Putnam (Eds.), *Philosophy of Mathematics” (1983), particularly, Part II “The Existence of Mathematical Objects,” and Part III, “Mathematical Truth.” Also, Stewart Shapiro (Ed), *The Oxford Handbook of Philosophy of Mathematics and Logic* (2007) For a more reader friendly discussion of these issues—for those lurkers who are fascinated with this discussion, but don’t really understand it--I would suggest, Stewart Shapiro, *Thinking About Mathematics* (2000).

Thanks, again for your response. I will give you the last word if you care to further comment.

The last word infinity.

>“OK, fine. I could do the algebra. What I wanted to know was where in bloody hell did the original formula come from? It's not like you can just pull something like that out of the air. The math induction proof of its correctness tells you not a thing about where the original formula came from.”

>COMMENT: Well, this is indeed a philosophical problem, but it should be noted that the problem lies at the root of number theory and the foundations of mathematics. Mathematics represents an axiomatic system(s), and the principle of induction follows from such axioms. The rigor of mathematics is dependent upon such axioms within the context of whatever level of mathematical system one is dealing with. It is well known that mathematics cannot stand on its own within a self-contained purely logical structure.

>So, if you have heartburn about mathematical induction, it seems to me you should also have heartburn about mathematical foundations generally, and not just certain inductive proofs.
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I don't have a problem with whether math induction proofs work or not, though there are mathematicians who do argue that the foundations of math induction are not nearly as rock-solid as most people think they are. I am not among them. My complaint is the while verifying a formula using math induction, while it works like a charm, tells you nothing about where the formula came from, which is the more difficult and interesting problem.

As a pandemic project, I came up with as many essentially different derivations of the formula for the sum of the first n squares. I came up with 5, including the induction proof, which was the least informative of the 5. I found a geometric proof that shows exactly where the formula comes from. It turns out to be the product of two simpler formulas. If I can find a diagram online I will post a link. Trying to explain a geometric proof in a text-only format like RFM is too painful to contemplate.
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>“Same with 0.9999... = 1. The key is that the number of 9s is infinite. As someone earlier in the thread said, if they are not the same, give me one number that is larger than 0.99999.... and smaller than 1. If you can't do it, they are the same number, even if they look radically different.”

>COMMENT: The key is that the number of 9999s *never* equals 1. Here is a number larger than 0.99999 and smaller than 1: 0.999999. What am I missing here? Obviously, the difference eventually becomes trivial, but it would seem to me that in "pure" mathematics, where "pure” logical rigor is called for, the distinction needs to be acknowledged and remembered.


I said the key is that the number of 9s is infinite, and then you rebut by giving an example of five 9s, and state that six 9s is more accurate. That is true, but neither five nor six are infinity. 0.999… = 1.0.

https://en.wikipedia.org/wiki/0.999...

Quoting from the article: "This number is equal to 1. In other words, "0.999..." is not "almost exactly" or "very, very nearly but not quite" 1  –  rather, "0.999..." and "1" represent exactly the same number."

The article is 12,000+ words long, and explains in mind-numbing detail what I put in much abbreviated form in my post. You can discount wikipedia, or throw around words like ontological all you want. 0.999… = 1.0, as per any freshman calculus course.

Sorry, I'm not budging on this one.


Interesting experiment I do on calculators now and then - ⅓ can't be exactly represented as a decimal fraction unless you allow for an infinite number of 3s (0.333…) nor can it be exactly represented in binary except as an infinitely repeating fraction.

So here's the experiment. Enter 1 in a calculator, divide by 3, hitting = to force the calculation to complete before going on. Then divide the result by 3.

Based on the internal representation of one third, the final answer should have a slight truncation error, and come out as 0.999999999 to however many 9s are the maximum number the calculator can store. On some (most? all?) scientific calculators, if you do that calculation, the answer you get is not 0.999999999, but 1.0.

Scientific calculators often have a rational arithmetic feature, where they will represent one third internally as a 1, a 3, and a symbol indicating that 1 is to be divided by 3. That is the only way you can precisely represent one third in a calculator, as a symbolic expression, rather than as a binary or decimal number. I do not use that feature in my experiment.

