Polymathematics

Refutations

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THE FINAL UPDATE!

Okay, so I'm still on vacation, but it's clearly time for me to step back into this.  Thanks to the people at scienceblogs, namely Goodmath/Badmath and the EvolutionBlog for taking up my cause.  And thanks very much to the people in the comments of my other posts for doing their best to explain.  Some of my explanations here use some of their ideas.

Let's do this by classifying the problems people have with my posts, from the most ridiculous to the least (roughly):

You must be a public school teacher, and I fear for your students.  You don't know enough math to teach it.  Stop filling their heads with nonsense.

Wow.  I actually teach at a private school.  Some of my former students have gone on to get perfect scores on SAT's, study advanced math at top-level universities, and place well at the national level in math competitions.  I have B.A. in math from a top-20 school.  All the information online (you can start here and here) agrees with me.  In addition, while I know that there are public high school teachers who are not very good with math, all of the ones I know who teach at, say, the pre-calculus level or higher also agree with me.  Please don't insult the entire profession just to discredit me.  If you're that worried, go to graduate school and study math, and teach it yourself.

.999... is clearly less than 1, but mathematics isn't advanced enough to handle infinity, so you can't prove it.  My intuition isn't flawed, math is.

This is another one that's so wrong that I barely know where to start.  Historically speaking, this debate is quite old.  In the 19th century, this apparent paradox (.999... is clearly less than 1 vs. .999... equals 1) was addressed by some great mathematicians (Cauchy and Dedekind, among others).  In particular, they formalized the notion that the real numbers are infinitely finely divisible, and in their formulations, all arithmetic operations worked the way they seemed like they should.  Using their formulations, all proofs result in .999...=1.  Those formulations are discussed more at the wikipedia site linked above.  Mathematics has been handling infinity well (using definitions that don't even require the use of the word 'infinity') for at least 100 years now.  Until you've studied that work, you might want to be careful about saying that math can't handle infinity.

Variations on:  1 - .666... = .444..., but .999... - .666... = .333..., so 1 and .999... can't be equal.

You'd think that people trying to argue with mathematicians would at least check their work:

1 - .666... = .444...?  If that's true then to check it, I should be able to add .666... and .444... and get 1.  But .6 + .4 is already 1.  So the sum of two larger numbers can't also equal 1.  The check fails.

Variations on:

10x = 9.5
- x =    .5
9x = 9

So x = 1 and also x = .5.  SEE!! I can use your stupid method to prove that 1 = .5!

If x is .5, then 10x is 5, not 9.5.  Your equations are unrelated, so of course you can prove something false.

(A lot of birds with one stone):

10 * .999... isn't 9.999..., it's 9.999...0
1 - .999... is .000...1
2 * .999... is 1.999...8

This is a common mistake among my students.  You're mistaking notation for mathematics.  Notation is not mathematics.  Mathematics is the study of ideas about patterns and numbers, and we have to invent notations to communicate those ideas.  Just because you can write something that looks mathematical, that doesn't imply that what you wrote has meaning.  The number written as .999... has meaning:

9*(1/10) + 9*(1/100) + 9*(1/1000) + ...

Every 9 in the list means 9*(1/something).  But .000...1, for example, is an abuse of notation.  It doesn't correspond with any meaning, so it doesn't communicate anything.  If you think it means something (and putting 1 at the END of an ENDLESS list of zeros shouldn't mean anything), you're going to have to tell me what's in the denominator of the fraction represented by that decimal place.  If you can't tell me that denominator, you're not using the notation right.  If you tell me the denominator is 'infinity', then see the next entry.

1 - .999... is 1/infinity, which is a number bigger than 0.

Mathematicians don't really use the noun 'infinity' very much, and when they do, it's usually as a shorthand for an idea that is relatively easy to define without the concept of infinity.  They do use the adjective 'infinite' to describe lists and sets, and the adverb 'infinitely' to describe how the real numbers are divisible.  While some intuitive ideas can be captured by using the idea of 'infinity' as a kind of number, you have to be very careful with it, and standard arithmetic doesn't usually work.  But as an intuitive idea, anything that might be written as 1/infinity never behaves differently from the number 0.  I can't prove that, however, since 1/infinity doesn't really mean anything.  Using infinity as a number creates fallacies that even the doubters of .999... = 1 would disagree with.

