This post concerns the weirdest, and possibly the coolest, thing I ever learned when studying math in high school and briefly in college.
Did you all know that the number 1 equals the repeating decimal 0.999...? Or, another way of saying it, did you know that the repeating decimal 0.999.... denotes a real number equal to 1?
I'm not making this up. For a more detailed (and better) explanation, check out the wiki article.
If you haven't clicked through to the link yet, here's the short-hand explanation.
1/3 = 0.333...
3 x 1/3 = 3 x 0.333...
Thus,
1 = 0.999...
Yikes.
You may be asking why I'm writing about this. Truth be told, I've always been fascinated by this "problem," and I encountered it recently during some research I was doing for a sci-fi novel. I won't begin to try and explain the real-world ramifications of this wonderful puzzle. As a matter of fact, I'm not sure there are any. What could it mean? Is this a problem inherent in our base-10 system of mathematics, or would this appear in all mathematical systems?
It's always cool to reexamine things taken for granted in a new light. Good fiction does this in many ways, one of them being the plot twist. A good plot twist is a plausible turn of events that forces us to reevaluate everything we've seen.
Still though, it's just crazy to think of the number 1, one of the very first things we learn, as being the same as 0.999...
5 years ago
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32 comments:
I'm sure that has some crazy implications when you're talking about some formulae for quantum physics or something equally arcane and mind boggling.
It's always kind of a wake-up call that we're not as smart as we think we are when we can debate the definition of the number 1.
Well, I don't want to be a wet blanket here, but it seems to me that the statement 1/3=0.333... must necessarily be approximate. So in turn, one would approximately equal 0.999.
But the bigger issue is that you're apparently back from your vacation and blogging again. It's nice to see.
Nate,
Via facebook, my buddy Mak informed me that this happens in all base systems too. To quote him:
"for example, in base 5, 1/5 = .111..., 2/5=.222..., etc, so (1/5)*5=1, while .111...*5=.55555, or 1."
That's even more insane.
As for what it means, damned if I know. The good news is you and I will be hanging out in a bar tonight and again on Sunday, so we can ponder this over some brews.
Seana,
No, you make a good point that these are just approximations. That's, I think, what boggles my mind so much.
Since high school, I haven't had a problem accepting the fact that words are approximations. I think Conrad really spoke to this idea - he could never quite explain what was happening in a story, but he could give us an idea, and that I think was a testament to his brilliance. His writing was almost akin to impressionism.
But when you tell me that numbers themselves - including something so fundamental as the number 1 - are approximations too, my brain crashes. Maybe that's my problem: I never considered numbers words, I always saw them as representative of concepts. (Putting aside the really weird numbers out there, like "i" which is the square root of -1.)
I just have a hard time wrapping my brain around the fact that we can only approximate "one-ness" in our math.
And thanks for the welcome back, but I'm not sure that's the bigger issue here ;)
Mak, a much smarter man than myself, also told me to look up this problem:
-e^(i*pi)
I tried reading through an explanation, didn't understand a word of it, and then read this summary:
An irrational number (e), raised to the power of an irrational number (pi) multiplied by an imaginary number (i) will give us -1.
v-word: gamer. No doubt Gerard Butler had something to do with that.
It doesn't seem to me that one is indeterminate, it seems that the thirds are. I was thinking last night that of course you can cut a pie into thirds, but the question is, can you? True thirds, I mean. And it seems to me tht quite possibly you can't. And this was of real relevance when I was growing up, because I have two sisters...No wonder someone was always complaining that someone else's slice was bigger.
Seana,
Great example! That goes to show this problem does cause havoc in the real world.
I agree, Brian: the idea that numbers are approximations makes my brain melt.
I think it's what's so fascinating about it, too. There's the whole idea of .999999 and .5555555 being 1, the idea of infinite halves and so on.
I remember being in some high-school math class (calc, maybe?) and the teacher saying something about real numbers.
As a smart ass I said, "What, as opposed to imaginary numbers?"
And with a straight face she said, "Yes."
Nate,
LOL. I did a little digging, and to bring this blog somewhat back to writing and words, it turns out the phrase real number is a RETRONYM! How awesome is that? Apparently, it wasn't thought up until some mathematician came up with an "imaginary number."
Your school system lets you skip
logarithms, imaginary numbers, infinite series and calculus in High School?
