## What is Wolfram Alpha good for?

It’s early days and this isn’t meant to be a carefully considered review of Wolfram’s “computational knowledge engine”. Rather, I just want to point out, for the benefit of anyone who might not yet know, that one small part of what it does is genuinely useful in a certain circumstance that comes up from time to time. Suppose that for some reason you want a list of primes, or to know $e$ to 100 decimal places, or the 100th power of 2. Previously I would have used Google for the first two, banking on someone somewhere having put the information online, and I might have struggled to understand just enough Mathematica to do the third. (However, I have just discovered that powers of 2 can also be found quite easily with the help of Google, so a more complicated example might be needed.)

Anyhow, with Wolfram Alpha one can type in some reasonable text such as “The first hundred powers of 2″ or “pi to 100 places” and it works out what you mean and gives you the answer. That alone won’t change my life, but it is convenient and it will occasionally help me with things like preparing lectures for a general audience, which I think is just about enough to make it worth it to me to bookmark the site, though I haven’t yet done so. It will also sketch graphs and simplify mathematical expressions without one having to learn any special language to put them in — you just guess what to write and if your guess isn’t too perverse it can work out what you mean.

What else does it do? Typing in “father of Barack Obama” gives “Wolfram|Alpha isn’t sure what to do with your input”. Just typing “Barack Obama” gives you his full name and his date and place of birth. Typing “England” gives you various basic facts about England. Typing “capital of Uruguay” gives you Montevideo and various facts such as its population, current weather, etc. After noodling about like this for a short time, I did what any non-saint would do and typed in my own name. To be precise, I typed in “Gowers”. The result was “Wolfram|Alpha isn’t sure what to do with your input”, together with the helpful suggestion that perhaps I had meant “powers”.

I think that gives a fairly good idea of what it does and what it doesn’t do. Perhaps one should regard the latter as a truly positive and innovative aspect of Wolfram Alpha: a New Kind of Search Engine (or whatever it should be called) that doesn’t waste hours of your time by tempting you to look yourself up.

### 53 Responses to “What is Wolfram Alpha good for?”

1. Jason Dyer Says:

It took me a while to figure this out, but it also can plot in polar coordinates:

polar plot sin(3x) from 0 to 2pi

2. rssfever » What is Wolfram Alpha good for? « Gowers's Weblog Says:

[...] post by gowers aggregated by [...]

Great post. Informative about WA. Check out my post on WA also. I took what WA can do a step further and posed the question if it is even possible to compare WA to Google. http://www.smadamr.com/2009/05/too-early-to-d…-wolfram-alphatoo-early-to-decide-the-fate-of-wolfram-alpha/

4. Alejandro Says:

Yesterday I needed to find the roots of a non trivial function and, having no software installed, I simply went to wolfram alpha, wrote the function and it gave me the roots, a graph and other useful information. Maybe Wolfram Alpha isn’t very useful for the general public (yet), but for mathematicians, I think it is gold!

5. Giles Warrack Says:

A nice evualation of one Etonian in College by another!
(submitted by a Wykehamist, but not in College)

6. Rafael Says:

Just out of curiosity, why did u type ‘the capital of Uruguay’?

7. Ashutosh Mehra Says:

Wolfram Alpha appears to accept a variety of simple Mathematica expressions:

PolarPlot[Sin[3x],{x,0,2Pi}] (Jason’s example above, in Mathematica)
FactorInteger[2^100-1]
Series[Log[1+x],{x,0,10}]
FindRoot[x == Cos[x], {x, 1}]
Integrate[1/(1 + x^3), x]
Sum[1/i^2,{i,1,100}]

But Alpha _doesn’t_ seem to allow even slightly more complex expressions. For instance, if I try to multiply the last sum by a factor of 6, it gets confused:
Sum[1/i^2,{i,1,100}]*6

Overall, it is a very useful resource.

8. nicolas Says:

Depends for what.
Practical case today :
i wanted to compute my Body mass index, and wolfram is just the place.
I did not want the definition hidden in a page, or have to convert from metric system to the stupid system.

you type in bmi in wolfram, you input the fields, and there you have it.
that was such a breeze.

9. Mike Says:

Can I declare myself to be the compleate luddite, and say that this almost looks like what alta vista was trying to do in the last century. Did google but this or am I wrong?

