Wikipedia:Reference desk/Archives/Mathematics/2009 December 21

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December 21

Calculus History

I am reading books on the history of Mathematics and its development, specifically Calculus and I am now further confused. I thought I had it right but I am sure that I don't so maybe some experts here can help clear up a few things. First about the Bernoullis, I know that they were Swiss but I always thought that their background was Italian. Is that true? The article here doesn't really say anything about this. Was it like an Italian family? Is the name Italian? Is the name German or something? Were they originally Italian who then relocated to Switzerland or something?

Second, the more significant question, is about the actual development of Calculus. As I understand, Newton (and Leibniz) are credited with the "invention" of calculus because they proved the Fundamental Theorem of Calculus. But then I learn that Riemann was the one who redefined the integral (using the definition that a function is said to be integrable if for a given epsilon greater than zero, there exists a partition such that the upper sum and the lower sum over that partition are within epsilon of each other) which allowed Riemann to prove all the properties of the integral previously known (such as linearity) and he could now integrate function with discontinuities (even with an infinite...with measure zero as we now know...number of discontinuities) and then Riemann also proved that integration and derivatives are inverses operations with his newly defined integral. So isn't Riemann the one who proved the fundamental theorem of calculus? Why isn't it credited to him? I mean the form we see it in today came from him.-Looking for Wisdom and Insight! (talk) 00:59, 21 December 2009 (UTC)

Wow, good questions! My information (s:1911 Encyclopædia Britannica/Bernoulli) is that the Bernoullis were fleeing the Spanish when they cam to Switzerland about a hundred years before they became famous. It doesn't say whether they actually were Spanish or how they got the name which doesn't sound Spanish any more than it sounds German. I will note though that 1) Itallian is spoken in Switzerland, though generally not as far north as Basel, and 2) People were a bit more flexible about their names then than we are now, so the name they used might change depending on who they were talking to or they might use a Latin version (which you had so speak to be considered literate at that time). I worked on the Bernoulli articles and I basically had to go by birth and death years to tell them apart; they all used two or three first names and most of the names were used by two or three relatives.

If you're interested in the history of calculus I recommend The Calculus Wars By Jason Socrates Bardi. The short version is at Leibniz and Newton calculus controversy. Anyway, Newton and Leibniz invented calculus using something called fluxions or infinitesimals (depending on which side of the English channel you were on). By modern mathematical standards they were very non-rigorous and it wasn't until Riemann and Cauchy and their generation that it was all put on a firm footing, whence the Riemann integral etc. My understanding is that part of the motivation for doing this was a scathing criticism of infinitesimals by Bishop Berkeley. This is a case of methods being ahead of the proofs that they work, which happens a lot more than mathematicians would like to think. In this case the the methods, known as the methods to calculate infinitesimals, or the infinitesimal calculus, or nowadays just calculus, while not rigorous, at least seemed plausible, so people used them because they were useful. In any case, the development of calculus took place over thousands of years so deciding wh gets credit for it is going to be arbitrary anyway, but that's the way the history of science goes much of the time. A lot of that is my personal viewpoint so take it with a grain of salt, but it does seem to be a more interesting subject than you would think.--RDBury (talk) 05:47, 21 December 2009 (UTC)

Fréchet Second Derivatives and Taylor Series of matrix functions

Hi all,

Another one from me! I've got a distressingly long list of Christmas work (how cruel!) I've got a big long list of Taylor series to calculate for matrix functions, using the Fréchet derivative - however, my lecturer has failed to give any examples (helpful) nor can I find any on the internet, so I'd greatly appreciate it if someone wouldn't mind showing me an example before I start beavering away at the list!

Say, $f(A)=A^{3}$ for any n*n matrix A: $f(A+H)-f(A)=(A+H)^{3}-A^{3}=A^{2}H+AHA+HA^{2}+o(|H|)$, and so $Df|_{A}(H)=A^{2}H+AHA+HA^{2}$ is the Frechet derivative. Now how do I go about calculating the second (third etc) Frechet derivatives? (This is the first example on my list - I have the formula $f(A+H)=f(A)+\sum _{r=1}^{\infty }{\frac {1}{r!}}D^{r}f|_{A}(H)^{r}$, right?

Thanks very much for the help (again!), I think once I've got one example sorted I can get going on the rest!

