## 0899: "Number Line"

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snowyowl
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### Re: 899: "Number Line"

redearth1210 wrote:meerta

Yes I have, it still hasn't completely shown me that the two are equal. I'm fine with the idea, I just don't like supporting ideas without some solid understanding of it.

http://www.smbc-comics.com/comics/20100620.gif
"[rigorous and correct proof that some formula =x times y] Now that you've seen the proof, you know fully why it works."
"But... why?"
"=x*y"
"Right, but why?"
"=X*Y"
"But that's totally unsatisf-"
"=X*Y"

It's always difficult when our intuition tells us something different from the proofs.
The preceding comment is an automated response.

SirMustapha
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### Re: 899: "Number Line"

hurr durr I'll make a series of jokes that are actually jokes I've made before. the people love graphs with math jokes, right?
1) I'll bet you didn't laugh writing this.
2) it's totally unoriginal. you've MADE the e^pi joke before, and made a much funnier comic about it.
3) all of these are silly math... catechisms, might be the right word. silly little things that every math-fan knows, and tells to other math-fans, but they've all heard it before, so they force a chuckle.

2/10 dude.

But you forgot the positive points of today's comic! Here are they:

- The alt-text;
- ... wait, no, that's actually it.

Cranica
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### Re: 899: "Number Line"

K^2 wrote:Hm. I was thinking of it more in terms of re-assigning multiplication unit. Defining (Gird-3)*a=a would make new ring isomorphic to integers.

Since that makes Gird-3 a multiplicative identity, wouldn't that just give Gird = 4, since you can only have one mult. identity in a ring?

iqtestsmeannothing
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### Re: 899: "Number Line"

Hendecatope wrote:I've never actually had to do this but apparently it's common practice to look at every 2 case as an 8 case. (For one thing, 2 has trivial squares like 0 and 1. Moving to 4 and then 8 adds some flexibility)

Interesting point. The perfect squares in the 2-adic units are exactly those congruent to 1 modulo 8, whereas for other primes p, the perfect squares in the p-adic units are characterized by a congruence relation modulo p. However I'd guess the comic was just being silly about 8 being a prime.

Re:those wondering why 0.999... = 1

Just read the wikipedia article on the subject, it covers every aspect of it. If something in the article is confusing you can ask about it.

atomfullerene
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### Re: 899: "Number Line"

Any way you can add in gird and wind up with 8 being prime?

joshrob03
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### Re: 899: "Number Line"

I find it interesting that people seem to want "proof" of the properties of a notation (.99(bar)=1 argument). Notation is true by definition; it cannot be proven because it is only true by assertion. The properties of the notation are invented to make the notation consistent with logic, which can be proven.

For decimal notation, a real number has a repeating decimal representation if and only if it is a rational number (for non-math folks x is rational means x=a/b where a and b are some integers, i.e. it has a fraction equivalent). This is something I'm sure everyone learned in primary school, which is why I bring it up. Now for the intuitive argument: what is the fraction representation of .99(bar)? (Hint: it's 1) Go ahead and try to find another, it'll help your brain wrap around the idea.

Is this a proof? Obviously not. But several valid proofs have been posted to little avail. Sometimes the only way to correct a logically incomplete argument is with another logically incomplete argument (if you think hard enough that will make sense).

iqtestsmeannothing
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### Re: 899: "Number Line"

atomfullerene wrote:Any way you can add in gird and wind up with 8 being prime?

Unless you change the fact that 8 = 2 * 2 * 2, nope. (One of the proposed ways of adding in gird gives 8 = 3 * 3, but in that case 8 still is not prime.)

JoeZ
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### Re: 899: "Number Line"

phlip wrote:Add in the fact that "37" is the most common random 2-digit number, and I think it was just an invented figure.

It does contain the number 7, which we all know is used in made-up figures...
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Andrusi
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### Re: 899: "Number Line"

stardf29 wrote:The .99(bar) thing was amusing, as was the alt-text. Not much else, aside from one thing...

"If you encounter a number higher than this, you're not doing real math"

Next number: 9

Well, that explains Cirno's Perfect Math Class.

Zero is defined as the number of buses in Gensokyo. As such, math occasionally stops working properly when Yukari gets bored.
Not named Dennis Miller.

