Wikipedia:Reference desk/Archives/Mathematics/2008 November 13

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November 13

What is the purpose of a quadratic equation?

My Mathematics education ended after three years of (honors level) classes in high school. One of the "fun" things to do was to solve quadratic equations. For the rest of my life I have never seen or used one, though, and I never knew what they are for, when or why they are used. I recently read a book by Carl Sagan who described the series of learning steps that would be required before one could even understand the vocabulary of quantum physics. Is there a progression in math that was never taught to me, that would explain to me the point of leaning Quad equations, or the Calculus, other than for their own sake in school?Skinjohn (talk) 07:53, 13 November 2008 (UTC)

There are many real world situations where quadratic equations are used. probably the most common involves the relationship between distance speed and acceleration. If you are an engineer designing a production line in a factory, you will need to solve such equations to determine how to safely and quickly get product from A to B. There are millions of related examples. -- SGBailey (talk) 09:00, 13 November 2008 (UTC)

An ancient example from geometry: The area of a rectangle is 12 and the perimeter is 14, what are the sides? This problem leads to the two equations x·y = 12 and x+y+x+y = 14. From the second equation you get y = 7−x, which inserted into the first equation gives a quadratic equation x·(7−x) = 12. It is interesting, though, that your simple question is hard to answer. Solving quadratic equations is the mathematicians tool. A craftsmans tool is not appreciated by those who do not know the craft. You cannot explain a screwdriver to a person who do not know a screw. The progression of math, which you have never been taught, is the history of mathematics. The quadratic equation is from antiquity while calculus is from the renaissance. Bo Jacoby (talk) 10:31, 13 November 2008 (UTC).

The comparison with tools seems pretty apt to me. Certainly, one can get through life just fine without ever solving a quadratic equation (outside school, anyway), just as one can get by just fine without ever using a chisel or a hacksaw. Even so, on the off chance that you ever find yourself needing to solve a quadratic equation or carve a hole in a piece of wood, it's good to know that such tools exist and maybe even have some idea how to use them. Mind you, I'm a mathematics student myself, and I've never really got around to fully memorizing the quadratic formula. I probably should, but generally it's enough for me to know that there is a formula that I can look up (online or in a book) if and when I need it. I don't actually own a chisel, either — but if I need one, I know where to get one. —Ilmari Karonen (talk) 12:28, 13 November 2008 (UTC)

Look it up? It's quicker to just re-derive it. Algebraist 13:12, 13 November 2008 (UTC)

Or just complete the square - the quadratic formula is just a way to skip ahead to the final answer, there's nothing wrong with going through it step by step. --Tango (talk) 15:24, 13 November 2008 (UTC)

Quadratic equations have no real purpose in higher mathematics but can be useful for finding values relating to projectile motion or, in general, basic physics. It can also be useful in calculus. Topology Expert (talk) 07:24, 14 November 2008 (UTC)

I'm not sure I'd agree with that, I find quadratic equations come up all the time in various branches of mathematics. Just today I was doing some affine geometry which involved the characteristic polynomial of a 2x2 matrix - a quadratic equation. --Tango (talk) 19:54, 14 November 2008 (UTC)

I'd certainly disagree with that. The entire study of elliptic PDEs, especially generalized solutions to them, often involves comparisons with particular quadratic functions. RayAYang (talk) 04:54, 18 November 2008 (UTC)

Oldest known unsolved mathematical conjecture? (Possibly with legitimate references?)Taffycaker (talk) 13:12, 13 November 2008 (UTC)

Hello wiki volunteers, my question is: What is the oldest known unsolved mathematical conjecture?(Possibly with legitimate references?) I have Google searched this, and read through wikipedia and the internet which has many claims (ie goldbach, twin prime, etc) but rarely with the appropriate references so can verify their claims. One exception is for twin primes conjecture which seems to be located in Euclid, though searching for Euclid's references of it doesn't verify the claim, so could any of you please provide the proper answer since there's always some guy claiming to prove the oldest conjecture (ie Kepler's conjecture/Hales) but isn't really certain that it is the oldest, which is annoying. Thanks for your time. Taffycaker (talk) 13:12, 13 November 2008 (UTC)

Does an unproved assertion count as a conjecture? According to this, Nicomachus claims in his Introduction to Arithmetic that all perfect numbers are even, which would beat Goldbach and Kepler. Algebraist 13:44, 13 November 2008 (UTC)

The Guinness Book of Records (2001 edition) lists Goldbach's conjecture as the "longest standing unresolved maths problem". Hut 8.5 18:33, 13 November 2008 (UTC)

I don't know whether Euclid wrote down conjectures but I guess he (and other ancient Greeks) privately speculated about things closely related to his writings and tried to prove them, for example whether there are infinitely many Mersenne primes. He proved the infinitude of primes and proved that Mersenne primes generate perfect numbers, so it would appear unnatural to not at least think about infinitude of Mersenne primes and whether they generate all perfect numbers (it was later proven they generate all even perfect numbers and the existence of odd perfect numbers is unknown). PrimeHunter (talk) 18:56, 13 November 2008 (UTC)

pi

what is pi to the nearest 20 digits?Fern 24 15:58, 13 November 2008 (UTC)

Oh yeah, and in roman numerals to the nearest 5 digits?Fern 24 16:00, 13 November 2008 (UTC)

According to pi, π=3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510... Roman numerals don't seem to have involved a systematic way of writing fractions. Algebraist 16:04, 13 November 2008 (UTC)

Although this may seem a good question, it is not really much use to know what pi is. You do not need to know what pi is (correct to 20 d.p) to do mathematics and nor do you need to know it correct to 2 d.p. Perhaps something a little more non-trivial would be to show that there is no polynomial equation with rational co-efficients such that pi is a solution to that equation (try it!).

