Math Is Fun Forum

Hi jadewest,

The key bit is here:

In other words, if you've got an earthquake A measuring 1 on the Richter scale and earthquake B measuring 2 on the Richter scale, earthquake B has an intensity 10 times stronger than earthquake A. Does that make sense?

What would happen if earthquake A measured 2 on the Richter scale and earthquake B measured 4 on the Richter scale -- what would the difference in intensity be then?

More generally any cubic can be reduced to a depressed cubic by substitution -- and then the method above can be applied.

Hi Jeremy,

Welcome to the forum.

The reason that pi doesn't have a final digit is because it is an irrational number, and irrational numbers have decimal expansions which continue forever. In other words, to answer your question it suffices to:

(1) Prove that pi is irrational, and then
(2) Prove that all irrational numbers have an infinite, non-recurring decimal expansion.

There are quite a few proofs that pi is irrational, some of which are more complex than others. One of the most common proofs is Lambert's, where he essentially (a) shows that all infinite continued fractions are irrational and then (b) finds an infinite continued fraction for pi, which automatically implies that it must be irrational from part (a). If you're interested in continued fractions I've got some videos about it on my YouTube channel (although it doesn't discuss the continued fraction of pi). This takes care of (1).

Now for (2). Suppose instead that you could find an irrational number which didn't have an infinite, non-recurring decimal expansion. Let's say for example, the number 0.12345123451234512345... This decimal expansion has an infinitely recurrent pattern (the '12345' bit). But we see that if we let x = 0.12345123451234512345... then:

Multiplying both sides by 100000:

Subtracting x from both sides:

And finally, dividing both sides by 99999 gives us:

But hang on -- if we can express it as a fraction of two integers, it must be a rational number! (This is pretty much the definition of what it means for a number to be rational.) Similarly, if pi had a 'final digit' we could also express it as a fraction -- for example, if pi terminated after 5 decimal places, i.e. 3.14159, then we could write that as 314159/100000, which is a rational number.

Let me know if that makes sense -- happy to clarify anything if needed.

You can also type the pi symbol in text form by typing the following:

:pi

which produces π.

Hi Bob,

I think the two questions are intended to be separate (the condition that f, g are three-times differentiable is probably just there to ensure that f''' and g''' exist rather than bearing any relation to Q1). I am interpreting (f.g)^(3) as meaning the third derivative of (f times g) but happy to be corrected by the thread starter.

Thanks both -- I've edited the original post back to what it was (and it should stay that way).

In addition, Disfruta las Matemáticas is the Spanish version of the website.

Hi CurlyBracket,

Yes, that's right. In general:

which you might see on your scientific calculator.

Note that an important point is whether or not the order matters. For example, suppose you have four cards labelled (A,B,C,D) and you want to choose 2 out of those 4. How many different ways are there to do it? If we don't care about the order, the combinations are just:

(AB, AC, AD, BC, BD, CD)

(AB, BA, AC, CA, AD, DA, BC, CB, BD, DB, CD, DC)

Hi MaddSci3ntisT,

Welcome to the forum.

It is unfortunately not just a recent phenomenon: it has been happening daily since I became a mod here and probably before then too (based on the lengthy ban list). In most cases the decision to delete/ban is straightforward since many of the posters you mention have often already been reported multiple times on places like 'Stop Forum Spam' or otherwise have an e-mail address which a simple google search reveals that they've done the same thing on half a dozen other forums. In other cases it can be much more difficult because some new members will deliberately post something harmless and then edit in a link a few days later, or they might pose as an ordinary member only to start eventually posting ads. These (particularly the former) are generally harder to spot and are less easy to deal with: I generally avoid banning posters unless I have good reason to and prefer to have some evidence of wrongdoing if at all possible but these types are admittedly (somewhat) rarer.

New users have to click on an activation link sent to an e-mail address to register which cuts down the number of spam registrations significantly: without this we would easily have a hundred different spammers a day (this is no exaggeration). As I understand there are also restrictions which prevent most new members (with < x posts) from posting links, which is why we have lots of new daily registrations from spam accounts but of those only a small proportion end up posting ads, and there are also word censors which filter out a few too. Consequently the spam that does make it through is, IMO, annoying but not unreasonably so, given (a) the nature and number of spam posts, (b) the overall forum activity susceptible to clicking on said posts and (c) the administrative resource available. At the moment (a), (b) and (c) are reasonably balanced and therefore the level of spam is, in my view, manageable. Regrettably we have in the past had the occasional new member post some particularly awful content on here -- and I can't be certain that some of the people who visit this forum (members or not) didn't fall victim to it where we may not have been able to deal with the issue immediately -- but fundamentally the most determined spammers/advertisers will find a way around any precautionary measure and so we tend to operate on a best endeavours basis given the tools we have.

In the meantime please continue to report any posts you see like this as it helps us deal with them more quickly: we often check the forum a few times a day but there will always be the odd one or two that get through when none of the mods/admins are active.