AvatarIII
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### Re: 899: "Number Line"

iqtestsmeannothing wrote:
atomfullerene wrote:Any way you can add in gird and wind up with 8 being prime?

Unless you change the fact that 8 = 2 * 2 * 2, nope. (One of the proposed ways of adding in gird gives 8 = 3 * 3, but in that case 8 still is not prime.)

even in that case, if 8 were a prime, it wouldn't be an even prime, because it woulnd't be able to be halved into an integer, right?

humanalien
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### Re: 899: "Number Line"

Every 7 years = made-up? I guess pon farr is just Vulcans' sneaky way of getting laid... or avoiding sex when they don't want it... or something...

Actually, stuff seems to happen every 7 years in the Star-Trek-producers' world, too. Most of the series had seven seasons, and I think the latest movie came out seven years after the debut of Enterprise...

I discovered a distance between two points that's shorter than a straight line! It's called a straight line segment.

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Shay Guy
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### Re: 899: "Number Line"

Pfhorrest wrote:Is 0.000...[infinitely-many-zeroes]...0001 = 0?

How do you have something after an infinite sequence of zeroes? Redundant.

Midnight wrote:hurr durr

I consider the use of such phrases to be equivalent to a triggering of Godwin's law. Same for "But, but..."

K^2 wrote:If Gird is integer, then so is Gird-3. 3 < Gird < 4 => 0 < Gird-3 < 1. So the integer line is now ... -2, Gird-5, -1, Gird-4, 0, Gird-3, 1, Gird-2, 2, Gird-1, 3, Gird, 4, Gird+1, 5, ... Nothing technically wrong with that, but it kind of hangs on what Gird+Gird is.

So is Gird a natural number? If so, what's its successor? And what's it the successor of? Is it an integer, but not a natural number, by the definition involving equivalence classes of ordered pairs of natural numbers? If so, what ordered pairs define it?

meerta wrote:I'm still thinking that if these two numbers are equal they must all be equal, otherwise at what point do they become unequal?

They're not two numbers. They're two ways of writing one number.

K^2 wrote:In fact, there are as many real numbers between two real numbers as there are real numbers overall.

How do you prove that? I know how to prove that, say, there are as many even integers as there are integers. But what do you do when you're working with uncountable infinities? Does the same logic work?

For that matter, I've read that there are exactly as many real numbers as there are sets of natural numbers, and I tried to prove it once, but I failed. How do you do that?

redearth1210 wrote:Infinity is a concept, correct? It's not actually a number.

Well, numbers are concepts, too. In fact, you know what a hereditary set is? It's a set with no elements that aren't themselves hereditary sets. You can't get much more conceptual than that. And most any kind of number can be defined as one of those, though you get interesting consequences like the natural number "3" not being the same thing as the integer "3."

The really relevant concept, though, is that of limits -- e.g., "the limit of 1/x as x approaches infinity is 0." They're one of the first things you learn about in calculus, and they're integral to the whole "repeating decimal" notation. Once you accept limits, the rest falls into place.

joshrob03 wrote:For decimal notation, a real number has a repeating decimal representation if and only if it is a rational number.

I can grok that, but do you know a proof? 1/3 works because 3's a factor of 9, 1/7 because 7's a factor of 999999... let's see.

x has a repeating decimal representation, with m nonrepeating digits past the decimal point followed by n repeating ones.

x*10^m, therefore, has only the repeating sequence past its decimal point, and x*10^(m+n) has the same sequence.

Therefore, x*10^(m+n)-x*10^m = x*10^m*(10^n-1) is an integer, y.

Because m and n are integers, 10^m*(10^n-1) must be an integer, z.

Since x*z = y, x = y/z, making x equal to one integer divided by another -- i.e., a rational number. And it's easily adapted to any other number base.

QED! That was fun. So how do you prove that all rational numbers have repeating decimal representations?

iqtestsmeannothing
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### Re: 899: "Number Line"

Shay Guy wrote:How do you prove that? I know how to prove that, say, there are as many even integers as there are integers. But what do you do when you're working with uncountable infinities? Does the same logic work?