10. Mike Says:

11. stevewiilliams Says:

@Mike. Did Google purchase Alta Vista? No, they did not. Yahoo is the current owner of Alta Vista. Back in 2003, a company called ‘Overture Services Inc.’ purchased Alta Vista, and then Yahoo purchased Overture. To this day, Alta Vista remains the portal through which Yahoo’s search results are provided.
See the full Wikipedia article here: http://en.wikipedia.org/wiki/Alta_Vista

• mikeathome Says:

Thanks for that Steve. Great article, it’s amazing how after so few years the exact details are blurred. I’m pretty sure I first used Alta Vista pre- 1995, just based on where I was working. But we were all gushy about the power of the Alphas, that internet stuff appeared to be a passing trend at the time. I still miss usenet as it was in the beginning, I think I had eudora for news and something that read gopher sites. Roll on the semantic web!

12. Daniel de França MTd2 Says:

Try this:

http://www94.wolframalpha.com/input/?i=antarctica

It only lists only japanese movie instead of a continent. This is a shame.

• Lora Says:

“Assuming “antarctica” is a country | Use as a movie instead”

- Lora

13. disquisitionesmathematicae Says:

“A New Kind of Search Engine (or whatever it should be called) that doesn’t waste hours of your time by tempting you to look yourself up.”

Good one!

14. Lora Says:

WolframAlpha is great. It’s like a chatbot. ^^
http://www94.wolframalpha.com/input/?i=how+old+are+you%3F
http://www94.wolframalpha.com/input/?i=How+many+languages+do+you+know%3F

- Lora

15. disquisitionesmathematicae Says:

Try Goldbach. The output is some city in Germany.Witten is also some city in Germany. And Tao, unfortunately, is a drug ingredient.

But it got Gauss, Euler and (surprisingly) Grothendieck right. Tells you something about the standards Wolfram uses.

16. disquisitionesmathematicae Says:

And when I put in my name (Sameed), it gave a prompt:
“did you mean ragweed?”

17. josh g. Says:

Wolfram doesn’t claim that Alpha is a search engine.

From the FAQ:

Is Wolfram|Alpha a search engine?

No. It’s a computational knowledge engine: it generates output by doing computations from its own internal knowledge base, instead of searching the web and returning links.

So of course it uses different standards. I think that it’s still odd when it comes up with weird partial results on a topic that isn’t heavily mathematical, so they have some work to do there.

18. Andy Says:

Hi Tim,

I’ve been playing around with thi too and I love when I find something about it that surprises me (which keeps happening!) check this one: http://www94.wolframalpha.com/input/?i=how+many+roads+must+a+man+walk+down+before+you+can+call+him+a+man

All the best.

• gowers Says:

I’ve been surprised that it hasn’t got more famous texts in its database. For example, if I type “To be or not to be” then it recognises it, but not if I type the almost as famous “Tomorrow and tomorrow and tomorrow” or “All the world’s a stage”. It did recognise “et tu Brute”, though with no hint that it had anything to do with Shakespeare. It clearly doesn’t have the complete works of Shakespeare, the Bible, etc.

Another question I can’t help asking myself when it does work is who would actually find a lot of what it does useful. Some of it is certainly useful, but often you type something in and what you get out is some very basic data that you could easily get from Wikipedia, with the disadvantage that you can’t go on to explore it in more depth.

While I’m writing this, I might also mention that “Solve sin(x)=x/2″ gave just the solution x=0 and not the other two solutions, even though there was a nice plot of sin(x) and x/2 that had the three points of intersection marked in red. That seems like a glitch to me (and doesn’t happen if you slightly change it so that there isn’t a trivial solution).

Here’s another amusingly bizarre result.

• Lora Says:

That’s not weird… this is:
http://www32.wolframalpha.com/input/?i=person+Maxwell+silver+hammer

- Lora

19. Kenny Easwaran Says:

That search on “Maxwell’s silver hammer” is especially bizarre given that it uses the term “Mestizo” but not “Hispanic” or any apparent synonym, and that it gives multiplication as an alternative to a list.

On my first trials of the site, it didn’t understand me when I said “list of US states by population” and suggested “US states” instead, which just gave a list of the fifty states in alphabetical order. When I asked “population of los angeles” and “population of san francisco” it gave nice historical graphs as well as the current value, but for “population of edmonton” it only had the current value.

