Open-Ended Math Questions?

[GreedyAlgorithm]

Those who keep criticising the proof as not rigorous, etc., are viewing it from the perspective of calculus or real analysis. They are right in that context. They are wrong in the context of 7th grade pre-algebra. That's why I told them to please stop trying to redefine the 7th grade understanding of .999... in real analysis terms of series and limits.

I've never argued otherwise in this thread.

AS

We all agree! Hooray! (Everybody dance!)

Now check this one out, it's probably my favorite "math is broken" trick:

x^2 + x = -1 (note that x=0 is not a solution, so we can divide by it)
1/x + x = -1 (now set things equal to -1 equal to each other)
x^2 + x = 1/x + x (subtract x, then multiply by x, still valid since x isn't 0)
x^3 = 1 (and therefore...)
x = 1 (but...)
1^2 + 1 = 2, not -1.

 

[69dodge]

If I might turn the heat up just a tad.
Earlier in the thread you mentioned the simple concept of a repeating decimal.

Appreciate that you are dealing with a simpleton here.
I know what 9 is.
I can make a stab at .9, even at .9999 , but I have no idea at all what .999... is.
To me , 0.9 is "Nine tenths of one" and so on. From the context of the thread, .9999... appears to be an infinite number. (not infinitely large. Infinitely long). I don't understand how , without actually expanding it to infinity, we can know what , if anything , an infinitely long number is equal to.

Maybe there's some confusion here between a number and a representation of a number?

Would you agree that the number 1/2 equals the number 0.5 ? Even though the representation "1/2" clearly differs from the representation "0.5" ?

The thing that isn't infinitely large is a number. The thing that is infinitely long is a representation of a number.

What additional information could we learn about the number represented, from seeing all infinitely many digits of its decimal representation explicitly written out? (If that were possible. Which it obviously isn't.) We already know they're all going to be 9's, from seeing the very finite description: "a decimal point followed by infinitely many 9's".

Anyway, here's a cute little problem from What is Mathematics?, by Courant and Robbins: what's 0.333333... + 0.989898... ?

 

[69dodge]

broken math

x = 1
x = 1

Now set things equal to 1 equal to each other.

x = x

Hey look, now x can be anything, but before it could only be 1.

 

[AmateurScientist]

We all agree! Hooray! (Everybody dance!)

Or smoke a bowl together.

Now check this one out, it's probably my favorite "math is broken" trick:

x^2 + x = -1 (note that x=0 is not a solution, so we can divide by it)
1/x + x = -1 (now set things equal to -1 equal to each other)
x^2 + x = 1/x + x (subtract x, then multiply by x, still valid since x isn't 0)
x^3 = 1 (and therefore...)
x = 1 (but...)
1^2 + 1 = 2, not -1.

That's a trick and of course fallacious, as you recognize. My proof isn't. I hope non-mathematicians in this thread understand the distinction.

AS

 

[insomneac]

Anyway, here's a cute little problem from What is Mathematics?, by Courant and Robbins: what's 0.333333... + 0.989898... ?

The solution is '1.32...'

33/99 + 98/99 = 131/99 = 1 + 32/99 = 1.323232...

 

[rockoon]

I don't buy the idea of open-ended math problems. Call me a Math-Nazi, but math is not a fuzzy-wuzzy subject. When you apply math to a problem, it's not because you're trying to form an opinion. It's because you want an answer.

There are definately open ended Math problems. I believe you are refering to Arithmetic problems. Big difference.

Definately all "Whats the next number in this series' questions are open-ended.

There are no less than 4760 clearly defined mathematical sequences that have '1, 2, 3, 4' within them.. and clearly there are a lot more.

To list a few:

The natural numbers
Prime powers
Palindromes in base 10
Numbers n such that n divides the (right) concatenation of all numbers <= n written in base 2 (most significant digit on left)
3-smooth numbers: numbers of the form 2^i*3^j with i, j >= 0
Smarandache numbers
Niven (or Harshad) numbers
Number of groups of order n
Highly composite numbers
Digital root of n
Number of n-bead necklaces with 2 colors when turning over is not allowed
a = a(n-1) + a(n-3)
Nonzero multiplicative digital root of n
Least number k such that [ (n+1)^k / n^k ] > 2
Number of partitions of n into at most 3 parts; also partitions of n+3 in which the greatest part is 3; also multigraphs with 3 nodes and n edges.
Triangle in which k-th row lists natural number values for the collection of riffs with k nodes.
Primorial primes: n such that n-th Euclid number is prime
Armstrong (or Plus Perfect, or narcissistic) numbers: n-digit numbers equal to sum of n-th powers of their digits (a finite sequence, the last term being 115132219018763992565095597973971522401).
Smallest k such that kn+1 is prime
Multiplicative digital root of n
Numbers n such that 10^n + 9 is prime

need I go on? extremely open ended.