Formally speaking, one needs to show that there is a bijection between (a, b) and R for any real numbers a < b. One such bijection is the function f : (a, b) \to R given by f(x) = tan(pi * ((x - a) / (b - a) - 1/2))

Shay Guy wrote:For that matter, I've read that there are exactly as many real numbers as there are sets of natural numbers, and I tried to prove it once, but I failed. How do you do that?

It's messy, but the basic idea is that for each number x in [0, 1) you consider the set of natural numbers k such that the binary representation of x has a 1 in position k. This isn't an exact bijection, and you have to worry about which binary representation of a number you choose, and then you need a bijection between [0, 1) and R, but that's the gist of it.

Shay Guy wrote:So how do you prove that all rational numbers have repeating decimal representations?

Euler's Theorem shows that any positive integer n relatively prime to 10 divides 10^{\phi(n)} - 1. This shows that 1/n has a repeating decimal representation (with period a divisor of \phi(n)). Then you can show that multiplying a repeating decimal by an integer or dividing it by 2 or 5 still gives a repeating decimal, and that shows that all rationals have repeating decimal representations.

n2kra
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### Re: 899: "Number Line"

My first thought ran to http://en.wikipedia.org/wiki/Pentium_F00F_bug

duckshirt wrote:I feel like there are a lot of references I don't get...
".99... is .000000037" less than 1 - a reference to (single-precision) floating-point numbers?
Either that or I'm just trolled trying to figure out references, in which case, nice work.

When it should have been: What Every Computer Scientist Should Know About Floating-Point Arithmetic

tahrey
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### Re: 899: "Number Line"

hurr durr I'll make a series of jokes that are actually jokes I've made before. the people love graphs with math jokes, right?

Oh, bore off. I got a rise out of it. Maybe they have been done before but some of us don't have eidetic schizophrenia.

Regarding http://www.strangehorizons.com/2000/20001120/secret_number.shtm... ah, I figured there must have been a source for Theta-Prime... http://scp-wiki.wikidot.com/scp-033

Given Moffat's seeming propensity for nicking stuff from that parthenon - or at least, thinking along similar lines (173 and 055 just for two) - there's a good chance we may see something similar on Doctor Who in the next couple years?

(Please blame any thread-anachronism on me originally writing 95% of this several hours ago and being interuppted :-/)

Edit: also, though I don't really "get" the whole Touhou thing, have some Shift-JIS for free: ⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨

SirMustapha
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### Re: 899: "Number Line"

tahrey wrote:Oh, bore off. I got a rise out of it.

Also known as "Shut up, only TRUE FANS can speak here."

And also known as "Oh, you didn't like his post? Bore off. I agreed with it completely."

Fixblor
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### Re: 899: "Number Line"

Arrrgh!

The football in this case is any summarization of the numbers.
Randall, you're Lucy.

And the part of Charlie Brown will be played by my mouse clicking the hell out of here ...
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ManaUser
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### Re: 899: "Number Line"

0.333... isn't really 1/3 either, it's just the best approximation in the decimal system, good enough for all practical uses. In the same way, 0.999... is an approximation of 1 that you might use if your 1 key were broken, also good enough for all practical uses, although unlike the former, it will make people look at you funny.

bigjeff5
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### Re: 899: "Number Line"

K^2 wrote:That's because this isn't what 0.9(9) means. The bar/parenthesis mean "take limit of the series". In this case, it is the limit of 0.9 + 0.09 + 0.009 + ... And while you might have a philosophic discussion on whether or not you can actually carry out that sum to infinity, the limit exists, and it is exactly one. No "1/infinity" left over.

That's the way I look at this discussion. In a philosophical sense, 0.9(9) cannot be equal to one, it will be forever something less than 1 (i.e. as soon as you put the "0.whatever" in there it cannot equal 1 any more, no matter what is behind it).

In reality, if the difference between 0.9(9) and 1 is infinitely small, then the difference simply doesn't exist, that's just how infinity works. Any math you do with the numbers will come out exactly the same (not approximately the same, as some have said, but exactly the same). This is the same as 0.3(3), which is simply a different way of writing 1/3. Math using either one will come out exactly the same (barring the technical impossibility of actually calculating anything with a number that has an infinite number of digits), there is no difference. I don't really understand infinity, but I can grasp enough of it to see that this is so.