20. Wolfram|Alpha: A new way to search « The Economicist Says:

[...] · No Comments In case you’re wondering why I’m posting this, especially when Wolfram|Alpha was on the front page of WordPress.com today anyway, let me explain. I had the idea to blog about WA (Wolfram|Alpha, wolframalpha.com) earlier [...]

21. Top Posts « WordPress.com Says:

[...] What is Wolfram Alpha good for? It’s early days and this isn’t meant to be a carefully considered review of Wolfram’s [...] [...]

22. Daniel de França MTd2 Says:

If Wolfram cannot give statistics or generate about a whole continent, you can expect the rest of it to be pretty much useless and worthless.

“Assuming “antarctica” is a country | Use as a movie instead”

This is why it fails hard.

23. Kent Yee Says:

To some extent it’s failing on some sociological, anthropological, and paleontological data. But I guess this is a good start for WA already. The more we use it the more it will evolve to provide our needs.

Try searching Homo habilis, Homo sapiens, or Homo erectus. It does not return anything. I was expecting to see cranial sizes, or time/period these species exist or even the cladistic information.

Anyway, as I mentioned above, maybe the more we use it the more it will provide us relevant or useful information. Just like Google, it evolved well through out time because a lot of us are using it.

I am seeing more promises and good surprises from WA in the coming future.

24. nathanieljohns Says:

The problem with Wolfram Alpha right now is that it simply doesn’t know what to do for far too many inputs. It’s “neat” when it does work and I can see where it would be useful for people who are less comfortable with Maple/Mathematica/whatever, but considering about half of the inputs I try simply produce an error message, why shouldn’t I just try to find the information on Google/Wiki/Maple first and *know* that I’ll find the answer?

I would rather have a tool that does one or two things extremely well than have a tool that tries to do everything, and does 50% of them alright.

25. Anonymous Says:

Keep in mind that Wolfram has a long history of what could at best be considered extremely abusive manipulation of intellectual property laws. (For example, suing his employee Matthew Cook for revealing that Cook had proved that rule 110 was universal.)

Unfortunately, Wolfram|Alpha fits into that tradition by carefully obscuring everyone else’s contributions to the data. It doesn’t cite any sources unless you click the source information button. If you do, it tells you that the “primary source” is Wolfram|Alpha, and it lists a bunch of “background sources and references” without any clear indication of which parts of the data correspond to which source. (In fact, a little lawyerly note explains that this doesn’t mean any specific source was used for any specific query.)

The FAQ list specifically says that people should cite Wolfram|Alpha as a primary source. That’s appalling – for some computed results, it is indeed a primary source, but for most data it’s a secondary source.

Nobody who cares about credit or scholarly sources should contribute data to Wolfram|Alpha or make any scholarly use of it.

26. Simon Says:

“sum x^-1.5″ ( http://www65.wolframalpha.com/input/?i=sum+x%5E-1.5 ) gives the following:

“Input interpretation:
$\Sum_{n=1}^{\infty} \frac{1}{x^{-1.5}$

Result: Sum does not converge. By the limit test, the series diverges.”

Hmm.

(Was that the sound of a million of Analysis 1A supervisors crying out in terror, and being suddenly silenced?)

((interestingly, sum n^-(3/2) gives the right answer))

• Lora Says:

The link you got there has an input interpretation: “sum1/x^1.5″
Result: (sum does not converge)
Looks pretty cool with the graphic and stuff.

- Lora

• Nathaniel Johnston Says:

@Lora – The point was that it DOES converge, even though Wolfram says otherwise. It seems to not know how to handle decimal exponents.

• gowers Says:

And yet weirdly at the bottom of the page it gives the correct answer under the heading “regularized result”

27. Miguel Lacruz Says:

The input “integral x^-1.5 from x=1 to infinity” provides the right answer. There is a conflict with decimal exponents for infinite sums.

28. disquisitionesmathematicae Says:

On typing “Brun’s constant” you get the following output:

Input:
B_2

Exact form:
1.9021605831
Decimal form:
1.9021605831
Continued fraction:Fraction form
[1; 1, 9, 4, 1, 1, 8, 3, 4, 4, 2]

Like the search above, it seems to be having trouble with infinite sums.The exact form should have been an infinite sum and not the decimal expansion,though it would have been wonderful if they had an exact decimal representation!

I searched the Wiki as well (just to find out how the competitors are doing) and surprisingly I came across a bizarre error there as well.The article on Brun’s constant says,
“The best estimate to date was given by Pascal Sebah and Patrick Demichel in 2002, using all twin primes up to 1016:

B_2 ≈ 1.902160583104.