So how do some calculators return a 1 for the expression 1 / 3 * 3?

There must be a software check, and if the answer is the binary equivalent of 0.9999999999, the software substitutes the value for 1.0. The software check may be a bit more complicated than that, but that is basically how it would do that.

And what if the calculation legitimately should have returned a 0.999999999 and the calculator incorrectly changed it to 1? Then it introduced a slight error. Then the debate becomes which is the worse, or more likely error, misrepresenting a calculation like 1 / 3 * 3, or the random calculation that just so happens to come out 0.9999999999?

And that question is above my pay grade. I did just do the experiment on the freebie scientific calculator app on my iPad. It returned a 1.0 for 1 / 3 * 3



Edited 1 time(s). Last edit at 09/13/2022 02:25AM by Brother Of Jerry.

Here is the opening paragraph of your Wiki link:

"In mathematics, 0.999... (also written as 0.9, in repeating decimal notation) denotes the repeating decimal consisting of an unending sequence of 9s after the decimal point. This repeating decimal represents the smallest number no less than every decimal number in the sequence (0.9, 0.99, 0.999, ...); that is, the supremum of this sequence. This number is equal to 1. In other words, "0.999..." is not "almost exactly" or "very, very nearly but not quite" 1  –  rather, "0.999..." and "1" represent exactly the same number."

COMMENT: Without getting into the details, all of the supporting 'proofs' for this position appear to be based upon a host of mathematical assumptions and axioms, some of which are not firmly established. At the end of the day, they seem based upon mathematical utility, not rigorous foundations. But I do not want to get hung up on that here, since I am not qualified to defend this intuition in detail.

Instead, consider the following points:

1. First, note that for every (0.999 ...) (with however number of 9's involved) the ellipsis (...) is a notation having a substantive function within the expressed term; i.e. to represent infinity. Without the ellipsis, each number 0.9, 0.99. 0.999 etc. represents a unique, finite, rational, number. Thus, there is a reliance here that the infinite itself represents a number, in this case "1," completely distinct from 0.999 (without the ellipsis). In other words, (0.999) and (1) are different numbers, but (0.999 ...) and (1) are the same number. In other contexts, infinity might represent the convergence of a series to a different number. Moreover, how about a divergent series? Does infinity there also reduce to a number, as per Cantor’s infinite sets and transfinite numbers? If so, how is this holistic view of infinity defended mathematically, without resort to Platonism? According to mathematics historian, Morris Kline:

“Cantor defended his work. He said he was a Platonist and believed in ideas that exist in an objective world independent of man. Man had but to think of these ideas to recognize their reality. To meet criticisms from philosophers, Cantor invoked metaphysics and even God.” (Morris Kline, Mathematics: The Loss of Certainty (1980), p. 203 (Note that the mathematician-physicist Roger Penrose, is also a realist as to mathematics and infinity, and he also describes himself as a mathematical Platonist.)

In any event, this seems to me as a rather philosophical, and ad hoc view of infinity. In our case here "infinity" seems to operate as a magic wand that creates a finite number ("1") out of an infinite series that constructively and intuitively never reaches 1. This strikes me as a sort of mathematical 'slight of hand' irrespective of its formal utility, and its place in any formal ‘proofs’ within mathematics itself.

2. Here is a quote from Gullberg, which for me makes more sense:

“In ordinary conversation, *infinite* means something that is very great in comparison with everyday things. In mathematics, however, infinity is not a number but a concept of increase beyond bounds. . . A collection such as that of all counting numbers, which continues indefinitely, is *infinite*, and a process that may be continued indefinitely is also infinite. Although the concept of infinity is difficult to grasp, we may simply define it as *not finite,* where *finite* means something that -- at least in theory -- is completely determinable by counting or measurement.” (Gullberg, Mathematics: From the Birth of Numbers (1997) p. 30)

3. There is a substantial disconnect between mathematical Platonism, that is the idea of ‘infinity’ as a holistic number, and physics, where infinity is anathema to theory. Thus, physicist, Frank Close noted:

“[I]nfinity is transcendent, beyond measure, signifying a failure of understanding rather than a real answer. To put this into context, the probability of chance can range from zero . . . to an absolute certainty at 100 percent. "Infinity," by contrast, is boundless and immeasurable; it has no quantifiable meaning. In the context of the questions that the scientists were posing, the answer was nonsense, analogous to your computer giving you an error message: "computer violation" of "overflow." When this happens it is usually a hint that you have made some catastrophic error -- such as instructing the machine to divide by zero. . . For physicists, *infinity* is a code word for disaster, the proof that you are trying to apply a theory beyond its realm of applicability.” (Frank Close, The Infinity Puzzle: Quantum Field Theory and the Hunt for an Orderly Universe, (2011) p. 3-4)

This disconnect between mathematical infinity and physics was also emphasized by the famous formalist mathematician, David Hilbert:

“In summary, let us return to our main theme and draw some conclusions from all our thinking about the infinite. Our principle result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought. . . The role that remains for the infinite to play is solely that of an idea -- if one means by an idea, in Kant's terminology, a concept of reason which transcends all experience and which completes the concrete as a totality -- that of an idea which we may unhestitatingly trust within the framework erected by our theory.” (David Hilbert, “On the Infinite”)

Going back to your original comments, it seems to me that you at first acknowledged the need of Platonism to sustain the argument that 0.9999 . . . is equivalent to 1. You said: “As long as you are willing to accept the Platonic ideal of an infinitely long string of 9s, then 0.999 is indeed equal to 1.0.” This latest response, however, seems to claim that this result is deductively provable from mathematics alone, without regard to Platonism. Which is it? Are the rules used in the Wiki link proofs, dependent upon Platonism? I would say yes, but the author makes no such acknowledgement.

Finally, I want to acknowledge to you personally, and to any other readers here (if there are any), that I do not presume by any stretch to be your equal regarding issues in mathematics. As previously noted, my mathematics background is extremely limited, and everything I said in this thread is within a cloud of uncertainty, and open to refutation, correction or clarification. In this thread I have deviated from my usual practice of not opining on matters of which I am unsure, or ill-equipped to defend. Nonetheless, it is my nature to push back. As such, I especially appreciated your comments.

[Sorry, I did not give you the last word as promised, but there was a lot you added here.]

Oh, Henry.

>COMMENT: Without getting into the details, all of the supporting 'proofs' for this position appear to be based upon a host of mathematical assumptions and axioms, some of which are not firmly established. At the end of the day, they seem based upon mathematical utility, not rigorous foundations. But I do not want to get hung up on that here, since I am not qualified to defend this intuition in detail.

The "proofs" in the wikipedia article are not "proofs", they are proofs, in varying levels of formality. The point is to show that the problem can be approached in many different ways that all end up at the same conclusion.

The proofs are based on axioms that are indeed well established, some of them in the 17th century, and the full buttoning down of the real numbers was established by Dedekind and the gang from the mid 19th into the early 20th century. Yes, there are altered rules that can be used to modify the real number system, and infinite series may behave differently in those systems, or not make sense at all, but we are talking about the standard real numbers here, which is what the overwhelming majority of people are referring to when they think or talk about numbers.

The foundations are very rigorous. They are so precise they read like a credit card contract, that nobody ever reads because they make your eyes glaze over. Yes, there are mathematicians working in systems that have different rules, but in the real number system, 0.999… = 1.000… , not "sort of", but exactly.

You don't like processes with an infinite number of steps. Neither did the Greeks. Zeno's Paradox "proved" that motion was impossible because before Zeno could cross the room, he had to cross half of the room, and before he could cross half of the room, he had to cross a quarter of the room, and before he could …

Long story short, there were an infinite number of steps that had to occur before he could even start to move, and that would take an infinite amount of time.

The Greeks obviously didn't believe that motion is impossible. The whole point of the paradox was to show that any analysis involving infinite processes was not to be trusted. This is still a widespread opinion among people even today.