Much appreciated! Typeships17 (talk) 03:32, 21 December 2009 (UTC)

Yes, that expansion sounds like the evil laugh of your lecturer. The second derivative of $f(A):=A^{3}$ is the symmetric bilinear map 1/2(UAV+UVA+VAU+VUA+AUV+AVU) $\scriptstyle D^{2}f(A)[U,V]=UAV+UVA+VAU+VUA+AUV+AVU.$ Actually this holds in any Banach algebra; if it is commutative, you find $D^{2}f(A)[U,V]=6AUV.$ A very efficient way to prove that a map is C^k, and to compute its differentials is the converse of Taylor theorem: a map from an open set of $\mathbb {R} ^{n}$ to $\mathbb {R} ^{m}$ is of class C^k if and only if it has a polynomial expansion of order k at any point of the domain, with continuous coefficients, and with a remainder which is locally uniformly o(|h|^k). For k=1 this is the definition of C¹ of course.--pma (talk) 09:51, 21 December 2009 (UTC)

What is that 1/2 doing there? Algebraist 12:44, 21 December 2009 (UTC)

I wonder too... --pma (talk) 12:57, 22 December 2009 (UTC)

Hah, I wouldn't be all too surprised if he did laugh like that. That's great but how did you go about actually calculating it? I'm not sure I follow quite how to get from the first derivative to the second and so on, perhaps the concept of going from a linear to bilinear to trilinear etc map is bewildering me. What limit gave you your (1/2?)UAV+UVA+VAU+VUA+AUV+AVU? Many thanks again, Typeships17 (talk) 14:24, 21 December 2009 (UTC)

You just take the first derivative and perturb A again:$Df|_{A+K}(H)-Df|_{A}(H)=KAH+AKH+KHA+AHK+HKA+HAK+o(|K|)$. The Taylor series you end up with will of course just be what you get by multiplying out (A+H)³. Algebraist 16:13, 21 December 2009 (UTC)

That's great, I've got the idea now, thanks ever so much - now onto A^-1, this one should prove a bit more challenging! (If anyone has any tricks for the general form of the nth derivative, please feel free to let me know, I managed to batter my way through the first but no further...) Anyway, many thanks again to both of you :) Typeships17 (talk) 17:58, 22 December 2009 (UTC)

Invertible matrix (more generally, invertible elements of a B-algebra) are an open set, and the inversion map $\operatorname {Inv}$ is analytic: if $\|H\|<1/\|A^{-1}\|$ the element $A-H$ is invertible and you have the expansion (a real evil laugh):

$(A+H)^{-1}=A^{-1}-A^{-1}HA^{-1}+A^{-1}HA^{-1}HA^{-1}-A^{-1}HA^{-1}HA^{-1}HA^{-1}+\dots$

From this you can find all the differentials, simmetrizing. E.g $D\operatorname {Inv} (A)H=-A^{-1}HA^{-1}$ and $D^{2}\operatorname {Inv} (A)[H,K]=A^{-1}HA^{-1}KA^{-1}+A^{-1}KA^{-1}HA^{-1}$--pma (talk) 09:15, 24 December 2009 (UTC)

Vector cosine

I was looking at Amazon.com's "people who bought this item also bought..." algorithm and I noticed that they use vector cosine to group users. For example, if I bought items 1, 6, and 9 (the product ID for each item), my purchase vector would be {1,6,9}. If you bought {5,6,7}, the cosine of the two vectors would be 4.86 (if my math is correct). I know that when the vectors are identical, the cosine of the vectors is 1. What is the domain of vector cosine? Is there a limit that indicates "opposite", such as when comparing {1,2,3} to {3,2,1}? Is there a limit that indicates "nothing in common", such as when comparing {1,2,3} to {4,5,6}? I'm curious about how accurate it is to use vector cosine to identify how similar two vectors are. -- kainaw™ 05:30, 21 December 2009 (UTC)

You should check out the articles Collaborative filtering and Netflix prize. The vector cosine seems to be a term used by people who specialize in this area rather than most mathematicians, but my research indicates that it's simply the cosine of the angle between the two vectors. If two vectors are nearly the same direction then the angle between them is nearly 0 and the cosine is close to 1. If vectors aren't close to the same direction then the cosine is closer to 0 or even negative. It turns out that the cosine is easier to compute than the angle itself (See Angle#The dot product and generalisation so it's useful for doing computation.--RDBury (talk) 06:11, 21 December 2009 (UTC)