 

[Grundar]

We all agree! Hooray! (Everybody dance!)

Now check this one out, it's probably my favorite "math is broken" trick:

x^2 + x = -1 (note that x=0 is not a solution, so we can divide by it)
1/x + x = -1 (now set things equal to -1 equal to each other)
x^2 + x = 1/x + x (subtract x, then multiply by x, still valid since x isn't 0)
x^3 = 1 (and therefore...)
x = 1 (but...)
1^2 + 1 = 2, not -1.

My answer to broken math.

If you try to solve the original equation you can see that it have complex roots. All you do is introduce another solution by raising the grade of the polynom. If you try to get all three roots for the x^3-1=0 equation with help of polynomial division, you find yourself coming back to the original equation.

/Hans

 

[Wavicle]

We all agree! Hooray! (Everybody dance!)

Now check this one out, it's probably my favorite "math is broken" trick:

x^2 + x = -1 (note that x=0 is not a solution, so we can divide by it)
1/x + x = -1 (now set things equal to -1 equal to each other)
x^2 + x = 1/x + x (subtract x, then multiply by x, still valid since x isn't 0)
x^3 = 1 (and therefore...)
x = 1 (but...)
1^2 + 1 = 2, not -1.

Hmmm, interesting. Hadn't seen that one before. Your polynomial is irreducible over the reals (translation: all solutions are complex numbers) so you need to be careful with your arithmetic. Your second line is not true.

Consider one of the solutions: x = 0.5 - 3i.
1/x = (0.5 + 3i)/9.25 = (2 + 12i)/37
1/x + x = (2 + 12i)/37 + (0.5-3i)
= (2 + 12i)/37 + (18.5-111i)/37
= (20.5 - 99i)/37
<> -1

 

[69dodge]

Consider one of the solutions: x = 0.5 - 3i.

?

That number doesn't satisfy x² + x = -1.

 

[Jekyll]

factor of 1/2 missing in the ith term, ohh and a square root sign

 

[Soapy Sam]

Maybe there's some confusion here between a number and a representation of a number?

Would you agree that the number 1/2 equals the number 0.5 ? Even though the representation "1/2" clearly differs from the representation "0.5" ?

The thing that isn't infinitely large is a number. The thing that is infinitely long is a representation of a number.

What additional information could we learn about the number represented, from seeing all infinitely many digits of its decimal representation explicitly written out? (If that were possible. Which it obviously isn't.) We already know they're all going to be 9's, from seeing the very finite description: "a decimal point followed by infinitely many 9's".

Anyway, here's a cute little problem from What is Mathematics?, by Courant and Robbins: what's 0.333333... + 0.989898... ?

I think we have to agree on what is meant by "equals" in your question.

If you mean: Are " 1/2" and "0.5" the same?
Then the answer is clearly "no". A glance is sufficient to distinguish them.
That's the symbolism question.

If what you mean is " Does the operation 'one divided by two' yield the number represented by the symbol '0.5' , then the answer is "yes".

In this example the distinction is trivial. But what about this one?
Does 1/3 = 0.33?

The practical answer is "It depends.Are we talking currency , or what?"
But that's a question about quantity.

If we are talking pure numbers, detached from the real world, the answer , surely , is "No- 1/3 is NOT the same as 0.3" or .33 or .333333

If now we simply define 1/3 as being equal to (.333...) well, as AS says, you can't argue with an axiom in mathematics.
But you can and must in the real world. The supposition that spirit exists is not one I am willing to admit at the start of a discussion on life after death. It's what is to be proved. We must question assumptions. We must look for paradox, or for evidence that supports or contradicts the assumption.