Unlike "real" things, numbers are not approximations. However every base system is an arbitrary choice of notational system which lets you express any possible number as the sum of a series of fractions of a limited set of numbers.In an arbitrarily chosen integer base x, the numbers at your disposal are (0...x-1). In base ten, this means 0 to 9. 3,24 is 3* 10/10 + 2*10/100 + 4+10/1000. 10 in base 10 is merely the switch from 1 cypher to 2 cyphers notation and it is the highest possible infinite expansion of the numbers from 0 to 9 - that is, 9.99999 with an infinite series of 9s, or in other words 9*10/10 +9*10/100, etc.
Now, every terminating decimal really ends in a ten, so
1 4.23 2.00001
are really 1.0 or 4.2300 or 2.00010000 (any number of zeros you like)
and therefore can also be expressed with .999 notation, as
0.999... 4.22999... 2.00000999...
The "nines" are an infinite expansion -there's no 0.00...01 at the end. In the same way 0.333... with an infinite expansion of 3s is 1/3. No approximation.
Now, if you really want your mind to melt, search on wikipedia for
-Hilbert's paradox of the Grand Hotel
-Aleph number
-Burali-Forti paradox
-Continuum hypothesis
-Banach–Tarski paradox
-Galileo's paradox
and this wonderfully brain-torturing sci-fi novel I own
-White Light (novel)
or read this free short story by the same author, also downloadable in pdf :Jack and the Aktuals,or,Physical Applications of Transfinite Set Theory
v-word:warsisin and then warsup.
Scary
Well I dont want to be a wet blanket either, but I think what its really saying is that for all intents and purposes 1 = 0.99999.... but actually it doesnt because you dont have infinite time or space to write out all the nines.
One of the most stunning formulae in science however is Euler's Identity which has baffled me since I was child and almost made me believe in God once:
Here's what Wikipedia says about it:
Euler's identity is considered by many to be remarkable for its mathematical beauty. Three basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:
The number 0.
The number 1.
The number π, which is ubiquitous in trigonometry, geometry of Euclidean space, and mathematical analysis (π ≈ 3.14159).
The number e, the base of natural logarithms, which also occurs widely in mathematical analysis (e ≈ 2.71828).
The number i, imaginary unit of the complex numbers, which contain the roots of all nonconstant polynomials and lead to deeper insight into many operators, such as integration.
You can read more Euler's Identity here. Its also on T shirts, posters etc. Its pretty amazing.
I didn't do calculus, because frankly it was not necessary, and as was my way, I talked myself out of it and none of the teachers were able to talk me back into it. Unfortunately, I couldn't see the application. It's too bad, because though lazy, basically I liked math.
I will look at the that Euler's identity again, because after a day of counting change at the cash register, I temporarily don't find math very fascinating.
v word=bangers, which won't give you much consolation, Marco. Although if we think of it as a component of a popular English dish, it may be easier to take.
Marco,
"Your school system lets you skip
logarithms, imaginary numbers, infinite series and calculus in High School?"
No, I definitely remember all these terms from high school math. What I didn't remember was the explanation for 1 equaling .999...Either way, truly fascinating.
The Grand Hotel sounds familiar
-just checked wiki-
and yes, I remember this one too now.
Never heard of Banach-Tarski, but the idea that "a pea can be chopped up and reassembled into the Sun" is awesome.
And thanks for linking to that short story - I'm going to give it a read later today.
Adrian,
"but actually it doesnt because you dont have infinite time or space to write out all the nines."
Okay, then we need two teams of monkeys banging away on keyboards. One team to write Shakespeare, the other team to write out all the nines.
Nice linkage with Euler's Identity. I'd never seen that one before. That wiki article isn't short on praise either: "the greatest equation ever."
Speaking of cool t-shirts, the wife recently bought me a Chuck Norris one that reads: "After a night of heavy drinking, Chuck Norris doesn't throw up. He throws down."
Seana,
I'm somewhat in your boat regarding calculus. I did take it, but was more worried about doing well on tests than I was in actually learning it, as was my MO on far too many subjects. So I ended up cramming for each test/midterm, doing fairly well on the exams, and then not remembering a single damned thing about the subject when school let out.
