The Geniuses' Thread

Crayolastereo, not really sure where you need help with functions, but since you mentioned part (c), I'll give you some really basic examples of function terminology (it may sound a bit condescending if you already know it, but I'm not sure where exactly you stand) and then some ideas you need to know for those part (c)'s.

Basic Stuff
f(x) = y just means f is a function. I'm not sure if you remember from Junior Cert, but a function is like a machine; you put in a number, you get another number out.
We write f(x) to show that f depends on x.

Eg. [latex]f(x) = 2x^2 + 4x + 3[/latex]
Then if we put 2 into this function (we usually refer to this as 'substituting 2 for x'), we write it as f(2), and here:
[latex]f(2) = 2(2)^2 + 4(2) + 3 = 19[/latex]

So when we put x into f, we get [latex]2x^2 + 4x + 3[/latex] and when we put in 2, we get 19.

By the way, when a function has powers of x, added together and multiplied by numbers, like here, it's called a polynomial.
The numbers in front of the x's are called coefficients.

An important idea with functions is that of 'roots' of functions. A root is a number which, when substituted into a function, gives a value of zero. So, in the terminology of functions, x is a root when
[latex]f(x) = 0[/latex].

So, for our example of a function, x is a root of f(x) when
[latex]2x^2 + 4x + 3 = 0[/latex].
To find x, we need to solve this quadratic equation, which is a Junior Cert question.

Leaving Cert stuff
Your leaving cert book will have all this stuff in it, so I'll just tell you what you might need to do to apply it to the questions.

1. The Factor Theorem
This theorem basically lets you go between having a factor of a function of x, and having a root. Two ways it may be used are:

1. Guess a root of the function, and then factor the function to find the other roots
2. In proof questions, where:

You are given a root a, and you must prove something about f(x). You can say that (x - a) is a factor, and this means that it divides into f(x) with no remainder. So, you do a long division of (x - a) into f(x), and set the remainder at the end equal to 0. Which brings us to...

2. Long Division
You'll need to be able to do long division with polynomials for use with the factor theorem. It helps to be familiar with regular long division to do this, of course.

3. Stuff about quadratic equations.
A quadratic equation is one that looks like this:
[latex]ax^2 + bx + c = 0[/latex].
There are extra details about the sum and product of roots of a quadratic equation to get to grips with.


I hope that answers your question, since you asked what f(x) means, and specifically about part (c)'s. If you meant something different, or if you want to know more about a specific topic, or have some specific question you have problems with, don't hesitate to ask.

**Timbuk2**

State a simple theorem that works for all n (a natural number) except for 7, 17 and 256.

Any ideas? I couldn't get this for ages, until I looked it up had a moment of inspiration when I realised I knew the answer all along :P

**Timbuk2**

Sorry for the double post

A simple statement that holds true for any n except for 7, 17 and 256 can be as simple as.

[latex](n-7)*(n-17)*(n-256) \neq 0[/latex]

I felt a bit cheated though when I looked up the answer figured it out.

Here's another thing I'm wondering about.
Is there such thing as a 'test' for a prime number. Surely their must be, but I don't know of any. I was thinking this when I was wondering is 341 prime (it isn't, as 11x31 = 341)

The only possible thing I could think of is by using Wilson's theorem, which states that (p-1)! is congruent to -1 (mod p) - but that is obviously flawed as (p-1)! gets very large, very quickly.

What about Fermat's Theorem? n^(p-1) is congruent to 1 modulo p. I'm not sure this can be used a test for p being prime though.

Actually, is there any apparent reason why one number is prime and another isn't? Like if you go through the prime numbers, is there a pattern (eventually) so you can predict what the next prime will be?

**Timbuk2**

**Timbuk2** wrote: »

Sorry for the double post

A simple statement that holds true for any n except for 7, 17 and 256 can be as simple as.

[latex](n-7)*(n-17)*(n-256) \neq 0[/latex]

I felt a bit cheated though when I looked up the answer figured it out.

Here's another thing I'm wondering about.
Is there such thing as a 'test' for a prime number. Surely their must be, but I don't know of any. I was thinking this when I was wondering is 341 prime (it isn't, as 11x31 = 341)

The only possible thing I could think of is by using Wilson's theorem, which states that (p-1)! is congruent to -1 (mod p) - but that is obviously flawed as (p-1)! gets very large, very quickly.

What about Fermat's Theorem? n^(p-1) is congruent to 1 modulo p. I'm not sure this can be used a test for p being prime though.

Actually, is there any apparent reason why one number is prime and another isn't? Like if you go through the prime numbers, is there a pattern (eventually) so you can predict what the next prime will be?