Thank you. That is a good link. I guess I'm just doing cosine of vectors wrong since I get 4.86. I thought cosine was limited to the range -1 to 1. Perhaps I'm just adding or multiplying wrong. -- kainaw™ 07:20, 21 December 2009 (UTC)

I have no idea what Amazon does (a link to your source would be welcome), but the purchase vectors above should probably be $(1,0,0,0,0,1,0,0,1,0,\ldots )$ and $(0,0,0,0,1,1,1,0,\ldots )$. Their so-called cosine similarity (which is indeed between -1 and 1) is 1/3. It can only be negative when some of the entries are, which is impossible in this particular setting. Negatives in general indicate opposite directions, with -1 polar opposites. 0 indicates no common items here, or orthogonality in the general case. Also note that {1,2,3} and {3,2,1} are the same, not opposite. -- Meni Rosenfeld (talk) 16:12, 21 December 2009 (UTC)

I'm inclined to agree with Meni Rosenfeld, and the "cosine" reported to be 4.86 above must be a mistake: such a cosine cannot exceed 1. (There are complex numbers whose cosine is a real number greater than 1, but that doesn't apply here.) Michael Hardy (talk) 20:32, 21 December 2009 (UTC)

I did have some math mistake somewhere. The cosine is 0.91. The formula shown in all of the papers I've read is cosine(A, B) = (A·B)/(||A||*||B||). At first, I thought ||A|| was the length of A (how many items are in A). I then noticed that it was the square root of the sum of all the elements of A squared. The dot product is a bit of a problem - what if the vectors are different lengths? Just use zeros to pad the smaller one? I don't see how you can get 0 since all the vectors being used are positive integers greater than zero. -- kainaw™ 20:47, 21 December 2009 (UTC)

Again, I think you are confused about how vectors represent purchases. The simple way (which again, may or may not be what Amazon does) is to have a vector whose length is equal to the total number of items available for purchase, and which has 1 in indexes of purchased items and 0 elsewhere. With this encoding, the cosine similarity in the example you gave is 1/3, like I said. -- Meni Rosenfeld (talk) 05:05, 22 December 2009 (UTC)

None of the examples that I've seen use 0/1 representation. They all use a vector of integer identifiers. Many refer to it as Pearson product-moment correlation coefficient. I'm now reading about the "centering" involved to see how that affects the cosine. -- kainaw™ 05:15, 22 December 2009 (UTC)

If those examples are online, please provide a link. My guess is that they present a list of IDs for compactness but do the calculations with a 0/1 representation.

In any case, it should be crystal clear that what you have done - multiply out the IDs - makes absolutely no sense whatsoever. For starters, IDs are on a nominal scale, while multiplication requires a ratio scale (Level of measurement) (with centering, you only need an interval scale). Second, it creates completely absurd situations. {1,2,3} has <1 similarity with {3,2,1} although they are the same purchases. {1,2,3} had >0 similarity with {4,5,6} although they have nothing in common. The similarity between {1,2} and {1,3} is different than between {5,10} and {6,10} although they have the same structure. The similarity between {1,3,5,7,92678} and {2,4,6,8,93412} is very high although they have nothing in common, while the similarity between {1,2,3,4,87154} and {1,2,3,75642,5} is close to 0 although they have a lot in common. -- Meni Rosenfeld (talk) 05:45, 22 December 2009 (UTC)

See "Geometric Interpretation" in Pearson product-moment correlation coefficient. It uses {1,2,3,4,5,8} and {.11,.12,.13,.14,.15,.18}. I'm going to do some tests with centered vectors to see if the results make sense. According to the article, 1/-1 is highly correlated and 0 is no correlation. -- kainaw™ 06:08, 22 December 2009 (UTC)

This has nothing to do with purchases. Here each index represent a country, the first vector gives the GNP for each country and the second vector gives the poverty for each country. Taking the dot product (after centering) works, because you are multiplying matching quantitative measurements (the GNP of a country with the poverty of the same country).

In the purchasing scenario, you tried to multiply IDs (which of course cannot be multiplied) by matching them based on their position in the purchase list. So if the 8th item customer A purchased is a children's book (ID 134675) and the 8th item customer B purchased is a shotgun (ID 134677) (made up numbers), you count it as evidence for similarity. And if the 8th item customer A purchased is a children's book, while the 9th item customer B purchased is that very same book, you don't count it as anything.

