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Malmis
Sweden1569 Posts
On June 19 2005 14:02 BigBalls wrote: http://mathworld.wolfram.com/NaturalNumber.html america doesnt include it (or at least every teacher and professor ive ever had doesnt), belgium might Looked it up in a dictionary ( www.ne.se ). Apparently, 0 was not counted as a Natural number in the past. However nowadays counts as one. | ||
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Holorin
France227 Posts
big N with a doube vertical bar : Natural numbers 0,1,2,3,4,5,6,7,8,9... N* same without zero Big Z relative numbers ... -6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8 ... Z* same without zero , Z+ =N , Z- = Z\{N*} Anyway * means remove zero, + means keep only positive ones, - negatives ones. Big Q those which can be written as a/b (a,b) in ZxZ* , they have a simplified form p/q where q is in N*, p in Z and GCD(p,q) = 1 , grand common divider D decimal numers , one fuck of a useless thing, those who can be written as Sum(k=1..n, A(k)/10^k) n is finiteand 0<=A(k)<=9 and A(k) is in N R Real numbers ... they verify that any part X of R that verify : there is M such as for any x in X we have |X| < M and so there is a "inf" (borne inférieure in french) Also we have the included segments axioma : For any sequence of segments In such as In+1 is in In the intersection of all In as n->infinity is definied and non-null if the lenght of the segments tend to zero the limit is singleton C complex numbers : using i the square root of -1 we generate the maginay part of C i think it very beautifull to see C as surface. z is in C only if it can be written as z=a+ib with a and b in R. | ||
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Catyoul
France2377 Posts
On June 19 2005 20:06 Bill307 wrote: Question: On the planet Thassux, given two points {a, b} and {c, d}, what is the distance between them? Recall that on Earth, the distance between two points (a, b) and (c, d) is given by the square root of (a - b)^2 + (c - d)^2. You can leave your answer in terms of binomials like "(a - c)": it doesn't get any simpler if you expand them out. d² = (a-c)² + (b-d)² + sqrt(2) (a-c) (b-d) On June 19 2005 20:06 Bill307 wrote:Also, for people interested in more of a challenge, try going in the opposite direction: as you have seen, the formula on the planet Thassux is not particularly elegant. However, on the planet Mophux, the distance from a point {x, y} to the origin {0, 0} is given by the square root of x^2 + xy + y^2. Assuming that, like Thassux and Earth, the first coordinate is how many units you must move "to the right", describe how the coordinate system may work on Mophux. If you want, you can also come up with other solutions that ignore this assumption, which would be cool  .With the assumption, second coordinate is 1/2 unit to the right, sqrt(3)/2 up Without it, it's funnier, the family of solutions can be described like this : Let a and b be 2 arbitrary numbers with the following relation : a - b = +/- pi/3 + 2 k pi (with k an integer) Let u1 = cos a, u2 = sin a, v1 = cos b, v2 = sin b. All coordinate systems on Mophux are of the form : u1 to the right and u2 to the top for the first coordinate, v1 to the right and v2 to the top for the second coordinate. | ||
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lightman
United States731 Posts
On June 19 2005 23:27 loloko2 wrote: I totally agree with him, you cant argue that guys, imsorry, hes right... ^_^ K. First of all let's review all our history lessons. The name of our country is United States of AMERICA. Right? What the name of your country PaeZ and loloko2 ? It's Estados Unidos de Mexico or Estados Unidos Mexicanos, right ? When us americans (US citizens) refer to our country as America it's because the word America is within the name of our country right ? That is the reason why Bigballs says "America...." Same way, when you refer to your country, you do it as Mexico right ? That's why you say "Mexico ..." Don't be stupid and come up with all that Vespucio stuff. | ||
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inkblot
United States1250 Posts
South America is a continent. And on the natural numbers debate, I was taught that natural numbers are 0,1,2,3..... and counting numbers are 1,2,3,4.... | ||
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ieatkids5
United States4628 Posts
On June 20 2005 10:22 lightman wrote: Don't be stupid and come up with all that Vespucio stuff. He didn't come up with the "Vespucio stuff", it's true. | ||
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inkblot
United States1250 Posts