On a computer, you can simply check all of the numbers less than p to see if they divide p. You can speed this up by realising that you only need to check up to [latex]\sqrt p[/latex], because if it is composite, one of the factors is less than [latex]\sqrt p[/latex].
You can speed this up a bit more. The Wikipedia page for Primality tests has more efficient methods, some of which are probabilistic, some deterministic.
I don't think there are any simple methods which are significantly quicker than checking divisibility though.

As for the others, the best guess mathematicians have comes in two parts:

The Prime Number Theorem, which tells us the density of the primes, i.e. the proportion of numbers less than n which are prime, as an asymptotic limit, and

The Riemann Conjecture, which is one of the most fascinating unproven results in mathematics, which makes a statement about the roots of a complex analytical function, which is related to primes.
Proving the Riemann Conjecture is seen as the highest goal in pure mathematics, and it gives us a good understanding of how the primes are distributed, or at least as good as we can currently get.
It's one of the Millenium Prize problems for which there is a million dollar reward offered.

Richard Cranium

There are plenty of theories/theorems about primes in number theory, I don't know if you're doing any of it this year.

It's been a while since I've done any, but off the top of my head, there is always a prime between n and 2n, and the number of primes less than n is less than log n, or something. I think it was Euler who proved that, in the process of proving that there are an infinite amount of primes.

EDIT: Actually, I was thinking of Gauss proposing that [latex]\pi (n) \approx \frac{n}{log(n)}[/latex]
More details here, I really should have put a small bit more effort into that original post.

Also, interestingly, there can be a (finite) gap as large as you like between any two primes.

**Timbuk2**

Wow how interesting! I saw a book all about prime numbers in the library the other day, and I was going to get it - but it looked ridiculously complicated - I'm only an empty-brained firsty :P I completely forgot about the computer-approach - that makes a lot more sense than what I was trying to say

Pygmalion

**Timbuk2** wrote: »

Sorry for the double post

A simple statement that holds true for any n except for 7, 17 and 256 can be as simple as.

[latex](n-7)*(n-17)*(n-256) \neq 0[/latex]

I felt a bit cheated though when I looked up the answer figured it out.

Here's another thing I'm wondering about.
Is there such thing as a 'test' for a prime number. Surely their must be, but I don't know of any. I was thinking this when I was wondering is 341 prime (it isn't, as 11x31 = 341)

The only possible thing I could think of is by using Wilson's theorem, which states that (p-1)! is congruent to -1 (mod p) - but that is obviously flawed as (p-1)! gets very large, very quickly.

What about Fermat's Theorem? n^(p-1) is congruent to 1 modulo p. I'm not sure this can be used a test for p being prime though.

Actually, is there any apparent reason why one number is prime and another isn't? Like if you go through the prime numbers, is there a pattern (eventually) so you can predict what the next prime will be?

Re: prime number testing and stuff.

If there were a fast way to factorise large numbers (which, by definition would also be a fast way to test primality) most of the encryption we rely on at the moment would immediately become useless, as a large part of the difficulty of breaking it is due to it being infeasible to factor large semi-primes.

Cows Go µ

**Timbuk2** wrote: »

Wow how interesting! I saw a book all about prime numbers in the library the other day, and I was going to get it - but it looked ridiculously complicated - I'm only an empty-brained firsty :P I completely forgot about the computer-approach - that makes a lot more sense than what I was trying to say

You should read The Music of the Primes by du Sautoy, (or at least I think that was his name) its not too hard to understand and it gives a fairly extensive back ground to number theory involving primes. Its easily one of my favourite popular science books.

I'll second The Music of the Primes. Du Sautoy is a great author. He's actually a really nice guy; I've met him.

You should also check out Dr. Riemann's Zeroes by Karl Sabbagh. It builds up to explaining the Riemman Hypothesis but goes through the history of our thought about primes.

KnifeWRENCH

I'm trying to decide what Masters I want to apply for at the end of the year. I wanna leave Cork but don't think I'm prepared to leave Ireland yet. At the moment I have two on my radar:

Medical Physics at Trinity, or
Nano BioScience at UCD.

I haven't yet ruled out Mathematics or Meteorology, though I'm not sure I have enough interest in Maths to keep me going for a year long Masters. And Meteorology, interesting as it is, is a pretty inexact science and has quite restrictive career options.

Ugh, decisions, decisions.....

**Timbuk2**

*cough*Actuarial Science Postgraduate in UCD*cough*
[/unhelpful answer]

Medical Physics sounds fascinating though! Ask yourself what you would have liked to do had you not been in your current course. It's a tough decision!