There are a lot of these kinds of philosophical arguments out there (not just in math, obviously), in which the difference only exists in the definition (from certain perspectives only, usually), but not in any application (practical or impractical) of the ideas.

tommm wrote:...
drixoman wrote:Not educated enough in math to get this sadly...

i'm an undergrad studying maths and i don't get it either.

That's really, really sad. I only went as far as Calc 2 (and barely skated by, at that - I've forgotten most of calculus by now) and I got most of the jokes, and even thought a few of them were pretty clever. The 0.99bar joke is especially rich, since it is obviously wrong in just about every way possible.

I could understand you not finding them funny, but not getting them at all? That's downright depressing to hear from a math undergrad.

Rokker
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### Re: 899: "Number Line"

Brian-M wrote:I like to think of 0.9999.... as being equal to 1-(1/∞). The real debate should be over whether 1/∞=0 or 1/∞→0.

I registered on this forum purely to tell you that what you wrote there is absolute nonsense.

In our decimal system, 0.999... = 1. They are equivalent ways of writing the same thing. Similarly, 024999... = 0.25. 0.33999... = 0.34. 13.999... = 14. 201.010101999... = 201.010102. Etcetera.

1-(1/∞) has no meaning whatsoever. You cannot treat ∞ as a number. The "debate" you speak of is not a debate at all. "1/∞" does not equal anything. "1/∞" does not tend towards anything. Now, in the limit as x tends to ∞, 1/x = 0. That has meaning.

May I ask, to what level of maths have you studied?

ManaUser wrote:0.333... isn't really 1/3 either, it's just the best approximation in the decimal system, good enough for all practical uses. In the same way, 0.999... is an approximation of 1 that you might use if your 1 key were broken, also good enough for all practical uses, although unlike the former, it will make people look at you funny.

Oh dear. No, 0.333... = 1/3. 0.333... is the decimal representation of the rational number 1/3. The two are identical. 0.999... is NOT an approximation of 1. 0.999 is. 0.9999 is. 0.99999 is. But 0.999... is not. See the difference? Once you cut off at a certain number of decimal digits, whether it be 3, 4, 5, or one googolplex, the two are not equal. But 0.999... IS equal to one and is NOT an approximation.

I'd like to stress that EVERY terminating decimal has another representation in our usual decimal system. The rough and ready rule would be to replace the last digit with one less than that digit, and then have an infinite number of nines. The case 1.000... = 0.999... is just one example of this.

bmonk
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### Re: 899: "Number Line"

Degenx3 wrote:My favourite argument for the 0.99 = 1 thing is using Order Theory. The set of real numbers is a dense ordering, therefore 0.99 recurring is equal to 1.

The calculus argument (using limits) is also pretty effective. 0.99bar can be shown to be arbitrarily close to 1.00--and so there is no difference.
Having become a Wizard on n.p. 2183, the Yellow Piggy retroactively appointed his honorable self a Temporal Wizardly Piggy on n.p.1488, not to be effective until n.p. 2183, thereby avoiding a partial temporal paradox. Since he couldn't afford two philosophical PhDs to rule on the title.

Rokker
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### Re: 899: "Number Line"

bmonk wrote:
Degenx3 wrote:My favourite argument for the 0.99 = 1 thing is using Order Theory. The set of real numbers is a dense ordering, therefore 0.99 recurring is equal to 1.

The calculus argument (using limits) is also pretty effective. 0.99bar can be shown to be arbitrarily close to 1.00--and so there is no difference.

"can be shown to be arbirtrarily close" - no, they are not arbitrarily close, when you write 0.99bar you are writing the number 1.

What you mean is the sequence 0.9, 0.99, 0.999, 0.9999, 0.99999, ... gets arbitrarily close to 1, and therefore the "limit" of this sequence, 0.99bar, is 1.

In fact the above is a Cauchy Sequence in the real numbers and therefore converges in the real numbers (in particular to a unique limit). Since the sequence is tending to 0.99bar and also to 1, the two must be equal by that argument too.

bigjeff5
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### Re: 899: "Number Line"

ManaUser wrote:0.333... isn't really 1/3 either, it's just the best approximation in the decimal system, good enough for all practical uses. In the same way, 0.999... is an approximation of 1 that you might use if your 1 key were broken, also good enough for all practical uses, although unlike the former, it will make people look at you funny.