Vespucci was an explorer, a cartographer named the continent (at the time, it was not determined they were two continents) America after Vespucci's first name. Vespucci didn't name anything. The correct way to refer to both North and South America together is "the Americas." | ||
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Sorrow_eyes
United States1007 Posts
 math puzzle guys, not history  | ||
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lightman
United States731 Posts
My point is to the Argentinian and Mexican fellows that became pissed because BigBalls referred to the US they way we normally do. Comodity purpuses that's all. Want another example ?: "God Bless United States of America" turns into -> "God bless America". That is, us US Citizens call our country "America" because it's part of our name, United States of America, not because we think our country is the entire continent. You should agree with me, some people, we, even call it "the states". When we refer to USA as "America" or "the states", we are talking about our country, we're not referring to the whole world or anything like that. It's just a mere coincidence. Just as I stated earlier, the official name for Mexico is United States of Mexico, and mexicans refer to their country as "Mexico". Also for your information, there is another case in our continent: Venezuela's Official name (a southamerican country) for more than one hundred years used to be United States of Venezuela, until the mid 1950s if I am not mistaking, and I am pretty sure that they referred to their country as Venezuela. The correct way to refer to both North and South America together is "the Americas" or as plain and simple as "the continent America". Oh and returning to the previous argument 0 is not a natural number. Anyway, I think it's not like we are going to change the world with any of these arguments. | ||
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lightman
United States731 Posts
Let n be a natural number. A cube with an arist of lenght n can be divided into 1996 cubes which arists are also natural numbers. Find the minor possible value of n. | ||
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Holorin
France227 Posts
/ PI | | ln(a+b*cos(t)) dt / 0 a pure answer is usless, the method is interesting :p Another one (easy) let T be a regular tetrahedron (? four equilateral triangulars faces, right) and M inside T [interior to T] Proove that the sum of the distances from M to T's faces is a constant not depending of the position of the point M (all heavy(carthesians) methods are not welcomed | ||
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decafchicken
United States20060 Posts
2m,1m...(white ball) ...and red balls... 1m,5m... 2m,5m... 3m,5m 1m,6m... 2m,6m... 3m,6m 1m,7m... 2m,7m... 3m,7m The white ball is then shot at a random angle from 0 to 360 degrees. Just to make it clear, a ball is 'potted' if at least half of the ball is in area of the 'pocket' Assuming the balls travel indefinitely (i.e. no loss of energy via friction, air resistance or collisions), answer the following: a: What exact angle should you choose to ensure that all the balls are potted the quickest? b: What is the minimum amount of contacts the balls can make with each other before they are all knocked in? c: Same as b, except that each ball - just before it is knocked in - must not have hit the white ball on its previous contact (must be a red instead of course). d: What proportion of angles will leave the white ball the last on the table to be potted? The diagram for the probelm can be found here:http://www.skytopia.com/project/imath/imath.html#13 | ||
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Cambium
United States16368 Posts
![[image loading]](http://img213.echo.cx/img213/6766/8b3mf.png) In the diagram, two circles are tangent to each other at point B. A straight line is drawn through B cutting the two circles at A and C, as shown. Tangent lines are drawn to the circles at A and C. Prove that these two tangent lines are parallel. | ||
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LordOfDabu
United States394 Posts
On June 19 2005 23:24 Sorrow_eyes wrote: pm me w/ ur Email addr for a UMS map on it :D After sending my PM with my request, I was asked to save and host a replay of the map's demonstration. Those interested can find the replay here: http://s40.yousendit.com/d.aspx?id=2GE3ENU8Q0USI2H3OX6WF12I0G The demonstration is not perfect due to the rate at which Staredit goes through the triggers, but it is sufficient for grasping exactly what is going on. | ||
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chodingKIN
9 Posts
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Bill307
Canada9103 Posts