.999... effectively equals 1, but it doesn't actually equal 1.  It rounds off to 1.  You can never really .999... going on forever because you can't live long enough to write it.

These are all really arguments that claim that .999... isn't really a number, and that you therefore have to stop writing it or round it off at some point.  Look, either you allow the possibility that there could be infinitely many (note the use as an adverb!) decimal places or you don't.  If you don't allow it, you'll have a lot of trouble with proofs that pi or square-root-of-2 can't be written using a number that has finitely many decimal places.  If you do allow it, you have to be prepared to discuss what happens if they all equal 9 at the same time, and you have to discuss it without rounding or talking about when they end.

.999... = 1 if you allow limits, but not if you're just talking about numbers.  The limit of the series isn't the same as adding up the numbers.

The evolution blog linked at the beginning of this post has an excellent discussion about this.  The upshot is that once you admit that there's an infinite geometric series here (which you have admitted as soon as you merely write .999...), there is no difference between the limit and what the thing actually equals.  They have to be the same by any defintion that is internally consistent.

Your fraction argument only works if I admit that 1/3 equals .333....  I don't think it does, so I don't think your arguement works.  1/3 can't be precisely expressed with decimal numbers.

Well, at least the people who argue this are not abusing notation, and they're not attacking me personally, and they understand that the assumptions have to be correct in order for the argument to be correct.  So I'm giving some credit to this one.  But unfortunately, .333... really does equal 1/3.  If you think 1/3 equals some other decimal, you're going to have to tell me what it is.  If you think that you can't express it with decimals, then remember that the very word 'decimal' itself comes from our base 10 number system, and that's a biological coincidence due to our 10 fingers.  In a different base, 1/3 might be no problem, but 1/2 might be.  Any problem results from notation, not from the concept of 1/3.  Remember, notation is not math, notation just communicates math.

Okay, I really have to go now...I'M ON VACATION, PEOPLE!!!

Uhhhhh, wow.

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I really didn't think this would go this far.  I've gotten thousands of hits in the last couple of days, with links to the original post showing up all over the place on link farms.  I will post a longer response in a couple of day, but I don't have time right now because I'm on vacation.

I will, of course, continue to believe my original claim, and thanks to the math people who post to help me explain it.  I will not be able to answer every single comment (I can't even keep up with reading them).  I hope that some of the people who read the post, though, were enlightened at least a little bit by the formal view of this question (I know I didn't include all the formalities, but I wasn't writing for people that would want them;  the formalities are there, but I'm not going to try to explain them.  This wikipedia article does a good job, if you're interested).

Back in a few days....

The Saga Continues

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UPDATE, WITH REFUTATIONS!

First of all, thanks very much to the Goodmath/Badmath blog for linking to my .9999...=1 post, and thanks to all the people who posted on my blog, too.

There were more demonstrations in the comments of how .99999...=1, and I am sure there are more.  The original post was never intended to cover them all, of course.

But the original troll (removed by request) made the following comment that just has to be addressed (I only feel like it's okay to post it because he posted in the comments.  I won't use e-mailed text without permission):

I was going to post a rebuttal with complete proof from 2(two) ASU mathematicians (who both agree with me), but upon review of all your posts, I came to the ultimate conclusion that you don't need proof. You will go to your grave believing with the core of your being that .9999999... does, in your mind, equal 1. However wrong I, or anyone else may think you are will not matter. Trying to convince you otherwise is like trying to convince an atheist that God exists.

So I am sending him (his e-mail address suggests his gender) the following e-mail, noting to him that it is also posted here:

You seem to be misunderstanding the nature of mathematical proof and belief.  The belief that .9999...=1 is not in any sense a religious belief as you imply.  If I ever see a mathematically sound proof that .9999... does not equal 1, I will humbly admit my error, and become a vocal proponent of that proof.  I repeat what I said in an earlier comment:  if you send me a proof that is verifiably (and I will check it out) by a mathematician at an accredited university (ASU is an excellent choice) that uses the standard real number system, I will post it on my blog with sincerest apologies to you, and then I will explain it on the blog to the best of my ability, including the reasons my previous proof was incorrect.  Until then, I will continue to side with the overwhelming majority (100%, as far as I know) of the mathematical professional community who agree with me.