While 1.9 < B_2 is shown, no real number N is known such that B_2 < N."

• Nathaniel Johnston Says:

This is perhaps a silly question, but can I ask where the error is there in that wiki article? Although they have confidence intervals that provide extremely strong evidence that the first 7 or 8 decimal places of Brun’s constant as calculated to date are correct, as far as I know no upper bound has yet to be rigorously proven. If this is incorrect, the wiki article should of course be corrected.

29. gowers Says:

I haven’t checked carefully, but there doesn’t seem to be anything about the proof of Brun’s theorem (that the sum of reciprocals of twin primes converges) that would make it ineffective: the Brun sieve itself is very simple and direct, and the use of it doesn’t appear to rely on any ineffective results. So I’d have thought that with a bit of patience and not much difficulty one could just go through the proof and check that it gives an explicit upper bound. Maybe a proper analytic number theorist could confirm this or point out where it is wrong.

30. Emmanuel Kowalski Says:

I’m also sure that getting an effective upper bound should be no problem here. (E.g., with the large sieve, where there is no hidden constant, one needs just to do two or three summation by parts effectively to get an explicit upper bound of size n/(log n)^2 for the number of twin primes at most n, and then one gets an effective lower bound for the n-th twin prime.)

Maybe the problem is that one will get an upper bound way above the actual value (maybe by a factor 10); the best asymptotic upper bound for the number of twin primes is off by a factor 7/2 (if I remember right — it may have been improved since I learnt this) from the expected asymptotic, so one can’t really hope to get a decent precision…

31. lukas Says:

WA is quite handy for correcting tests on generating fuctions

32. gowers Says:

Kevin Buzzard sent me an email with a very interesting reason to suppose that Wolfram alpha may not be as useful as it hopes it will be. Here is the relevant part (which I reproduce with his permission).

Wolfram alpha: you say “here’s a great thing Wolfram alpha can do: if I want to know 2^100 then (blah)”. Ok let me explain another way of doing this that I conjecture will be more robust. Let me use the following example: let p_i denote the i’th prime. Recently I wanted to know an example of a prime p_i with p_{i+1}-p_i=366. I tried a maths package but clearly the prime I was after wasn’t going to be reached by a simple loop (it’s much too big).
So I wrote a simple stupid loop anyway, in order to find the smallest p_i with p_{i+1}-p_i=1,2,4,6,8,10,12. I could even have done this by hand. The answers are 2,3,7,23,89,139,199. We now have a “what are the next few numbers in this sequence?” question and you suggested Wolfram Alpha might be good at this (and indeed it might). But Sloane’s online encyclopaedia of integer sequences is *really* good at this—and I mean *really* good.
I went to

http://www.research.att.com/~njas/sequences/

and typed in the first few terms and I instantly get references and links like

http://www.research.att.com/~njas/sequences/b000230.txt

from which I could immediately get my answer, and so on.

To give another example: last month a friend of mine asked me if I could say anything about the positive integers n such that sigma(n)=sigma(phi(n)) with sigma the “sum of divisors” function and phi Euler’s totient function. Rather than trying to do mathematics I did exactly the same thing: wrote a little computer loop to search for small solutions, found

1
87
362
1257
1798
5002

, searched for this sequence in Sloane and boom! A big list of known examples, references, infinite families of examples, open problems and more. On the other hand, *currently* when faced with the input

sigma(n)=sigma(phi(n))

Wolfram can’t parse it. I don’t know enough about Wolfram Alpha to know if it can be persuaded to do what I did (which of course is that I just followed an algorithm!).

Kevin

33. Polar Coordinate Investigation Using Wolfram Alpha (help needed see bottom) « Adam Lavallee Says:

[...] good thing Wolfram Alpha isn’t a subsidary of GoDaddy!) landed me on this blog: Gower’s.  Somewhere in the comments, I found that a simple comma would allow my to graph both equations [...]

34. Week 4 – Google Squared vs Wolfram Alpha « Information Architecture & Content Management Says:

[...] What is Wolfram Alpha good for? http://gowers.wordpress.com/2009/05/22/what-is-wolfram-alpha-good-for/ [...]

35. axix Says:

i feel this is good for getting you stupid because it does everything for you and the more you get used to this the more ur gona get dumb…
P.S. i use this app for my itouch 4g