I've always enjoyed these kinds of "proofs." I saw my first one in my first algebra class, where you start with a^2 = b^2, manipulate the equalities until you can divide by a non-obvious zero, and come up with with 0 = 1. That's slight of hand trickery, and not very cool.More fun are other methods that show more of mathematical interest.
For example (this also involves some slight of hand) let's start out with the simple equation
x = 1 + 2x. No big deal, most could solve this with one hand tied behind their backs. But of these, some balk when I say that since this is an equality, I can substitute 1 + 2x for x however I like, and write x = 1 + 2 ( 1 + 2x ) = 1 + 2 + 4x. Of course, this is a perfectly valid operation, and I can do it again, as in x = 1 + 2 + 4 ( 1 + 2x ) = 1 + 2 + 4 + 8x. See the pattern? So I can continue doing this a and obtain x = 1 + 2 + 4 + 8 + 16 + 32 + 64 + ..... Clearly, the sum of all the powers of two diverges, and so is infinite. Therefore x equals infinity. On the other hand, we can solve the original equation for x = -1. Therefore -1 = infinity. Not exactly 0 = 1, but close enough, don't you think? Cute, but not exactly fair, since I hid a large factor of x off to the right of the ellipses. That's the slight of hand, which is cheating. Let's see if we can show that 0 = 1 without cheating.
Think about how taking the square root of a large number makes it smaller. For example, the square root of 2.0 is 1.414... And the square root of 1.414... is 1.189... Look at the following recursive formula (again, forgive the poor notation)
x = root ( 2 + root ( 2 + root ( 2 + root ( 2 + ....
Can you solve for x? It's actually easy, once you see the symmetry of the formula. As above, the left and right sides of the equals side are freely interchangeable, so you can write
x = root ( 2 + x )
x^2 = 2 + x
x^2 - x - 2 = 0
x = 2 (and of course also -1 but we'll only look at positive roots)
We can readily generalize this to solve for any x = root ( y + root ( y + root ( y+ ...
For example, root ( 3 + root ( 3 + root ( 3 + ... yields (1 + root 13 ) / 2 and root ( 1 + root ( 1 + root ( 1 + ... yield phi, the golden ration, ( 1 + root 5 ) / 2, or 1.618.... In general, from the quadratic formula, x = ( 1 + root ( 1 + 4y) ) / 2.
Consider, now, how taking the root of a small number makes it bigger. The square root of a quarter is a half,and the square root of a hundredth is a tenth. So what do we get when we ask for
x = root ( 1/2 + root ( 1/2 + root ( 1/2 + ... ? Well, this yields ( 1 + root 3 ) / 2 which is about 1.366... As y gets smaller, x approaches 1, from above. After all, y = 1 gave phi, well above 1. Think about it for a bit and I'm sure you'll agree it's intuitive.
So we can conclude that if we used an infinitely small y we would get 1 for our answer.
1 = root ( 0 + root ( 0 + root ( 0 + .... But of course the square root of zero is zero, and no matter how many zeroes you add up it's still zero. So root ( 0 + root ( 0 + root ( 0 + ... = 0 and there you are, once again, 0 = 1.
Where is the issue here? It's not in using only positive roots. What are your thoughts? I'll post more here myself later.
Wednesday, February 25, 2009
A most disturbing formula
It's hard to say exactly why I find this so disturbing. I continue to struggle with my understanding of what constitutes a real number, as can be seen in my other posting. This is connected to the idea of continuity, which has led to reading on the nowhere continuous Dirichlet's function, where f(x) is 1 if x is rational and zero otherwise. I had taken some comfort earlier in thinking this was a somewhat arbitrary distinction, but that was broken when I came across an algebraic formula for this function. It is f(x) = lim k->infinity ( lim n->infinity ( Cos( k! pi x ) ^ 2n ) ) Sorry for the poor notation - I'll have to look up better ways of typing formula in blogger. Anyway, let's take a look at this. It's a double limit of a cosine function, which itself is an infinite summation. All these infinities can make one uneasy, but they really don't do all that much. Let's look at the effect n has on the formula. The idea is that since the cosine function is bounded between zero and one, any exponentiation of its value will leave the one unchanged, but drive any value less than one closer to zero. The limiting process of that exponentiation is simply to restrict the value to either zero or one and eliminate all intermediate values. This is what introduces the discontinuity when the cosine equals 1, but otherwise the function would be continuous everywhere else. This is where the factorial of k comes in, and it's rather clever. k! assures that we can multiply out the denominator of any rational x, transforming it into an integer. That means we'll have an integer multiple of pi as the variable of our cosine, and since half the numbers in k! are even, it further means that we'l have an even integer multiple of pi, which of course assures us that the cosine produces a one at every rational number.