What is at issue is how the notion of .999...is derived. The infinity here is not the magnitude of the number, but the length of it's expression. There is an infinite number of "9" s implied in the definition. I completely agree with that. If the magnitude of .999... was infinite, you and other (evidently sane) people would not be saying it is equal to 1.

My problem with infinities is that they seem to undermine the most fundamental axiom of arithmetic, namely that one integer only can be assigned to one item only creating a one to one mapping between item and integer. That is what counting is. If, when counting beans, I assign an uncountable number of integers to each bean , my total count will be meaningless. Yet this is exactly what infinities do.

Here I go on a bean count- 1, 2, 3, I, 5, 6, 7 usw.
Now- how many beans have I got?
(Where I stands for an infinity).
The whole point of numbers is that they are used to count; I know I have a billion pennies if I can put a cardinal integer in sequence alongside each penny, each number one greater than its predecessor, and the last one is " one billion".
If we postulate a number which is uncountable, we short circuit the whole notion.

We can either have counting numbers or infinities, but not both in the same system.

Also, the universe does not supply evidence for infinities.
Remember the "black body catastrophe"?

Now here are my two open math questions:
1. Does "number" when detached from quantity, retain actual meaning?
2. Are numbers themselves quantised?

The conventional defence in favour of infinities is that no matter how big a number, we can always add one more. Note that this assumes infinities to always be very large numbers. (As opposed to very small numbers, or simply "uncountable" numbers). Why must this be?

If we redefine an infinity as a number which is simply not countable, then there is no actual requirement for it to be a big number. For example, the number "x" is not countable. But does .999x = 1? I have no idea, because the "uncountable" definition does not carry the information required to get that answer.

Sorry if all this seems like / is total nonsense. It may well be. But The present idea of infinity seems to me to be as nonsensical as "spirit"- indeed possibly more so. This is why I do not accept that .999...=1, even while appreciating that the infinity in the expression is not one of magnitude.

I hope this does not seem too stupid.


And your question-
what's 0.333333... + 0.989898... ? You will appreciate, given my beliefs, that it would be sacreligious to attempt an answer. Indeed I demand you apologise for asking it. I propose to riot in the street and burn you in effigy.

 

[DeviousB]

That's a trick and of course fallacious, as you recognize. My proof isn't. I hope non-mathematicians in this thread understand the distinction.

Neither, incidentally, is GA's proof "...99999 = -1". As has already been pointed out, this is perfectly true if you represent negative numbers using a 'tens-complement' methodology (anyone know why we don't?).

Actually, it is axiomatic to the decimal expansion of real numbers that 0.999... = 1. So your proof is basically that "given that 0.999... = 1, 0.999... = 1" (http://en.wikipedia.org/wiki/Construction_of_real_numbers).

On the other hand, I have made it as clear as I know how that I first learned the algebraic proof in the 7th grade, when I was 12, and that I have been discussing it in the context of pre-algebra students learning it.

[...]

Those who keep criticising the proof as not rigorous, etc., are viewing it from the perspective of calculus or real analysis. They are right in that context. They are wrong in the context of 7th grade pre-algebra. That's why I told them to please stop trying to redefine the 7th grade understanding of .999... in real analysis terms of series and limits.

Really?

Because when Zombified wrote:

Good enough for pre-algebra, but to make that proof formal, you have to define operations on infinitely expanded digit strings, again in terms of sequences, and by the time you've done that you've covered more than enough math to make the limit argument.

You replied:

No, you don't. That is a formal proof.

You don't have to define any operations on an infinitely repeating decimal. It's quite simple. Multiplying by 10 means you move the decimal point one place to the right. Done. Everything else remains exactly as it was.

You're covered no more math than the most basic of functions on algebraic equations and simple multiplication by 10 and subtraction, which are basic arithmetic functions.

Limits are not required to prove that .999... = 1. You can even do it using fractions. Take 1/9, or 1/3, for example. 1/9 = .111... Multiply both sides by 9. You get 1 = .999... 1/3 = .333... Multiply each side by 3 and you get 1 = .999... yet again.

Simple.

Unfortunately, it is not a formal proof, and you do need to define your representaion of real numbers, etc. It may be (is) a perfectly good proof for a 12 year-old, but it is no more correct than using KE=(1/2)mv^2 to prove you can travel faster than light given enough fuel.