On June 20 2005 07:29 Catyoul wrote: d² = (a-c)² + (b-d)² + sqrt(2) (a-c) (b-d) Wow, I need to spend more time in this topic. The first time I worked out the answer to my own question I got it wrong  . So I guess the formula on the planet Thassux isn't so bad after all. And yes, that is the correct answer .On June 20 2005 07:29 Catyoul wrote: With the assumption, second coordinate is 1/2 unit to the right, sqrt(3)/2 up Without it, it's funnier, the family of solutions can be described like this : Let a and b be 2 arbitrary numbers with the following relation : a - b = +/- pi/3 + 2 k pi (with k an integer) Let u1 = cos a, u2 = sin a, v1 = cos b, v2 = sin b. All coordinate systems on Mophux are of the form : u1 to the right and u2 to the top for the first coordinate, v1 to the right and v2 to the top for the second coordinate. With the assumption you are correct: the second coordinate is angled at 60 degrees (pi/3) up from the x-axis. My friend actually worked this out in the "forwards" direction a few months ago, which is what gave me the idea for this question. (ok, I'm gonna move into some linear algebra terms now  )The last result is also interesting. I haven't actually tried to do it myself yet so I'll just take your word for it  . I guess it means that so long as the angle between the two basis vectors is 60 degrees (pi/3), the formula will come out like that.I wonder if the formula is always of the form a^2 + xab + b^2 ... ... ... and after working it out, I see that in general, the formula is: a^2 + b^2 + 2 (cos theta) ab = a^2 + b^2 + 2 a.b [dot product] dot product makes sense because a and b are really just lengths of vectors, and the angle between them is the angle between the two basis vectors And of course it doesn't matter what the actual directions of the basis vectors are, since the distance will be the same in any case . Therefore, this will come out to a^2 + b^2 + ab whenever the cosine of the angle between the vectors is 0.5 . Which is precisely at +/- pi/3 + 2 k pi, which is what you said. Therefore, your final answer is also correct .Last of all, this distance stuff reminds me of something I did in my final year of high school involving vectors, dot product, and distances. Wish I could remember what it was that I learned . | ||
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Catyoul
France2377 Posts
On June 20 2005 15:05 Cambium wrote: Here is a good one: In the diagram, two circles are tangent to each other at point B. A straight line is drawn through B cutting the two circles at A and C, as shown. Tangent lines are drawn to the circles at A and C. Prove that these two tangent lines are parallel. It's gonna become a habit of mine to answer puzzles after parties I guess :p Tangent lines run perpendicularly to the radius at the tangent point. Let's name the centers of the circles O and O'. The triangles OBA and O'BC are isosceles (and similar btw), the angles OBA and O'BC are the same and are also the same as OAB and O'CB because the triangles are isosceles. Thus the angles between the tangent lines and the AC line are the same, proving their parallelism. | ||
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Catyoul
France2377 Posts
On June 20 2005 13:03 lightman wrote: Back to our interesting and nice thread. Here's a hard one: Let n be a natural number. A cube with an arist of lenght n can be divided into 1996 cubes which arists are also natural numbers. Find the minor possible value of n. Hmmmmmmmm The smaller cubes all have an arist of at least 1, which also means a minimum volume of 1 per cube, thus a minimum volume of 1996 for our big cube. The smallest cube that can contain 1996 is with n=13, making a 2197 volume cube. We're still 201 too big if filled only with arist 1 cubes, so we'll replace some with bigger ones. If we replace a cube of arist 1 with a cube of arist 2, we increase the volume by 8-1=7 If we replace a cube of arist 1 with a cube of arist 3, we increase the volume by 27-1=26 If we replace a cube of arist 1 with a cube of arist 3, we increase the volume by 64-1=63 If we replace a cube of arist 1 with a cube of arist 4, we increase the volume by 125-1=124 With one cube of arist 4, one of arist 3 and two of arist 2, we've got our count of 201 and we've proved that n=13 is possible. Since it's also the smallest possible, it's the one we were looking for. | ||
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Holorin
France227 Posts
On June 20 2005 15:05 Cambium wrote: Here is a good one: In the diagram, two circles are tangent to each other at point B. A straight line is drawn through B cutting the two circles at A and C, as shown. Tangent lines are drawn to the circles at A and C. Prove that these two tangent lines are parallel. easyly : by drawing angles we get the answer : ![[image loading]](http://img260.echo.cx/img260/2630/easy5ab.jpg) | ||
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