Pygmalion

KnifeWRENCH wrote: »

I'm trying to decide what Masters I want to apply for at the end of the year. I wanna leave Cork but don't think I'm prepared to leave Ireland yet. At the moment I have two on my radar:

Medical Physics at Trinity, or
Nano BioScience at UCD.

I haven't yet ruled out Mathematics or Meteorology, though I'm not sure I have enough interest in Maths to keep me going for a year long Masters. And Meteorology, interesting as it is, is a pretty inexact science and has quite restrictive career options.

Ugh, decisions, decisions.....

This.
A no-brainer really.

KnifeWRENCH

**Timbuk2** wrote: »

*cough*Actuarial Science Postgraduate in UCD*cough*
[/unhelpful answer]

Hmmm Arcturial Science, eh? Would a normal Maths degree (or in my case, half a normal Maths degree) be enough to do that, or would you need a financial maths degree? I've done Measure Theory and Martingales this year and am doing Stochastic Modelling after Christmas, so would have some financial maths background. Not sure an actuary would suit me though; the money's great but it sounds very dull (no offence! :P)

Medical Physics sounds fascinating though! Ask yourself what you would have liked to do had you not been in your current course. It's a tough decision!

I've asked myself that question every time I've gotten pissed off with my course and thought about dropping out. Nothing ever came to mind, hence my decision to stick with the course. My relationship with Maths and Physics is a love-hate one; I find the subjects fascinating, I just wish I was better at them. :mad: I reckon I'm more right-brained than left-brained tbh; I'd probably be better suited to an Arts degree or something. I was always better at "wordy" things than "numbery" things, I just happened to prefer "numbery" things to "wordy" things. (As the preceeding sentence demonstrates, I am quite the lyrical wordsmith! :pac:)

Going back to Masters courses, I've had medical physics in mind since about 1st year. The UCD Nanoscience course was one I only discovered recently.

Tbh I'd prefer to go to Trinity than UCD, just for the location (also UCD is too big; it scares me! :pac:) But right now I'm leaning ever-so-slightly towards Nano Bioscience; seems a bit more varied than medical physics. Ultimately, the right course is more important than the location.

But I'm pretty liable to be indecisive; this time next week, I may have my heart set on a Masters in basket weaving and martial arts in Azerbaijan.*

*note: course may not exist.

**Timbuk2**

KnifeWRENCH wrote: »

Hmmm Arcturial Science, eh? Would a normal Maths degree (or in my case, half a normal Maths degree) be enough to do that, or would you need a financial maths degree? I've done Measure Theory and Martingales this year and am doing Stochastic Modelling after Christmas, so would have some financial maths background. Not sure an actuary would suit me though; the money's great but it sounds very dull (no offence! :P)

None taken

I should have posted a link
http://www.ucd.ie/statdept/aactstat/actuarialweb/postgraduate.htm

Entry Requirements wrote:

Applicants will normally be expected to have a very good foundation in mathematics and/or statistics, and should have a 2.1 honours degree in a quantitative area such as mathematics, statistics, computer science, engineering, or economics and/or finance.

KnifeWRENCH

Pygmalion wrote: »

This.
A no-brainer really.

I like cyrptography. But computer science is icky. I don't care how my internet-box works, all I care about is that it does work.

**Timbuk2** wrote:

Applicants will normally be expected to have a very good foundation in mathematics and/or statistics, and should have a 2.1 honours degree in a quantitative area such as mathematics, statistics, computer science, engineering, or economics and/or finance.

Cheers for that. I'd be lucky to get a 2.1 tbh - fecking 3rd year results went against me, having missed too much college to catch up properly. Still gonna aim for it though.

Medical Physics sounds very interesting, but the Nano Bioscience might be a little more general.

If you didn't want to go the actuarial road, there are other jobs in finance. Being a trader or a quant could be pretty cool.

jumpguy

Medical physics is meant to be fairly ace. I remember at the radiography day in the Mid-West Regional in Limerick the radiographer was going on about how she was *kinda* jealous of the medical physicists. Any idea what's the best path to take into that?

KnifeWRENCH

I can't decide my future. For the past few months, I've been trying to decide whether to go into Medical Physics or Nanotechnology....now I'm considering abandoning physics and pursuing postgrad study in maths instead. UCD seem to have a really good taught masters program., but I'm not sure I really want a career in maths....of course, I'm not sure I want a career in physics either.

Blergh, why can't someone just tell me what to do with my life? I hate making my own decisions....

EDIT: Moved from The Den, so as not to disrupt the flow of new 'albums' that everyone is making! :pac:

Pigwidgeon

KnifeWRENCH wrote: »

I can't decide my future. For the past few months, I've been trying to decide whether to go into Medical Physics or Nanotechnology....now I'm considering abandoning physics and pursuing postgrad study in maths instead. UCD seem to have a really good taught masters program., but I'm not sure I really want a career in maths....of course, I'm not sure I want a career in physics either.