The challenge, of course, is open to anyone.  This space is available for such a proof, if any should appear.  I'll update again if anything turns up, but I'm still not gonna hold my breath.

No, I'm Sorry, It Does.

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UPDATE!!:  The saga continues at this post.

MORE UPDATES, WITH REFUTATIONS!

THE FINAL UPDATE!

Every year I get a few kids in my classes who argue with me on this.  And there are arguers all over the web.  And I just know I'm going to get contentious "but it just can't be true" whiners in my comments.  But I feel obliged to step into this fray.

.9 repeating equals one.  In other words, .9999999... is the same number as 1.  They're 2 different ways of writing the same number.  Kind of like 1.5, 1 1/2, 3/2, and 99/66.  All the same.  I know some of you still don't believe me, so let me say it loudly:

Do you believe it yet?  Well, I do have a couple of arguments besides mere size.  Let's look at some reasons why it's true.  Then we'll look at some reasons why it's not false, which is something different entirely.  The standard algebra proof (which, if you modify it a little, works to convert any repeating decimal into a fraction) runs something like this.  Let x = .9999999..., and then multiply both sides by 10, so you get 10x = 9.9999999... because multiplying by 10 just moves the decimal point to the right.  Then stack those two equations and subtract them (this is a legal move because you're subtracting the same quantity from the left side, where it's called x, as from the right, where it's called .9999999..., but they're the same because they're equal.  We said so, remember?):

Surely if 9x = 9, then x = 1.  But since x also equals .9999999... we get that .9999999... = 1.  The algebra is impeccable.

But I know that this is unconvincing to many people.  So here's another argument.  Most people who have trouble with this fact oddly don't have trouble with the fact that 1/3 = .3333333... .  Well, consider the following addition of equations then:

This seems simplistic, but it's very, very convincing, isn't it?  Or try it with some other denominator:

Which works out very nicely.  Or even:

It will work for any two fractions that have a repeating decimal representation and that add up to 1.

Those are my first two demonstrations that our fact is true (the last one is at the end).  But then the whiners start in about all the reasons they think it's false.  So here's why it's not false:

• ".9 repeating doesn't equal 1, it gets closer and closer to 1."

May I remind you that .9 repeating is a number.  That means it has it's place on the number line somewhere.  Which means that it's not "getting" anywhere.  It doesn't move.  It either equals 1 or it doesn't (it does of course), but it doesn't "get" closer to 1.

• ".9 repeating is obviously less than 1."

Hmmmm...it might be obvious to you, but it's not obvious to me.  Is it really less than 1?  How much less than 1?  No, seriously...tell me how much less?  What is 1 minus .99999999...?

Really???? Infinitely many zeros and then after the infinite list that never ends, there's a 1????  Surely that's stranger than the possibility that .9 repeating simply does equal 1.  Or for something even stranger, consider this:  if .9 repeating is less than 1, then we ought to be able to do something very simple with those two numbers:  find their average.  What's the number directly between the two?  Or for that matter, name any number between the two.  Let me guess:  the average is .99999...05?  So after this infinite list of 9s, there's the possibility of starting up multiple-digit extensions?  Doesn't that just raise the obvious question:  What about .9999999...9999999...?  Namely, infinitely many 9s, and then after that infinite list, there's another infinite list of 9s?  How, exactly is that different from the original infinite list of 9s?  If you saw it written out, where would the break between the lists be?

I'm afraid that if you apply the "huh??" test of strangeness, you get a much higher strangeness factor if you say that .9999999... is not 1 than you do if you say it is 1.

• "Uhhhhh, I'm sorry, but I still don't believe you.  .99999... just can't equal 1."