Since irrational numbers cannot be expressed as the ratio of integers, no integral multiplier, no matter how large, can change an irrational into an iteger. Therefore the above cosine for irrational x can never be given a multiple of 2 pi and so will always produce less than one, which the exponentiation can drive to zero. This gives us exactly Dirichlet's function where all rationals map to one and all irrationals map to zero.
The disturbing thing isn't that an irrational multiplied by infinity is still an irrational, because that's not how the limit process works. Consider any finite but large integer k multipling an irrational. The result will still be irrational. The nature of integer times irrational doesn't change just because k gets bigger. The fact that we're using k! means that k! is way bigger than k, but finite integer k gives just as finite an integer k! The limiting process doesn't change this essential characteristic. The use of lim k-> infinity of k! is an ingenious way to convert all rationals to integers, and it does that well. Note that both the integers and the rationals are of the same cardinality. They are both countably infinite, and this is somehow comforting.
(Side thought: However, nothing in the formula explicitely defines k as a sequence of integers, though it is certainly implicit in the factorial operation, which is only defined on integers. Would extending he formula to use the Gamma function change the conclusion? Certainly Gamma operating on an integer gives an integer, and so does Gamma operating on a (countably infinite) number of other values produce integers. I wonder if Gamma only produces an integer when operating on a rational. Is that even an important consideration? Let me think about that later and write more in a reply...)
Somewhat easier to see but more discomforting is the use of the limit in exponentiation. Again, it seems simple enough on the surface. For any small number, exponentation makes it smaller. The limiting process of that is zero. It strikes me as essential that zero is the identity element, because that means any modifications of a value by this element leaves the value unaltered. Essentially, there is no trace that such an operation was performed. To me this says that when zero is the result of a limiting process, continuity may be lost, because a series of increasingly untraceable operations drive the limit. This argument certainly lacks rigor, and may be deeply flawed. It's kind of at the heart of my whole difficultiy with understanding what a real number is. But I digress.
In fact, I've digressed more than enough. Since this is just a blog entry and not a paper, I'l try to collect my thoughts and make a follow up post later. In the meantime, your comments would be very appreciated.
Since irrational numbers cannot be expressed as the ratio of integers, no integral multiplier, no matter how large, can change an irrational into an iteger. Therefore the above cosine for irrational x can never be given a multiple of 2 pi and so will always produce less than one, which the exponentiation can drive to zero. This gives us exactly Dirichlet's function where all rationals map to one and all irrationals map to zero.
The disturbing thing isn't that an irrational multiplied by infinity is still an irrational, because that's not how the limit process works. Consider any finite but large integer k multipling an irrational. The result will still be irrational. The nature of integer times irrational doesn't change just because k gets bigger. The fact that we're using k! means that k! is way bigger than k, but finite integer k gives just as finite an integer k! The limiting process doesn't change this essential characteristic. The use of lim k-> infinity of k! is an ingenious way to convert all rationals to integers, and it does that well. Note that both the integers and the rationals are of the same cardinality. They are both countably infinite, and this is somehow comforting.
(Side thought: However, nothing in the formula explicitely defines k as a sequence of integers, though it is certainly implicit in the factorial operation, which is only defined on integers. Would extending he formula to use the Gamma function change the conclusion? Certainly Gamma operating on an integer gives an integer, and so does Gamma operating on a (countably infinite) number of other values produce integers. I wonder if Gamma only produces an integer when operating on a rational. Is that even an important consideration? Let me think about that later and write more in a reply...)
Somewhat easier to see but more discomforting is the use of the limit in exponentiation. Again, it seems simple enough on the surface. For any small number, exponentation makes it smaller. The limiting process of that is zero. It strikes me as essential that zero is the identity element, because that means any modifications of a value by this element leaves the value unaltered. Essentially, there is no trace that such an operation was performed. To me this says that when zero is the result of a limiting process, continuity may be lost, because a series of increasingly untraceable operations drive the limit. This argument certainly lacks rigor, and may be deeply flawed. It's kind of at the heart of my whole difficultiy with understanding what a real number is. But I digress.
In fact, I've digressed more than enough. Since this is just a blog entry and not a paper, I'l try to collect my thoughts and make a follow up post later. In the meantime, your comments would be very appreciated.
Saturday, January 10, 2009
What's Real?
Lately, I've been thinking a lot about the nature of the real numbers. There are several issues that play into this, like how to have an intuitive understand of continuity in the reals when there is no notion of a "next" real to a given one. Much of my thinking was an attempt to understand how for every two rational numbers there is an irrational between them, and for any two irrations there is a rational between them. Yet there are far more irrationals than rationals. So what are the reals, really?