When calculus came along, it proved that a half plus a quarter plus an eighth plus … is a convergent series whose sum is equal to 1. Not arbitrarily close to 1. If you add the entire infinite series its value is 1. Zeno is not only able to walk almost all the way across the room, but never actually reaches the opposite wall. He gets to walk clear across the room, period. The Greeks simply did not have the correct framework for dealing with infinity.

BTW, 1.0 also has an infinitely repeating fraction, 1.000…
Infinitely repeating zeroes are considered trivial, and they do have the nice feature that they can indeed be exactly represented in any number base at all. That's why we prefer 1.0 over 0.999… . We can represent 1.0 exactly and with far fewer digits, even though in theory they are the exact same number.

BTW 2 - The wikipedia article is not some mathematical quackery that as soon as a real mathematician notices, they will correct the egregious error and show the proper proof that 0.999… and 1.0 are not the same number.

For all your thought, the one thing missing is a proof that they are different numbers. That would be easy enough to do. In the real number system, between any two distinct numbers, there is always a third number that is greater than one of the numbers and less than the other. That is the precise reason there is no "first" positive real number in the real number system. Whatever number you pick as the first number greater than zero, it is always possible to find another number smaller than the the first number, but greater than zero. That is closely tied to the result that the infinity of real numbers is "bigger" than the infinity of the integers. Georg Cantor, long story.

It would be easy to prove 0.999… is not equal to 1.0. Just find a number, any number, that is between them (without chopping the tail off the sequence of 9s). You can even use infinitely long fractions yourself, if you want. The number just has to be bigger than 0.999… and less than 1.0

;)

All of this convinces me that language was invented to lie. The computer you write these posts with approximates numbers with floating point. Usually with decent precision, but they are still approximations. Few apps are integer-only. The number 10 is not really 10, we just call it 10. We accept the lie. If you subtract two floating point numbers that are close in value, the lie is magnified to where it can become a problem so the consequences of FPU approximation need to be understood if you code.

If I pull up a calculator and ask it for the square root of 2, it displays a huge number of digits. That is yet another lie. Those kinds of numbers simply do not exist in physical reality. They are an homage to the Math Gods. The Platonic ideal is so important that the garbage digits must be displayed.

The wonderful thing about Physics problems is that they teach you to tell white lies. They are intractable until you toss out terms that don't matter. Do you want a Platonic ideal or do you want a working equation?

This ties into religion. It's there to be practical. Everyone goes along with the lies because where would we be without them? Of course, the devil is in the details as Mormonism demonstrates. Mormons in the modern world are an insular people whose "faith" can only survive in an echo chamber. An echo chamber is a nice place to go insane if you're into that. Some of us aren't.



Edited 1 time(s). Last edit at 09/14/2022 12:08AM by bradley.

Actually, the number 10 (ten) is really 10, even in binary floating point format. It looks a lot different from integer ten internally, but there is no conversion error, like there is, for example, with the representation of one third in binary, or even decimal, floating point.

As fate and the vagaries of life would have it, I was a non-voting member of the IEEE 754 floating point standard committee for a few years, and got to see the sausage being made. I was quite impressed. I thought they did an outstanding job. I was also very impressed when Apple was one of the first companies to adopt the standard lock, stock and barrel when it was released around 1985. No surprise, really. They were active participants in the design.

I’m rusty now, but there was once a time when I could convert to/from binary fractions in my head. Knowing entirely too much about fractions came in handy in this thread! :)

> . . . I was a non-voting member of the IEEE 754 floating
> point standard committee for a few years. . .

Wow. You must have been a hit with the ladies!

> ∞ and < 0.

That would be the infinite sum of 1+2+3+4+... which you would think would be infinity, and a "proof" that it is actually -1/12.

https://www.youtube.com/watch?v=w-I6XTVZXww&t=86s

The proof plays fast and loose with rules about divergent series, but it turns out it is not completely bogus, and the two following rather lengthy explanations point out.

https://www.youtube.com/watch?v=YuIIjLr6vUA

https://www.youtube.com/watch?v=jcKRGpMiVTw

zero

it's empty, null, void and yet it's an even number by definition

365