I think your confusion comes from the fact that we cannot, in a practical sense, use 0.3(3) to do any calculations, because the equipment to make a calculation based on an infinitely repeating number will never exist. This is a limit of our ability to deal with infinites, not the numbers themselves. 0.3(3) is not approximately 1/3, it is equal to 1/3.

1/3 = 0.3(3) every time, you never see an approximation symbol for that equation because it is not an approximation.

An approximation would be 1/3 [imath]\approx[/imath] 0.333333333333, which is what you end up having to use (depending on your sig-figs) when you try to calculate with 0.3(3). Again, a limit of our abilities, not the number.

This is beaten to death in the math forums far more thoroughly than it has been here, so much so that there is a forum rule not to discuss it any further.

Edit: Switched 1/3 and 0.3(3) in my first paragraph - oops.
Last edited by bigjeff5 on Mon May 16, 2011 6:03 pm UTC, edited 2 times in total.

Stanistani
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### Re: 899: "Number Line"

This comic was a successful troll snipe landmine.

Quake
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### Re: 899: "Number Line"

"largest even prime"

Hehe.
Difficult?
Bananas (smiley happens to be an "8" followed by a " ) " )

Lol

skeptical scientist
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### Re: 899: "Number Line"

For the trolls in this thread, as well as the genuinely confused.

Is 1 = 0.999...?

Longer answer: To fully understand this question, you need to start with the rational numbers, which are fractions of two whole numbers. Any fraction of two whole numbers (with nonzero denominator) is a rational number, such as 1/1, 1/2, 3/2, or 2/4. It is possible, however, for two rational numbers to be the same, for example 1/2 and 2/4 represent the same number, usually written as 1/2. To get a unique representation, we have to pass to what are called "least terms", where the numerator and denominator have no common multiple.

Using rational numbers, we can define what we mean by finite decimal expansions, which always represent rational numbers. The notation 17.243 is just a convenient shorthand for the number 1*10+7+2/10+4/100+3/1000. So .9=9/10, .99=9/10+9/100=99/100, .999=999/1000, and so on. That means .999...9 will be less than one, for any finite number of 9s.

What then do we mean by an infinite decimal expansion? To understand that, we have to use what are called limits and sequences. A sequence is just what it sounds like: a list with a first item, a second item, a third item, and so on. More formally, a sequence is a function whose domain is the natural numbers (aka the counting numbers: 1, 2, 3, and so on). 10, 20, 30, 40, ... is a sequence, as is 0, 0, 0, 0, ... as is 1, 1/2, 1/3, 1/4, .... Importantly, a sequence is an infinite list, so it has a first item, but no last item. Even though a sequence has no last item, it may have a "limit": a number that the elements of the sequence are approaching, whether or not they actually get there. We say a sequence a1, a2, a3, ... of numbers converges to a number a if for any small positive distance ε, the terms of the sequence are eventually all within ε of a (that is to say, all but the first N terms are within ε of a, for some number N, possibly very large). If a sequence converges to a, we say that a is the limit of the sequence.

So for the examples above, the sequence 10, 20, 30, ... doesn't converge to anything, because if ε is small, say 1, then you can't have both the nth term (10n) and the n+1st term (10n+10) both be within ε of the same number. The sequence 0, 0, 0, ... converges to 0, as for any ε>0, all of the terms of the sequence are within ε of 0 (since they are all equal to 0). Finally, the sequence 1, 1/2, 1/3, ... also converges to 0, although this one is trickier. However, if ε>0, then there is a whole number N>1/ε, which means 1/N<ε. So after the first N terms of the sequence, all later terms are within ε of 0, since they are all smaller than 1/N, which is less than ε. So this sequence also converges to 0, even though it never actually gets there. Finally, you can show that if a sequence converges to a, it can't also converge to b, if b is different from a. This is because if ε<|a-b|/2, a number can't simultaneously be within ε of a and within ε of b. So if a sequence has a limit, that limit is unique.