I don't mean to sound disrespectful, but it seems you are biting a bit more than you can chew here. You shouldn't try understanding collaborative filtering algorithms if you've not yet mastered basic topics like Pearson's correlation coefficient. -- Meni Rosenfeld (talk) 06:33, 22 December 2009 (UTC)

I see my mistake now. In collaborative filtering, the term "similarity" is often used to mean "correlation". In actuality, those are two very different terms. I was trying to see how cosine produced a similarity when all it produces is a correlation. So, my initial assumption that cosine does not produce a valid similarity is correct if the definition of similarity is not rationalized to mean correlation. -- kainaw™ 12:01, 22 December 2009 (UTC)

That's not quite right. For sure, "correlation" is one thing and "similarity" is another. Indeed, the correlation between GNP and poverty has nothing to do with similarity. But nobody tries to imply that one means the other. Rather, it is claimed that the correlation between the features of two items can indicate similarity between the items. This may or may not be valid, depending on the features we choose.

In the case of Amazon, the "items" are customers. The features are the products they purchased, or more specifically, the ith feature is a 0/1 variable indicating if a customer purchased product i. It is claimed (or not. Where is the link to Amazon's algorithm?) that correlation between the features of customers indicates similarity between the customers. For example, customer 13 purchased products 1,2 but not 3, and customer 25 also purchased products 1,2 but not 3. This is used as evidence that customers 13 and 25 are similar (have the same shopping preferences, or whatever).

Of course, there are countless other ways to approach the problem of similarity, but representing items as feature vectors is very powerful. Even then, cosine similarity is just one of the ways to compute a correlation metric between feature vectors - and hence, by assumption, a similarity metric between items. -- Meni Rosenfeld (talk) 12:32, 22 December 2009 (UTC)

The length of A can be the number of non-zero coordinates in some contexts, e.g. coding theory, but in this case it means length in the Euclidean sense. The cosine formula includes the lengths of the vectors to allow for varying lengths of the vectors involved.--RDBury (talk) 05:08, 22 December 2009 (UTC)

Calculi(Non Newtonian)

Everybody, I don't seem to understand what's going on the pages "Other Calculi" here[1]. Can anyone explain it to me? Thanks!The Successor of Physics 06:21, 21 December 2009 (UTC)

I gather the idea is a variation on the definition of derivative using multiplication rather than addition. The result is something like a logarithmic derivative. Did you have a specific question?--RDBury (talk) 06:52, 21 December 2009 (UTC)

RDBury, I know that. Maybe I should restate my question. I meant that the bijective function φ there should have two inputs e.g. addition, x + y, x and y are two inputs. How come in those pages, the function φ only has one input?The Successor of Physics 08:03, 21 December 2009 (UTC)

The function φ is not supposed to be addition in ordinary derivatives and multiplication in multiplicative derivatives. Rather, it is the transformation that is applied to transform ordinary derivatives to new derivatives - it is the identity for ordinary derivatives (no transformation), and the exponential function for multiplicative derivatives (exponentiation transforms addition to multiplication - $e^{a+b}=e^{a}e^{b}$). -- Meni Rosenfeld (talk) 10:27, 21 December 2009 (UTC)

Thanks, Meni!The Successor of Physics 14:09, 21 December 2009 (UTC)

To make sure my conception is correct, so what you mean if the f is the function with two inputs e.g. multiplication, $ab=f(a,b)$ then $\phi (a+b)=f(\phi (a),\phi (b))$. Am I correct?The Successor of Physics 14:18, 21 December 2009 (UTC)

Precisely. -- Meni Rosenfeld (talk) 15:51, 21 December 2009 (UTC)

Thanks!The Successor of Physics 04:03, 22 December 2009 (UTC)

Resolved

Article introducing complex numbers

I would like to request a new article introducing the concept of complex numbers. The current article does not introduce them in a way that's accessible to someone who does not already have a great deal of mathematical knowledge. After looking up educational resources elsewhere on the web I found the concept fairly straightforward and logical but I'm not qualified to write it myself.

Just the introduction to 'Complex number' contains 27 links to other articles, of which at least half are similarly dense and inscrutable. As it stands there's no way for someone to develop an understanding of these concepts from reading the wikipedia because there's no starting point, you just wind up clicking between articles full of thick and unelaborated jargon.