Well, let's look a little more carefully at what we really mean by .999999...:

This equation isn't really up for debate, right?  It's simply the meaning of our place value system made explicit.  That thing on the right hand side is called an infinite geometic series.  They have been studied extensively in math.  The word "geometric" means that each term of the series is the identical multiple (in this case 1/10) of the previous term. The definition of the sum of an infinite geometric series (and other series, too, but we won't get into those) goes something this:

1. Start making a list of partial sums:  the sum of the first one number, then the sum of the first two numbers, then the sum of the first three, etc.
2. Examine your list closely.  In this case the list is: .9, .99, .999, .9999, .... (Note that the actual number .99999.... is not on the list, since every number on the list has finitely many 9s.)
3. Find some numbers that are bigger than every single number on your list.  Like 53, 3.14, and a million.
4. Of all the numbers that are bigger than every number on your list, find the smallest possible such number.  I think we can all agree that this smallest number is 1.
5. That smallest number that can't be exceeded by anything on the list is the definition of the sum of the geometric series.

Notice that I keep putting the word definition in bold face.  (See, I did it again!)  That's because it's a definition, which isn't really up for debate.  It is the nature of a mathematical definition that once you acccept it, you have to agree to its consequences.  In other words, .99999... = 1 by the definition of the sum of a geometric series.  It's also true if you use the popular formula

a/(1 - r) with a = 9/10, and r = 1/10.

We're left with this:  merely saying ".99999... doesn't equal 1" admits the fact that this number .99999... exists.  And if it exists, it equals 1 by definition.  The only way out for you now, if you still don't believe it, is to have a different working definition of the sum of an infinite series (go talk to some math professors, and see how far you get) or to deny the very existence of the number .9999....  I have seen a lot of people doubt that the number equals 1, but very few of them are willing to deny the very existence of that number.  If you want to play "there's no such thing as infinitely long decimal representations," I'm afraid you won't get very far, because there's always the number pi to worry about, too, you know.

Okay, so there's my rant.  .9 repeating equals one.  No, I'm sorry, it does.

A Good Deal, a Bad Deal, a Terrible Deal

The Good Deal:

The movie "Akeelah and the Bee".  Sure, it's a feel-good sappy movie that's mostly for kids and that's good because adults can enjoy it, too.  But it has many more things to recommend it:  it deals with race fairly honestly (at least an 11-year-old's world of race);  it deals with white privilege pretty honestly (affluent kids can take Latin, and their moms have time to take them to stuff);  it has its unpredictable moments, which is unusual in this kind of movie;  it has a strong, smart (eponymous) Black girl as a protagonist (I just had to use that word, because I could);  and most importantly, it acknowledges the importance of community, which is relatively rare in kids' movies.  Sure, sure, it has its drawbacks, but most are because it had to be written so that kids wouldn't miss any of the plot subtleties.  So go see it.

The Bad Deal:

I recently bought a car for my wife.  At the signing of all the papers, when the business manager was explaining the warranty, he said that I could extend the 3 year warranty on the non-power-train portion of the care for a mere $1200, and that (how could I pass this up?) the warranty extension had a money-back guarantee.  If I don't use it in the extended period, I'll get my money back.  WOW, is that a bad deal.  So, let's see.  Suppose I give them $1200 now, and 4 years from now, something goes wrong on the car.  "Oh, hurray," I think, because I remember that I've purchased the extension, and this will be covered.  But the bill comes to $600.  Should I pay it out of my own pocket?  Or should I let the warranty cover it?  If I let the warranty cover it, then I've basically paid $1200 for a $600 repair.  This will only pay off if there's another $600 worth of repairs (remember, this car is brand new...things wrong with the manufacturing will likely need repair quickly, within the original warranty period).  So maybe I should pay it out of pocket...hmmmm.  If I do that, and there are no other problems, I lose the use of and interest on my $1200, which I will at least get back.  But if there are other problems, I'm still in exactly the same dilemma.  The only scenerio that makes this worthwhile is if there is a repair in the extension period of approximately $1200 or more.  I was not able to think all of this through at the time, but luckily it seemed fishy enough that I refused.