And we did try to tell you...

 

[T'ai Chi]

b o r i n g.

 

[DeviousB]

Anyway, here's a cute little problem from What is Mathematics?, by Courant and Robbins: what's 0.333333... + 0.989898... ?

Someone's already posted the answer (solved using quotients, note), so here's another one: what is (0.333...)^2?

Well, .333... = 1/3, (1/3)^2 = 1/9, and 1/9 = 0.111... so the answer is 0.111..., simple.

Only...

(0.3)^2 = 0.09
(0.33)^2 = 0.1089
(0.333)^2 = 0.110889
(0.3333)^2 = 0.11108889
(0.33333)^2 = 0.1111088889
(0.333333)^2 = 0.111110888889
(0.3333333)^2 = 0.11111108888889
:
(0.333...(x oo)...)^2 = 0.111...(x oo)...0888...(x oo)...9 ?!

I.e. slightly less than a 1/9th.

 

[drkitten]

In this example the distinction is trivial. But what about this one?
Does 1/3 = 0.33?

The practical answer is "It depends.Are we talking currency , or what?"
But that's a question about quantity.

Then the practical answer is wrong. And, for that matter, "impractical." A bank that tried, for example, to take 1/3 * $1,000,000 and got 330,000 would have an auditably wrong answer. (And I suspect the local authorities -- certainly the banking commission, and possibly the Queen's Bench -- would have words to say to the bank. Words beginning with "Will the defendant please rise?")

If we are talking pure numbers, detached from the real world, the answer , surely , is "No- 1/3 is NOT the same as 0.3" or .33 or .333333

If now we simply define 1/3 as being equal to (.333...) well, as AS says, you can't argue with an axiom in mathematics.

Fortunately, we don't simply define 1/3 as being equal to 0.333...; instead we define 1, 3, the idea of real numbers, and "divide" (when applied to those reals). You can divide in other fields than the real numbers; in the field of integers, 1/3 has no value. In the field of Z/5Z (the field of integers mod five), 1/3 is equal to 2.

All according to the definitions of of 1, 3, and divide.

What is at issue is how the notion of .999...is derived. The infinity here is not the magnitude of the number, but the length of it's expression. There is an infinite number of "9" s implied in the definition. I completely agree with that. If the magnitude of .999... was infinite, you and other (evidently sane) people would not be saying it is equal to 1.

My problem with infinities is that they seem to undermine the most fundamental axiom of arithmetic, namely that one integer only can be assigned to one item only creating a one to one mapping between item and integer.

That's not an axiom of arithmetic. It's not even true; the integer 2 is also the item "4/2." One integer, several "items." Or perhaps expressions would be a better term, but you defined the vocabulary for this discussion, so by the rules of mathematics, I have to use yours.

If, when counting beans, I assign an uncountable number of integers to each bean , my total count will be meaningless. Yet this is exactly what infinities do.

That's not what infinitities do.

As you pointed out, "counting" is the assignment of an integer, in sequence, to a set. But since "infinity" isn't an integer -- you got that right, at least -- you'll never assign "infinity" to anything.

Here I go on a bean count- 1, 2, 3, I, 5, 6, 7 usw.

This is no more valid a "bean count" than the following -- 4, 4, "Buick Century," 31, "Oxfordshire," 2.

When counting, you only get to use integers and you only get to use them in order.

We can either have counting numbers or infinities, but not both in the same system.

Half right. You're right that infinity is not a "counting number," but that doesn't mean that we can't have both in the same system. We just have to know how to use them.

Now here are my two open math questions:
1. Does "number" when detached from quantity, retain actual meaning?
2. Are numbers themselves quantised?

Neither of those questions are meaningful as phrased. You'll need to unpack them a bit.

The conventional defence in favour of infinities is that no matter how big a number, we can always add one more. Note that this assumes infinities to always be very large numbers.

No, it doesn't. In fact, it explicily assumes the opposite -- infinities are not numbers.

If we redefine an infinity as a number which is simply not countable, then there is no actual requirement for it to be a big number.

This is incorrect as phrased. If the definition of "infinity" is "a number that is not countable," then it is by definition "a number."

The problem is not with the idea of "infinity." The proof that 0.9999 equals 1 does not rely on the notion of "infinity" at all. "Infinity" is extremely problematic in mathematics -- and for that reason, a lot of 18th and 19th century mathematics worked very hard to avoid using "infinite" quantities at all.