Marco,
From the short story: "“A Zeno speed-up,” murmured Jack, who’d often pondered the ancient philosopher’s paradoxical observation that any unit is an endless sum of the form 1/2 + 1/4 + 1/8 + 1/16 + etcetera. Every stretch of time held an actual infinity of intervals although, yes, most of these intervals were below the Planck scale. But the Planck hobgoblin seemed to have little force in Alefville."
Is this related to fractals in a way? Depending upon how you measure something, the smaller and smaller your measuring stick, the more your measurement approaches infinity. Like the old shoreline example. Let's say if we measure it in miles, it's 10 miles. But then if we use a yardstick, it measures longer because now we can take into account the rises and falls of the sands and the dunes. Then if we use a ruler, we can measure even tinier changes, thus making the shoreline longer. etc, etc
Or am I misremembering fractals and butchering this idea?
No, Brian, we aren't in the same boat, because you actually didtake calculus and learn it, even if it seems irretrievable now.
It's funny, because I actually did have a good woman math teacher in high school, who was very interested in girls going further with math, and this was still at a time before the computer era had really taken hold, so this was a bit unusual. But neither my sister or I could ever quite see it's relevance. We were laughing about all this the other evening, because as it turned out, she did a double major in computer sciences and dance.
As for me, there have been far too many times in life when someone should have said, okay, you win the debate. Now shut up and do your homework.
Also, I must just say once again that you have a very understanding wife.
What I meant is that here, unless you choose a technical school, you can't skip calculus. Curricula are fixed-they vary according to the type of high school (classical, scientific, linguistic) not the choice of the student, but even classical high schools have obligatory math.
Basically, in the kind of schools which are meant to give you a preparation for university (and not for a specific job) you can't escape calculus.
Fractals are the equivalent of periodic fractions -they are sequences which periodically repeat themselves, like snowflakes which reproduce the same form macroscopically and seen at the microscope.
The reference is to Zeno's paradox
I'm sorry - I realize the story is full of incomprehensible jargon.
my good sounding v-word:sinvony
Marco,
I dug the story, man. The jargon, though often incomprehensible, worked for me and immersed me in the story. Very cool.
and yet you complained about Primer!
He complained about it, yes, but it was more for the sake of blog material. Surely you understand.
Adrian,
I liked the jargon of Primer, it was just that last act that bugged me. I'm still holding out hope for Carruth to make another movie. That guy's got skills to pay the bills.
(It's a good thing there are no Erics that frequent this blog - v word: baneric.)
Seana,
The truth comes out, as it usually does.
a simple postulate- numbers are imaginary, they cannot be real for they were invented by man in the abstract. therefore anything equals anything in this imaginary continuum that man created. however if they are real, then each number has an absolute value. the socialists among us believe they are not real while the capitilists fervently believe in absolutes. This is truly the defining difference between collectivists and capitalists
Anon -
Wow, you really went Warp 8 there.
I'm going to break trend and not comment on the mathematics and instead say that I very much like your difinition of a plot twist. I'm going to share it with my writers group.
I had to go back and look at that plot twist definition, but you're right, J.M.. It's a great way of looking at it.
JM,
Thanks for that, though I certainly can't claim I came up with the idea. I remember reading it somewhere, can't say where though.
I also like the idea of a theme twist, too. I don't know if there's a more formal term for this idea, but it seems to happen a lot in science fiction. It's like a plot twist, and sometimes it's revealed through a plot twist. There are probably many examples of this, but the best one I can think of is the ending of Planet of the Apes. Great plot twist that not only makes you reevalute the story but also your approach to the theme of the story too.
Seana,
For the life of me, I can't remember where I read that definition of plot twist, but it seems to fit really well for good plot twists. In the case of The Sixth Sense, once you see the twist, in your mind you go back over the entire story and see it from the perspective that (SPOILER ALERT) Bruce Willis was dead for 95% of the film.
I do actually love anything that makes me feel like I have been completely gullible and asleep for the course of the book or the movie. Don't ask me why.
v word=putypie, which I quite like, again for inscrutable reasons.
Planet of the Apes and The Sixth Sense are both excellent examples of the plot/theme twist that, as you said, causes one to reevaluate the entire story. I love that. The Sixth Sense in particular "twists" with great subtlety for maximum astonishment at the reveal. Thanks for reminding me of that film. I'm going to watch, analyze and I hope learn I how can achieve something similar in my own book.
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