We are finally prepared to show that .999...=1, because we are finally prepared to define what we mean by an infinite decimal expansion, like .999.... The number represented by an infinite decimal expansion b.a1a2a3... is simply the limit of the finite expansions. In other words, b.a1a2a3... is, by definition, the limit of the sequence b.a1, b.a1a2, b.a1a2a3, ... (Yes, that is really how decimal notation is defined; consult any textbook which actually has a full definition of decimal notation which includes infinite decimal expansions.*) So, in particular, .999... is, by definition, the limit of the sequence .9, .99, .999, .... And just as the sequence 1, 1/2, 1/3, ... converges to 0, this sequence converges to 1. This is because the distance between the nth term of this sequence and 1 is 1/10n, and given any ε>0, this distance is eventually less than ε.

Just as is this case with 1/2 and 2/4 in the rational numbers, these are two different representations of the same number by decimal expansions. To get unique representations, you would have to add a rule that decimal expansions can't end with an infinite sequence of 9s, just as to get unique representations using fractions, you have to add a rule that the numerator and denominator can't have a common factor.

*You may find such a book surprisingly difficult to find, but probably most college calculus textbooks will suffice; Spivak's calculus is one example.
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Louis XIV
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### Re: 899: "Number Line"

The largest even prime is actually 1763042.

RogueCynic
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### Re: 899: "Number Line"

I've always said math was a figment of the imagination, and now I have proof.Thank you Randall. As a side note, my computer is telling me there are no JsMath TeX fonts found. More proof. Oh, and read my sig.
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Cauchy
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### Re: 899: "Number Line"

I figured gird was something similar to http://en.wikipedia.org/wiki/Zero_sharp, that is, a real number whose existence is not determined by ZFC and whose existence or non-existence would imply certain things about set theory, such that "orthodox" mathematicians take its existence as "canon" so as to have access to those implications.
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Shay Guy
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### Re: 899: "Number Line"

iqtestsmeannothing wrote:It's messy, but the basic idea is that for each number x in [0, 1) you consider the set of natural numbers k such that the binary representation of x has a 1 in position k. This isn't an exact bijection, and you have to worry about which binary representation of a number you choose, and then you need a bijection between [0, 1) and R, but that's the gist of it.

That's what I tried, but I ran into the old .999... problem -- for instance, {2,4} and {2,5,6,7,8,9,...} stand for the same number under the mapping I was using. Like you said, not an exact bijection. So all I really managed to prove was that the cardinality of the rational numbers was no more than that of the power set of the natural numbers.

iqtestsmeannothing wrote:Euler's Theorem shows that any positive integer n relatively prime to 10 divides 10^{\phi(n)} - 1. This shows that 1/n has a repeating decimal representation (with period a divisor of \phi(n)). Then you can show that multiplying a repeating decimal by an integer or dividing it by 2 or 5 still gives a repeating decimal, and that shows that all rationals have repeating decimal representations.

Nifty.

tahrey wrote:Given Moffat's seeming propensity for nicking stuff from that parthenon - or at least, thinking along similar lines (173 and 055 just for two)

I think "Blink" predates it (and probably inspired it).

tahrey wrote:I don't really "get" the whole Touhou thing

I'm not sure fans do.

ManaUser wrote:0.333... isn't really 1/3 either

Yes it is. That's what the notation means. "0.333..." means "the sum of the infinite series 0.3+0.03+0.003+...+3*10^-n+...", which is precisely 1/3. Just like how pi is exactly equal to 4/1-4/3+4/5-4/7+4/9-...

bigjeff5 wrote:That's the way I look at this discussion. In a philosophical sense, 0.9(9) cannot be equal to one, it will be forever something less than 1 (i.e. as soon as you put the "0.whatever" in there it cannot equal 1 any more, no matter what is behind it).

What sort of philosophical stance is that, and how does it interpret infinite sums? Does it have anything to say on the equality of 1 and 1.0, or on the value of pi?

iqtestsmeannothing
Posts: 7
Joined: Mon May 16, 2011 12:26 pm UTC

### Re: 899: "Number Line"

Shay Guy wrote:That's what I tried, but I ran into the old .999... problem -- for instance, {2,4} and {2,5,6,7,8,9,...} stand for the same number under the mapping I was using. Like you said, not an exact bijection. So all I really managed to prove was that the cardinality of the rational numbers was no more than that of the power set of the natural numbers.