The usual approach, instead, involved the idea of "limits." For example, "real" numbers are defined as the "limit" of a sequence of terminating rational numbers; pi is somewhere between 3 and 4; it's somewhere between 3.1 and 3.2, and so forth. (If you really want to see how this works, the magic term to look up is Dedekind cut, but I do have to warn you that this is getting into fairly heavy mathematics.) Similarly, Newton's nonsensical notion of the "instantaneous change" has been re-formalized via a limit process to create the calculus. (It's not "instantaneous," but merely the limit as the time of change becomes shorter and shorter, approaching zero.)

And, while you may not be able to manipulate 0.999.... directly on a digit by digit basis, you can manipulate it as a process of limits. In which case you discover fairly quickly that it's the same as 1.

I hope this does not seem too stupid.

Not stupid at all. Perhaps underinformed. The problem really is that you're trying to connect very low-level (elementary school) mathematics with very high-level (college or post-graduate school) mathematics, and you may not have sufficient grounding in secondary school mathematics to understand the connections.

But there's a reason that high school students typically aren't taught about the formal properties of "infinity," or the set-theoretic basis underlying the number system. They don't need them, and it can easily confuse the unwary.

 

[AmateurScientist]

Neither, incidentally, is GA's proof "...99999 = -1". As has already been pointed out, this is perfectly true if you represent negative numbers using a 'tens-complement' methodology (anyone know why we don't?)

Actually, it is axiomatic to the decimal expansion of real numbers that 0.999... = 1. So your proof is basically that "given that 0.999... = 1, 0.999... = 1" (http://en.wikipedia.org/wiki/Construction_of_real_numbers).

I think you have little idea what you're talking about. BTW, your wiki link goes to a non-existent entry.

Really?

Because when Zombified wrote:


You replied:

Unfortunately, it is not a formal proof, and you do need to define your representaion of real numbers, etc. It may be (is) a perfectly good proof for a 12 year-old, but it is no more correct than using KE=(1/2)mv^2 to prove you can travel faster than light given enough fuel.

And we did try to tell you...

Your stance is a nice example of someone's being educated beyond the point where they can understand truths in simple terms (or perhaps an example of a little learning's being a dangerous thing -- who knows?). My proof isn't some special case in which you can use it to trick 12 year olds about algebra. Your analogy suggesting that we can prove to 12 year olds who don't know better using Newtonian mechnics that faster than light travel is possible is inapt and untrue.

Sorry you don't get the elegance of the algebraic proof and that you believe one must define real numbers in order to get it.

I'm not going to continue to butt my head against the wall for you.

Read more about it yourself.

AS

 

[drkitten]

Actually, it is axiomatic to the decimal expansion of real numbers that 0.999... = 1. So your proof is basically that "given that 0.999... = 1, 0.999... = 1" (http://en.wikipedia.org/wiki/Construction_of_real_numbers).

I think you meant this link here
.

And, no, it's not axiomatic. Just as a simple example (from that page)

Construction from Cauchy sequences

If we have a space where Cauchy sequences are meaningful (such as a metric space, i.e., a space where distance is defined, or more generally a uniform space), a standard procedure to force all Cauchy sequences to converge is adding new points to the space (a process called completion). By starting with rational numbers and the metric d(x,y) = |x - y|, we can construct the real numbers, as will be detailed below. (If we started with a different metric on the rationals, we'd end up with the p-adic numbers instead.)

Let R be the set of Cauchy sequences of rational numbers. Cauchy sequences (xn) and (yn) can be added, multiplied and compared as follows:

(xn) + (yn) = (xn + yn)
(xn) × (yn) = (xn × yn)
(xn) ≥ (yn) if and only if for every ε > 0, there exists an integer N such that xn ≥ yn - ε for all n > N.

(Formatting lost in cut-n-paste, my apologies).

This is all I need to prove that 0.999... = 1.

By the definition of comparison above (which is axiomatic), any Cauchy sequence for 0.99999... ≥ any Cauchy sequence for 1. Similarly, 1.0 ≥ 0.999.... But if a ≥ b and b ≥ a, then a = b; hence 0.999.... = 1.