FTA: Complex numbers form a closed field somehow with real numbers. OK, so what's a closed field? Don't know, go to the article. OK, it's some type of field, what's a field? Go to the article, and before I've left the introduction I'm wondering what 'quintic relations' are or an 'integral domain' and if I'd only read the wiki I still wouldn't know what a complex number is or what it has to do with anything. Now I'm not averse to learning all this, I'd love to understand it, but clicking from article to article isn't helping. It's frustrating in a way that other areas of the wikipedia aren't, I don't experience this in the physics or computer science sections for example, if understanding one area depends on understanding another one can usually just click through and read the prerequisite article without falling down the rabbit hole. —Preceding unsigned comment added by 196.209.232.87 (talk) 14:39, 21 December 2009 (UTC)

Hmm. I can't actually see a Reference Desk question anywhere in your complaint. You could add your request to Wikipedia:Requested articles/Mathematics, or you could take it to Wikipedia talk:WikiProject Mathematics. Gandalf61 (talk) 14:56, 21 December 2009 (UTC)

your complaint belongs in Talk:Complex_number. Ask short questions here and get short answers. Say, Question: "what is a complex number?" Answer: "a complex number is an expression of the form a+ib where a and b are real numbers and i·i=−1". Go on, ask your next question. Bo Jacoby (talk) 14:59, 21 December 2009 (UTC).

@Gandalf61 - that is what I needed to know, will do, thx —Preceding unsigned comment added by 196.209.232.87 (talk) 15:10, 21 December 2009 (UTC)

Unfortunately, we do not write multiple articles on a particular concept (in accord with the guideline that Wikipedia is an encylopedia, and not an introductory comprehensive textbook). However, I do not mind explaining the terms you have mentioned.

Before I proceed furthur, I would recommend you to read the article on rings, for this provides a reasonably basic introduction to the theory of rings, integral domains and fields (it would be appropriate for you to read from this section onwards).

The set of complex numbers, together with its two operations (addition and multiplication) may be defined as follws:

$\mathbb {C} =\{a+bi:a,b\in \mathbb {R} \}$, where $\mathbb {R}$ is the set of real numbers, and i is the "imaginary unit"; it satisfies the relation $i^{2}+1=0$ or $i^{2}=-1$ (intuitively, it is a "root of −1").

If $x+yi,w+zi\in \mathbb {C}$, we define their sum as $(x+yi)+(z+wi)=(x+z)+(y+w)i\in \mathbb {C}$.

If $x+yi,w+zi\in \mathbb {C}$, we define their product as $(x+yi)\cdot (z+wi)=x\cdot (z+wi)+yi\cdot (z+wi)=xz+xwi+yzi+ywi^{2}=(xz-yw)+(xw+yz)i$.

Hope this helps, and be sure to read this article from this point onwards. --PST 15:10, 21 December 2009 (UTC)

@above - thank you, clearer now. —Preceding unsigned comment added by 196.209.232.87 (talk) 15:50, 21 December 2009 (UTC)

I don't think people should need to read the Ring article to understand the Complex number article so there is definitely an issue with the Complex number article. Math articles tend be be written by mathies for mathies and unfortunately (and contrary to WP:MOSMATH) that sometimes includes articles that should be (at least partly) understandable to typical high school students. It's not a good idea to create new articles to solve this; some people have tried this with articles that have names like 'Introduction to X'. Not only do they amount of content forks but, judging from the amount of heat they generate in AfD discussions, they cause more problems than they solve. The correct solution is to have an non-technical, jargon free introductory section in each article that non-mathies are likely to come across. For the moment it would be a good idea to go over the Complex numbers article with an eye to making the introductory section more accessible, but maybe an more general review is in order.--RDBury (talk) 05:53, 22 December 2009 (UTC)

This is a discussion for Talk:Complex number or WT:WPM, not here. Algebraist 13:23, 22 December 2009 (UTC)

Principles of Mathematics

Is Principles of Mathematics a standard reading in math degrees? Is it still worth reading?--ProteanEd (talk) 17:36, 21 December 2009 (UTC)

Definitely not standard reading. It's worth reading from a historical perspective, but not as a way of learning logic. Mathematical logic has come on a long way in the last 100 years, so modern books are a better choice. It's also a very difficult read - I only got about half way through! --Tango (talk) 17:43, 21 December 2009 (UTC)

(edit conflict) It certainly wasn't mentioned in my degree programme. I haven't read the work, but from glancing at it, it doesn't seem to be a work of mathematics per se, but rather the philosophy of mathematics, which is not normally taught to mathematics undergraduates in any serious way. Even within the philosophy of mathematics, I believe Russell's logicism is rather out of fashion nowadays, though there are certainly still people around who are logicists in some sense. Algebraist 17:47, 21 December 2009 (UTC)