Blergh, why can't someone just tell me what to do with my life? I hate making my own decisions....

I know the feeling. It's no fair.

On the same note, I plan on talking properly to both my parents about dropping out of college and starting again next year. I can't handle another year and a half of this. But I can't do it without them because I'll need help money-wise, obviously I'll pay as much as I can but still. They don't seem to get that the longer I stay the more I'll have to pay when I go back which I will be no matter what. Bleugh.

Carl Sagan

KnifeWRENCH wrote: »

I can't decide my future. For the past few months, I've been trying to decide whether to go into Medical Physics or Nanotechnology....now I'm considering abandoning physics and pursuing postgrad study in maths instead. UCD seem to have a really good taught masters program., but I'm not sure I really want a career in maths....of course, I'm not sure I want a career in physics either.

Blergh, why can't someone just tell me what to do with my life? I hate making my own decisions....

Well just remember you can always change what you're doing. It'll be awkward but you it's possible at any time.

Duffy the Vampire Slayer

I just discovered Hark, a Vagrant a webcomic based on history and literature. I love it

**Timbuk2**

Has anyone used Minitab before? We will be starting Stats labs in it next week. Up to now, I have been using R, which is great. But Minitab looks more complicated, and isn't free (like R is)


What's more is I couldn't get it working through Networked Applications on the UCD computers, so I have never even used it.

sock puppet

**Timbuk2** wrote: »

Has anyone used Minitab before? We will be starting Stats labs in it next week. Up to now, I have been using R, which is great. But Minitab looks more complicated, and isn't free (like R is)


What's more is I couldn't get it working through Networked Applications on the UCD computers, so I have never even used it.

Yeah I have. Well by used I mean I've entered the data in and stared blankly at my regression output for the other 45 minutes of the lab. I've never used R though so don't know how it compares. It's actually easy enough to use.

**Timbuk2**

In one of my stat modules, one of the questions was based on regression outputs. In the labs, we fitted a regression model (using the least-squares method) using R, but in the exam, it would show an output from various different statistical software - I found them ok to read, but the whole language is probably very different in Minitab, than in R.

You should download R - it's free, easy to use, and quite good - you can make extremely nice graphs with it :P

sock puppet

**Timbuk2** wrote: »

In one of my stat modules, one of the questions was based on regression outputs. In the labs, we fitted a regression model (using the least-squares method) using R, but in the exam, it would show an output from various different statistical software - I found them ok to read, but the whole language is probably very different in Minitab, than in R.

You should download R - it's free, easy to use, and quite good - you can make extremely nice graphs with it :P

Eh no thanks. I've had quite enough for one year. Yeah in our exams outputs are always from different packages. They all look pretty much the same anyway. They'll all have the information you need there.

Also we never typed in the commands ourselves with Minitab. We entered data onto a spreadsheet and then did whatever we had to do from the menu options. R's like programming isn't it?

**Timbuk2**

Oh right, that's a bit different then. R is loosely like programming, as in you enter commands. You import data, for example, and then use functions (which take parameters) to do things with this data - plot it, summarise it, etc.

It'll be interesting learning another package. The labs are designed to be introductory in nature so I don't think a prior knowledge of Minitab is required.

R is programming, but most people only use built-in functions and packages which is why it only seems "like" programming. You can do pretty much anything in it that you can in other languages.

Minitab should be grand Tim. It's a bit more awkward in my opinion. I don't think you access Minitab through Networked Applications these days. There's another similar thing on the UCD computers, but I can't remember how to get to it off the top of my head. SAS is in there anyway, so I assume that's where Minitab is.

EuropeanSon

Just stumbled across this article. A wonderful piece of writing, it beautifully explains precisely what is wrong with the manner in which maths is taught and perceived in our society.
http://www.maa.org/devlin/LockhartsLament.pdf

Insect Overlord

EuropeanSon wrote: »

Just stumbled across this article. A wonderful piece of writing, it beautifully explains precisely what is wrong with the manner in which maths is taught and perceived in our society.
http://www.maa.org/devlin/LockhartsLament.pdf

That's brilliant. Wish I'd read it five and a half years ago! I might have had more of an inclination towards Honours Maths in the Leaving Cert if I'd thought about it differently.

Lawliet

That article was pretty cool, particularly this bit:

We are losing so many potentially gifted mathematicians—
creative, intelligent people who rightly reject what appears to be a
meaningless and sterile subject. They are simply too smart to waste
their time on such piffle.

I now feel smug about my incompetence

...Although on second thoughts I'd probably be awful no matter how well it was taught.