I'd never present this proof in a secondary school classroom, of course.

Two Cauchy sequences are called equivalent if the sequence (xn - yn) has limit 0. This does indeed define an equivalence relation, it is compatible with the operations defined above, and the set R of all equivalence classes can be shown to satisfy all the usual axioms of the real numbers.

This allows another quick proof, of course. 1.0 - 0.9999... demonstrably has limit 0.


Of course, if you dislike Cauchy, there are many other definitions of real numbers that will work. But in any case, the equivalence is not a simple axiomatic fiat, but a propert of the definition you choose. And if you want to make an alternate definition that doesn't make 0.999.... equal to 1, you may -- but I'll place a small wager that I can prove your definition does nto satisfy "all the usual axioms of the real numbers."

 

[Soapy Sam]

DRKitten-

That's not an axiom of arithmetic. It's not even true; the integer 2 is also the item "4/2." One integer, several "items." Or perhaps expressions would be a better term, but you defined the vocabulary for this discussion, so by the rules of mathematics, I have to use yours.-DRK

My bad English. By "Item" I meant each of a number of physically real objects (beans, whatever) being enumerated. One bean corresponds to one integer, in sequence. If not a "fundamental axiom of arithmetic", I should think it's at least a fair description of what counting actually is.

If infinity is an actual number found in that sequence, it must correspond to an actual infinite pile of beans- but since there are (in my example) already 3 smaller piles of beans, the 4th element of the series can't be infinite. This is essentially standing the "We can always add one" argument on it's head, saying no number can be infinite , because you can always add one, therefore infinity can't exist.

If infinity is not a number in that sequence,(as I think we agree it is not)- then what actually is it?

This is incorrect as phrased. If the definition of "infinity" is "a number that is not countable," then it is by definition "a number."

You've lost me there. Is there by any chance a "not " missing?

The problem really is that you're trying to connect very low-level (elementary school) mathematics with very high-level (college or post-graduate school) mathematics, and you may not have sufficient grounding in secondary school mathematics to understand the connections.

It may be even worse than that . I think in words. My mental definition of infinity is essentially the dictionary one of something that goes on forever (I quite liked AS' expression of the repetition of nines as a temporal effect).
I simply don't believe any such quality is inherent in anything, including human imagination. When I hear someone use infinity in an argument my response is similar to what I feel when I hear someone invoke the spirit.

This is (I assume) of no mathematical significance whatever, but does make me wonder about my automatic responses to some paranormal claims.

Believe me- I know my limits mathematically. My interest in questions like this has far more to do with the psychology of belief than numerics. Can't hurt if I occasionally understand the odd mathematical fact though.

On open question one- How would you personally define a number? One for example. (Without reference to external objects like cows or Buick Centuries? ). What properties does one have?

 

[Rasmus]

My mental definition of infinity is essentially the dictionary one of something that goes on forever (I quite liked AS' expression of the repetition of nines as a temporal effect).
I simply don't believe any such quality is inherent in anything, including human imagination. When I hear someone use infinity in an argument my response is similar to what I feel when I hear someone invoke the spirit.

But nobody here is invoking infinity to prove a point.

If you doubt that inifity exists, then arguing what an infinite string of 9's means on the right side of the common comma in a number is pointless.

My mind cannot truly encompass 'infinity'. Who else but Douglas Adams coulkd have a better explanation here?

But if you take a pen and a piece of paper and divite 1 by 3, it should be obvious that the result involves an infinite strings of 3s after the comma. You can't write it down to the end, but you can see that there is no other way the number could develop. There will be more 3s and nothing else.

Likewise, you cannot complete on a piece of paper all the steps it would take to figure out what 0.333... and 0.333... and 0.333... is. But you can easuly figure out that the result will be 0.999... no matter how far you go. You can see that if you were to write the number down that way it would require an infinite string of 9s.

And if 1/3 + 1/3 +1/3 = 1, then so must 0.999...

Rasmus.

 

[Wavicle]

?

That number doesn't satisfy x² + x = -1.

Ugh, you're right. I guess that's the danger in post at 2am, I should resolve not to do that. Clearly I cannot do very simple quadratics at that time. Anyway, the solution should be x = -0.5 +/- sqrt(3)/2i. And the error I think works out to the same spot. x + 1/x != -1.