So here are the remaining details. There are two ways I know of to repair the hole. The elegant one is: using binary representation, we get an injection from [0, 1) to P(N), as described. Using ternary representation, we can get a surjection from [0, 1) to P(N). Then by http://en.wikipedia.org/wiki/Cantor%E2%80%93Bernstein%E2%80%93Schroeder_theorem there exists a bijection between them.

More elementarily, just observe that the injection only misses countably many points, so you can repair the defect by unioning a countable set to [0, 1), which it is easy to show does not change the cardinality. (You need to show that [0, 1) has at least countably many points, which is easy because of 1/n for n > 1 integer, and that the union of two countable infinite sets is countable infinite.)

JoeZ
Posts: 22
Joined: Thu Jan 20, 2011 3:35 pm UTC

### Re: 899: "Number Line"

Easy proof

1/3 = 0.3333333333333333333333333333333333333333333333333333333333333333333333333333333333...

0.9999999999999999999999999999999999999999999999... = 3 * 0.33333333333333333333333333333333333333333333333333...

3 * 0.33333333333333333333333333333333333333333333333333... = 3 * 1/3

3 * 1/3 = 3/3 = 1

therefore 0.9999999999999999... is exactly equal to 1, because it is exactly three times (the decimal representation of) one third.
I can't go, this is too important. Someone is wrong on the internet!

screen317
Posts: 252
Joined: Mon May 16, 2011 7:46 pm UTC

### Re: 899: "Number Line"

JoeZ,

The same people would question that 1/3 = .3...

screen317
Posts: 252
Joined: Mon May 16, 2011 7:46 pm UTC

### Re: 899: "Number Line"

let x = .999...
*10 *10

10x = 9.999...
-x -x

9x = 9

x = 1

.999... = 1

Hirg
Posts: 18
Joined: Mon May 16, 2011 10:04 pm UTC

### Re: 899: "Number Line"

Did no one else see the real point of this comic?

'The Wikipedia page "Lists of numbers" opens with "This list is incomplete; you can help by expanding it." ' I don't know about you, but I see that as a call to complete that "List of numbers" page.

clanders
Posts: 47
Joined: Thu Jan 14, 2010 12:07 am UTC

### Re: 899: "Number Line"

Randall tragically died two years ago, but he coded a program to automatically draw stick figures and attach to them jokes in the form of a tri-weekly comic. Unfortunately, Randall was not half the programmer he thought he was.

sykes1024
Posts: 2
Joined: Wed Nov 14, 2007 11:10 pm UTC

### Re: 899: "Number Line"

redearth1210 wrote:It would be 0.000bar with the idea holding that after an infinite number of zeros come a 1.

>after an infinite number of zeros come a 1
>after an infinite number
>after infinite
/facepalm

skeptical scientist
closed-minded spiritualist
Posts: 6142
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Location: San Francisco

### Re: 899: "Number Line"

JoeZ wrote:Easy proof

1/3 = 0.3...

0.9... = 3 * 0.3...

3 * 0.3... = 3 * 1/3

3 * 1/3 = 3/3 = 1

therefore 0.9... is exactly equal to 1, because it is exactly three times (the decimal representation of) one third.

That's not actually a proof, since you would first need to prove that .3...=1/3. But kudos for using 239 more 3s and 9s than you needed to make your point.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

fimzo
Posts: 33
Joined: Mon Sep 06, 2010 2:48 am UTC

### Re: 899: "Number Line"

I think I'm going to use one of the proofs given here to convince my friends that .9(bar)=1, probably skeptical scientist's. I'm just finishing 8th grade, and I'm in an advanced math class with two other students in my school. Neither of them believed my when I explained it to them with the .3(bar) explanation, and one of them made the claim about .3(bar) being infinitely less than 1/3. We asked the class professor about it, and she agreed with him, that .3(bar) doesn't equal 1/3. After that, I posted the same question on the class's help forum, where the head of the program answered, saying he agreed with me. So we still disagree about it.
-Fimzo