Topology Expert (talk) 07:19, 14 November 2008 (UTC)

But do not expect to succeed. Algebraist 11:53, 14 November 2008 (UTC)

As for the Roman numeral approximation, ³⁵⁵/₁₁₃, should do it, as that's good to 7 significant figures. In Roman numerals, that would be ^CCCLV/_CXIII. StuRat (talk) 19:17, 16 November 2008 (UTC)

Did the Romans write fractions like that? --Tango (talk) 20:22, 16 November 2008 (UTC)

In fact, we have a section on Roman numeral fractions, Roman numeral#Fractions. According to that, pi would be approximately III•ЄƧ»»»»», that is 1+1+1+1/12+1/24+1/72+1/1728+1/1728+1/1728+1/1728+1/1728≈3.1418. --Tango (talk) 20:32, 16 November 2008 (UTC)

Dominoes game strategy

There's a game I have on my iPhone called Dominosa and I can't think of a strategy to use to solve it besides random guessing. You start off with an grid of numbers (0-3, 0-6, 0-9 depending on difficulty) and you need to create a grid of unique dominoes. They'll turn red if you've got a duplicate piece already, and it will look like this. Any tips? -- MacAddct1984 ^{(talk • contribs)} 16:22, 13 November 2008 (UTC)

I don't know the game, but are you putting the numbers on the grid or are the numbers there and you are pairing them off? Assuming the latter, which is a pencil and paper game I have come across, I always start by (a) listing all the dominoes (see triangular numbers of 4, 7 and 10; (b) look for the unique doubles; (c) look for each other domino being unique; (d) Look for places that already placed dominos force the shape of another domino. (When you scan the grid for (say) domino 1-2, there may be three places it could go. Once another domino is placed, this scan may be affected and you have to redo it. In some puzzles, you have to say "1-2" can be here or there. If here then this and this follow if there then that and that follow. One of the two will probably lead to a dead end, so ideally you'd do this in your head to prevent redness. -- SGBailey (talk) 23:18, 13 November 2008 (UTC)

Go through the 28 patterns from (0,0) to (6,6) one at a time and see if each appears more than once on the grid - if so, pass on. Mark those that are unique on the grid and repeat this cycle until all have been found. With random positioning of the 28 dominoes it's unlikely that any other strategy is worthwhile, given that it's a fairly small problem anyway. That's for the 0-6 case, if I've understood you correctly, the others are similar. It's not random guessing, it's systematic search. In the empty grid you gave, (0,0) for example appears only once. Enclosing that pair immediately removes one (0,1), one (0,4), two (0,7)s and two (0,9)s from consideration…81.132.235.188 (talk) 23:41, 13 November 2008 (UTC)

And there's only one (8,8) pair (in the lower right corner), which immediately implies (2,5) to the right of it.
And there is only one place, where one end of (3,3) can be (it's the third digit from top and from left edge of the board), so — although we can't say in advance which 3 will be the other end — we know for sure the (3,0) pair is somwhere else. :) --CiaPan (talk) 11:17, 14 November 2008 (UTC)

Just for your convenience: in the version I've played, from Simon Tatham's Portable Puzzle Collection, you can add lines between numbers by right-clicking, indicating that you know these two can't form a domino together. This helps a lot with solving. Perhaps the iPhone's version has this option, too? Is it actually the same implementation, ported to the iPhone? Oliphaunt (talk) 17:36, 14 November 2008 (UTC)

Oh wow, that's awesome, I'm going to have to grab that for my computer. Yup, it's pretty much a direct port of that; Simon Tatham is [listed as one of the people] behind the project. And yup, you can draw lines splitting up possibilities. -- MacAddct1984 ^{(talk • contribs)} 19:04, 14 November 2008 (UTC)

I guess it really is simply a matter of looking for unique combinations and keep eliminating possibilities. It just seems like such a tedious way to go about it, especially for the largest grid size. -- MacAddct1984 ^{(talk • contribs)} 19:04, 14 November 2008 (UTC)

Curve Sketching Program

Does anyone know a good computer program for curve sketching (graphing). I use the description curve sketching because I am interested in a program that is smart enough to know the important features of a graph (e.g. intercepts, stationary points, critical points, etc.) and mark them on the graph rather than just generating a curve and putting unimportant numbers on the axes.AMorris (talk)●(contribs) 23:30, 13 November 2008 (UTC)

I suggest you either use a graphics calculator (hate those animals) or use NuCalc (maybe useful for calculus).

Topology Expert (talk) 07:21, 14 November 2008 (UTC)