The Terrible Deal:

If you have been paying veeeeery close attention to this blog, you will know that my wife and I have been working on an adoption of a girl from China since before the inception of the blog.  When we started the process (by applying to an adoption agency) in December of 2004, the wait between the arrival of our papers in China and the referral (a picture, a name, setting travel plans in motion) was approximately 6 months.  In the interrim, the wait has gotten longer, and longer, and longer, and now we know that the wait will be at least a year from when our papers got there (September of 2005), and very possibly longer.  Yet there's no real way to know when, exactly, so my wife and I can't make definite plans for the fall or later.  My wife (who has wanted to adopt for most of her life) is being driven insane by the unpredictable wait.  Even I, with my almost relatively unchanging schedule (teaching, rehearsing, working out, etc.) would like to just get some real information so I know what to look forward to.  We were not planning on it taking this long, and our lives feel on hold until we get that referral.  If we had known the wait would be this long, we would likely have considered adopting from a different country.  In addition, our choice of agency turned out to be somewhat poor.  While they have a good reputation for getting paperwork done, they have been distinctly unhelpful (or worse) in getting us (and their other clients) accurate information about the whole situation.  I'm usually a pretty patient guy about waiting for stuff, but it's starting to get to me, too.  A terrible, horrible, no-good, very bad deal.

Coolest Math Fact Ever

I can't believe I haven't blogged about the coolest math fact ever yet.  It is not easy to prove—which is another way of saying that I couldn't prove it.  I solicited proofs on an internet board, though, and I got one that I could understand.  Which I might post in a couple of weeks when I have time over my spring break.

So.

Divide the circumference of a unit circle into n equal arcs with points P_0, P_1, ...P_(n-1).  From P_0, draw the n-1 chords that connect it to the other points.  Since it's a unit circle and the points are equally spaced, the lengths of these n-1 chords depend only on n.  Now find the product of the lengths of the n-1 chords.  Try to guess at the punchline, which I'll type between the asterisks below in white type...you can highlight it to see the theorem when you're ready.

*The product of the lengths of the n-1 chords always equals n.*

I know...for you non-math people, that doesn't sound amazing.  But...it's soooOOOOooooOOO hard to describe how improbable that result is.  I'm sure the mathies in your life will back me up on that.

Tan, tan, and more tan

No, not the color.

The trig function.

______________________________________________
Exercise for the advanced pre-calculus student:

Prove that if A, B, and C are the 3 angles of a triangle,

    tan A + tan B + tan C = (tan A)(tan B)(tan C).
______________________________________________
Exercise for much more advanced students:

Why the hell is that really true?  I mean, beyond the basic proof using trig identities.  I admit I don't know how to approach the question, but I'd appreciate any explanations you can give, although I don't claim I will be able to understand them if they go beyond (very) basic group theory or complex analysis, which I suspect they might.

A Pi Question

Okay, time for another math post.  Here's my question about pi:

We all know, of course, that pi can't repeat because it's irrational (transcendental, even).  But that just means that it can't repeat the same string over and over and over forever.  There's nothing to prevent any particular string of symbols from repeating once, starting right after the string ends.  So the question is, for some n, do the first n digits of pi repeat (once, in positions n+1 through 2n)?

Here's the only progress that I've made so far.  For a random list of numbers (and I'm pretty sure pi actually hasn't been proven to be totally random, but I could be wrong about that...please let me know if I am), the probability that a specific n consecutive digits will appear at a given location is clearly 1/(10^n).  So for that random string, the probability that the first n digits repeat as described above is 1/(10^n). Call that event E_n.  I think that the set of events {not(E_n) for all positive integers n} is a set of independent events—meaning that if one set of n digits doesn't repeat, then it has no bearing on whether any other set repeats or not.

Therefore, the probability of none of the events E_n occuring is the product of the indvidual events:

[1—1/10][1—1/100]...[1—1/(10^n)]...

which I'll call P, and which doesn't equal 0, since none of the factors (thanks vlorbik) are 0;

and the probability that a string of n digits will repeat for some n must be:

1 —P.

But I sure don't know how to compute that product, and since so many digits of pi are known, I can't decide if this calculation should apply to pi or not.

Any ideas?  Am I way off base here?

UPDATE:  my spreadsheet tells me that P is surely .9, which makes 1-P=.1, which sounds like a semi-reasonable probability that a random string of digits has that property.  but nothing definitive.

Just a Theory

Remainders are just wrong.