It's not simple to obtain even a given real. We start with an integer, and the simplest integer is zero, obtained by a count of the null set. That is, 0 = NullSet (I don't see how to get the proper symbol for the null set in this editor). The next integer is formed by a set containing a single instance of the null set, that is, 1 = { NullSet }, and 2 = { NullSet, { NullSet } }, etc. There are a countably infinite number of integers, and each one is unique. Personally, I find it completely amazing that this is the same zero that's the identity element of addition and even more amazing that this one is the identity element of multiplication, but that's a subject for another post.
With the integers established, we can build the rationals by the introduction of a unary reciprocal operator. This gives an inverse over multiplication for all elements except zero (and the necessity of this exception is an interesting point worth another post). The rationals are formed from dividing one integer with another (except zero). This allows us to solve any linear, or first order equation, of the form 0 = ax + b, where a and b are integers. It should be noted that the integers are not exactly rational numbers, though this is a very fussy point. The rational number 4/1 has the same magnitude as the integer 4, but it's not quite the same. A similar distinction may be more interesting later.
It's been known since ancient times that the integers and rationals weren't adequate, since there is no ratio of integers that can produce two when squared. Only recently have I understood why this so upset the Pythagorians, and I'm sure my distress concerning the real numbers is only a fractions of what they felt then. As far as I know, there are only two ways to form a meaningful irrational number, Dedekind cuts and Cauchy sequences. I haven't read Dedekind's papers yet, and I don't know much about his approach. I'm more familiar with Cauchy sequences, which uses an infinite number of rationals to approximate an irrrational, which is obtained as the limit of the infinite sequence. Technicaly, a Cauchy sequence is a series of rational numbers (a0, a1, a2, a3, ... ) where there exists some N such that for any two distinct i, j > N, there is a delta (I'll have to use d) 0 < | aj - ai | < style="font-style: italic;">The Calculus Gallery that the rationals are very poor approximators of the irrationals, from Liouville's number. That is, for any irrational x, any and all rational numbers are a measurable distance away from it
More later.
Notes about builds from integers to rationals to Cauchy Sequences and irrationals. Divide the irrationals into algebraics and transcendentals, the difference being the infinitude of their formula. Distinguish further those transcendentals which can be fully specified with a finite expression and those that cannot. The later class is the unnameable transcendentals. Expand on that.
It's not simple to obtain even a given real. We start with an integer, and the simplest integer is zero, obtained by a count of the null set. That is, 0 = NullSet (I don't see how to get the proper symbol for the null set in this editor). The next integer is formed by a set containing a single instance of the null set, that is, 1 = { NullSet }, and 2 = { NullSet, { NullSet } }, etc. There are a countably infinite number of integers, and each one is unique. Personally, I find it completely amazing that this is the same zero that's the identity element of addition and even more amazing that this one is the identity element of multiplication, but that's a subject for another post.
With the integers established, we can build the rationals by the introduction of a unary reciprocal operator. This gives an inverse over multiplication for all elements except zero (and the necessity of this exception is an interesting point worth another post). The rationals are formed from dividing one integer with another (except zero). This allows us to solve any linear, or first order equation, of the form 0 = ax + b, where a and b are integers. It should be noted that the integers are not exactly rational numbers, though this is a very fussy point. The rational number 4/1 has the same magnitude as the integer 4, but it's not quite the same. A similar distinction may be more interesting later.
It's been known since ancient times that the integers and rationals weren't adequate, since there is no ratio of integers that can produce two when squared. Only recently have I understood why this so upset the Pythagorians, and I'm sure my distress concerning the real numbers is only a fractions of what they felt then. As far as I know, there are only two ways to form a meaningful irrational number, Dedekind cuts and Cauchy sequences. I haven't read Dedekind's papers yet, and I don't know much about his approach. I'm more familiar with Cauchy sequences, which uses an infinite number of rationals to approximate an irrrational, which is obtained as the limit of the infinite sequence. Technicaly, a Cauchy sequence is a series of rational numbers (a0, a1, a2, a3, ... ) where there exists some N such that for any two distinct i, j > N, there is a delta (I'll have to use d) 0 < | aj - ai | < style="font-style: italic;">The Calculus Gallery that the rationals are very poor approximators of the irrationals, from Liouville's number. That is, for any irrational x, any and all rational numbers are a measurable distance away from it
More later.
Notes about builds from integers to rationals to Cauchy Sequences and irrationals. Divide the irrationals into algebraics and transcendentals, the difference being the infinitude of their formula. Distinguish further those transcendentals which can be fully specified with a finite expression and those that cannot. The later class is the unnameable transcendentals. Expand on that.
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