Reducible Polynomials With All But Constant Coefficient Equalling 1

Resolved

Is there any established theory for determining for what C>1 the polynomial xⁿ+x^n-1+...+x+C, n even, is reducible? I was previously unaware of the facts that 1) for n=4 you get a reducible polynomial for C=12 and 2) for n=8 you get a reducible for C=20. Empirically, it appears these are the only cases (I ran a PARI/GP program to C=1000 and n=240, but it doesn't generate a related result if there is a smaller odd n with reducibility in addition to some even n, so this list might be a little short--it seems unlikely it is).Julzes (talk) 18:52, 21 December 2009 (UTC)

Reducible over what? Z? Algebraist 19:15, 21 December 2009 (UTC)

Yes, over Z. Thanks for reminding me.Julzes (talk)

I decided to just mark this as resolved. If anybody reading this happens to have known about these two oddballs, let me know, but it just looks like a nice problem to prove their uniqueness, and I'm sure there is no theory to them.Julzes (talk) 19:20, 21 December 2009 (UTC)

googol

A friend of mine & I decided to look for the googol as a power of 2 (even moderately smart people get bored). We never thought to use the Google calculator (we didn't even know it existed). So, the TI-83. Is 2^332.1928094886 really EXACTLY one googol? Seems amazing... —Preceding unsigned comment added by 174.18.161.113 (talk) 21:28, 21 December 2009 (UTC)

No, it can't be, because log₂(10) is a transcendental number. See the Gelfond-Schneider theorem. --Trovatore (talk) 21:31, 21 December 2009 (UTC)

I think that wins the prize for biggest hammer used to crack a nut. Algebraist 21:34, 21 December 2009 (UTC)

Mm, fair enough, especially given that when I looked it up and thought it through, I realized that to apply G-S, you first need to show that log₂(10) is irrational, which already suffices to answer the question. At first glance I didn't see an easier way of showing log₂(10) is irrational than using G-S, but actually it follows easily from the fundamental theorem of arithmetic. --Trovatore (talk) 21:47, 21 December 2009 (UTC)

To me, the question does not seem serious. The person asking the question is well aware of the fact that if there are more digits the calculator cannot show them. In fact, I imagine that in order to get 13 significant figure on a TI-83, one must subtract (or divide) out the whole part of the exponent, and this process seems too advanced for someone who did not know the answer to the question asked.Julzes (talk) 22:55, 21 December 2009 (UTC)

Here's a really simple way to see this: Suppose

$2^{m/n}=10^{100}=2^{100}5^{100}\,$

where m, n are integers. Then

$2^{\ell /n}=5^{100}\,$

where

${\frac {\ell }{n}}={\frac {m}{n}}-100,$

and so

$2^{\ell }=5^{100n}.\,$

But that is impossible because it says an even number equals an odd number. Any high-school student will understand that one—no Gelfond–Schneider theorem needed. Michael Hardy (talk) 00:12, 22 December 2009 (UTC)

Yes, that's what Trovatore and I were alluding to above. Algebraist 00:37, 22 December 2009 (UTC)

Well, it is a bit simpler than the argument I had in mind, as it doesn't need the full FTA. --Trovatore (talk) 10:12, 22 December 2009 (UTC)

Michael Hardy, you could use this simpler method to prove it is transcendental

$10^{100}=2^{100}5^{100}=2^{100}2^{{\frac {ln5}{ln2}}100}=2^{100+{\frac {ln5}{ln2}}100}$

which is impossibly transcendental.The Successor of Physics 04:19, 22 December 2009 (UTC)

I wasn't trying to prove it was transcendental. But as far as "simpler" goes, the fact is any high-school student can understand my argument, whereas yours would have to rely on more sophisticated results such as Gelfond–Schneider. What exactly did you have in mind as your grounds for inferring that that number is transcendental? Gelfond–Schneider? Or something else? In order to use Gelfond–Schneider, you'd need to know that ln 5/ln 2 is irrational, and the proof of that is just what I gave. I suspect your comments lack all merit. Michael Hardy (talk) 05:11, 23 December 2009 (UTC)

To answer what I think the OP intended to ask - yes, $10^{100}=2^{x}$ exactly, where x is approximately 332.19280948873623478703194294. -- Meni Rosenfeld (talk) 05:00, 22 December 2009 (UTC)