By which I mean both mathematically and morally.  Division is such a natural idea that I can no longer accept that it might not "come out evenly".  It is still acceptable to have division result in a decimal, but not in a remainder.  Here is my logic:

I call it my theory of Intelligent Division.  A search of the text of the bible reveals that the word "remainder" is only used to describe the outcome of subtraction, never division.  This is a clear indication that a "remainder" upon division is not an acceptable result.  Rather, we must postulate that in all instances of division, an intelligence greater than ourselves is guiding the outcome, and such an intelligence would never allow something as sloppy as a "remainder".  Look, for instance, at the decimal result of any division.  There is a clear sign of an intelligence at work, because the numbers after the decimal point show non-randomness to a high degree.  To ignore that pattern (in favor of "remainders") would be to ignore the obvious intelligent hand guiding the process of division.

Our culture has clearly lost its way when so-called "number theorists" discuss the notion of ugly "remainders".  Such immorality has spawned completely unnatural processes such as "modular arithmetic" in which 350 times 2 equals 34 ("modulo 666", a clear sign of unscrupulousness) when we all know it must equal 700.  These "mathematicians" tell everyone that "remainders" are an established fact in number theory, and that they explain all sorts of facts about numbers.  But they never explain to you that number theory is only that:  just a theory!

And this theory has huge flaws.  The most glaring of these is the remainderist idea that "you can't divide by 0".  Any theory with such huge gaps in the evidence cannot possibly stand up to intense scrutiny.  The bible contains the word "divide" 129 times (the word "add" only 53 times, "subtract" not at all, and "multiply" only 48 times), and surely it is impossible that such a holy operation would be so fatally flawed.  The time is clearly ripe to overthrow the intellectual hold that remainderists have had on our education system for years.

America's children are being taught about these "remainders" from as early as 8 or 9 years old, and it is time for change!  I demand that Intelligent Division be taught along side "remainder" division so that the 8- and 9-year-olds can make an informed decision about what is mathematically correct.  We cannot let the remainderists set the curriculum for our youth.  If you live in a family-oriented school district (especially in Kansas), please let the school board know that there is an alternative.  They can demand that students be exposed to all ways of looking at division, not just the ways the remainderists insist upon.  The future of country is at stake here.

Reflexive Education

I recently taught several of my classes about the three major properties of relations:  reflexive, symmetric and transitive (which I'm going to assume you are familiar with...if not, you probably don't care much about this post anyway).  In my math class experience, they were taught as:

Reflexive

for all x:  x = x

Symmetric

for all x,y:  if x = y then y = x

Transitive

for all x,y,z:  if x = y and y = z then x = z

To which I had always responded with an internally muttered, "ummm, duh".  But what they didn't tell me was that it's just not a big deal that 7=7.  Of course it does.  That's not what the reflexive property is really all about.  Relations can be any kind of true/false comparison between any kinds of objects.

6 = 9 (false, compares numbers with "equals")

8 < 12 (true, compares numbers with "less than")

Angle X is complementary to angle Y (compares angles with "complementary)

p implies q (compares facts with "implies)

and more.

And the three properties might apply to all of them.  So as the sets get more fun and the comparisons get more interesting, the kids actually get excited about deciding which properties apply.  Here are the examples I use (and the properties they have):

comparisons on numbers:

equals (R, S, T)
greater than (T)
not-equal-to (S)
[note:  not-equal-to isn't transitive since the first and third numbers could be the same]
divides-evenly (R, T)

comparisons on angles:

congruent (R, S, T)
complementary-to (S)

comparisons on facts:

implies (R, T)
[note:  the fact that it isn't symmetric is a good review of how the truth of a conditional says nothing about the truth of its converse]

comparisons on students in school:

has-a-class-with (R, S)
has-English-class-with (R, S, T)
[note:  because you can only be taking one English class at a time]
has-tied-the-shoes-of (R)
[note:  gets laughs]
has-kissed-on-the-mouth (S)
[note:  i tell them this is PG-13 rated.  gets big laughs and giggles and "eeew"s (when considering transitivity especially)]

comparisons on all people:

sibling-of (S)
[note:  strictly defined as full biological or adoptive sibling...no half- or step-.  not transitive for the same reason not-equal-to isn't]
brother-of (none!)

I tend to think that a lot of people don't understand that math vocabulary and symbols are supposed to reflect the patterns of thought that go on in our heads.  Math is not about an arbitrary set of rules to apply to bizarre symbols.  The students think this is barely math, but I tell them it's very much math--it's just that they never realized what's really math.

And the next day, they all know exactly why every segment is congruent to itself:  "Because congruence is reflexive!"