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see that mr. Donald has only made conjectures about the nature of the problem, not about the solution. You all were just wasting my time, because it's clear that you all have no idea about how to solve this problem. You were pointing obviousnesses such as "discussing continuity includes the possibility that there is discontinuity." or and trying to figure out what I was asking like here "When x+y = 0, the value of (x+y)/(x+y) is undefined. Therefore it is not a limit, and it is unclear what it means to ask for its limit (if that is the question of he first part)." (then sin(x)/x when x-: -->
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blue curve is the purple curve translated by (-5/3,-50/27), and that the blue curve is the purple curve rotated 180 degrees about the point (1/2,-2). Similar points apply to B, and it's possible he was trying to show something in connection with that. I agree that C appears to be pointing out both the growth and the decline of a cubic on scaling, and would like to add that D appears to show that a curve that starts out below can end up above, for the same reason. I agree with CM about E, and with
Capuchin regarding his artistic skill.
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times I can find that you have stated, suggested or intimated before that you are interested in taking the limit along the straight line y=-x is exactly 0 (zero, zilch, niente, nada), which is a bit less than one thousand. But here goes. On that straight line (at least according to what you wrote, namely "I have a function which is ... 0 when x+y=0"), the function value is
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Since the original asker has abandoned the question, I'll go ahead and ask my related question, only for the sake of my curiosity. What do you call the concept of finding the limit points of a multidimensional function as the function of points on a curve or a set of curves defined parametrically on
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I guess calling people lazy, stupid and wasting MY time with your incompetence goes against WP:CIVIL and that assume good faith thing, I guess. It's not a homework question, you
Einstein, since I rarely have homework on holidays. If you could read, because at this point I'm really doubting so, you'll
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It's hard to say for sure what he was going for, but he does seem in general to be displaying characteristics of various cubics. In A, the two curves are very symmetric to each other. The cryptic remark about "two, one" is unhelpful, but it is the case that they intersect at (1,0) and (0,2), that the
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0 isn't a limit either by your ""logic""), which was pretty clear from the beginning, if you have an idea of the topic, of course. Next time, try to answer to questions that, at the very least, you can understand, thank you. It's not that hard to understand that function has problems when x+y=0, and
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at the origin. You did not state what the points of interest are, but it is so obvious that the function is not continuous at, for example, (1,–1), that the idea did not cross my mind that this might be an issue. For the origin, I have supplied a solution. Maybe I have miscounted, but the number of
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I have two questions. One's on the talk page of that article, if anyone wants to have a shot at it. The other is, where online can I find out more about them? I searched Google of course, but almost every single site that mentioned them had a verbatim copy of our current article instead of original
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Well, okay, a multidimensional limit normally has to cover approach from all directions. However, you could take the limit along a particular plane, like x=0 or y=0 or x=y or something like that; it seems this would be a compact way to describe it. "Limit as (x,y) approaches (0,0) for f(x,y) where
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Lets take 1,-1 then. evaluating points like g(1.001,-0.999)=1000.001 and g(0.999,-1.001)=-1000.001 gives evidence that all along y=-x, (except x=y=0) the function is divergent or asymptotic (if either term makes sense for functions on planes), which means that setting g(x,y) = 0 for y=-x does not
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Lambian, I was just speculating about the problem that was offered (although my surmise was correct about the missing part, at least), I didn't actually do any of the math to see what happens with that condition. Taraborn, discussing continuity includes the possibility that there is discontinuity.
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A principal feature in the multidimensional case is the additional freedom in how we approach a point. In the one-dimensional case, we can only approach from above or below, though these can already give different answers. In your plane example, we can approach along many different paths. What we
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When x+y = 0, the value of (x+y)/(x+y) is undefined. Therefore it is not a limit, and it is unclear what it means to ask for its limit (if that is the question of he first part). The second part is more clear, but should properly be stated as follows: "What is the limit of (x+y)/(x+y) as (x,y) →
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Taraborn, mathematicians like it when the statements and notation are nice and tidy. You have used notation which, to my knowledge, is highly nonstandard (if at all sensible), which is the source of much conufsion. What I understand is that you ask not what happens when you move along the line
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It did not help that you managed to completely mangle up the question as you initially posed it. If you study my reply above to Donald Hosek, you'll find it contains the answer. But the issue is: DO YOUR OWN HOMEWORK. If you are too lazy for that, do not waste our time by posting your homework
375:. If the limit of f(x,y) as (x,y) → (0,0) exists, the value of f(x,y) should be everywhere defined in a neighbourhood of (0,0) with the possible exception of the point (0,0) itself. However, arbitrarily close to (0,0) there are points (x,y) such that x+y = 0, so there is a little problem . --
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Okay, this means I got the problem right. My apologies to everyone that might have been offended by my tone, especially
Lambiam. Sorry for not making the problem clear enough, I thought it definitely was but, as Meni Rosenfeld pointed out, I'm not a mathematician and I can't speak "in your
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operation where you take the derivatives with respect to x and y separately in order to find the components of a vector representing that function's change (though again, if the function isn't actually continuous from all directions, the gradient doesn't really exist). -
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710:... was that an insult? I'm not an idiot, and I'm quite aware of that. By saying "calculating a limit" I considered either giving its value or proving its nonexistence. I don't think the problem is that hard to understand, seriously. I tried doing the limit when x-: -->
892:. Usually, constant functions are rather continuous, and this one is no exception. For a continuous function, to find a limit, you can just take the function value. In this case, it is 0. I have used some algebra, in particular the fact that y = –x implies x+y = 0. --
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is a constant. So if every term of the sum works out to the same thing, you could collect all the terms together to remove the sigma. More generally, perhaps the sum can be simplified to remove the explicit sum (even if the terms are not all constant).
1262:? Does this concept have a name? I know once you've defined the curve parametrically, it essentially becomes the problem of finding the limit of a function of a single variable.... but it still seems interesting and noteworthy enough to have its own
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I guess the 2n means that the fact it's even is significant, but I can't really see how. The factorials in the answer given seem to suggest the binomial coefficients come in to play there, but, again, I can't quite see how I can get an expression for
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x=y"? Now, I don't really see the mathematical significance of a limit that is the same from both axis-aligned planes (i.e. x=0 and y=0), so I can't say that I think there's a particular name for it. It does, however, remind me of the
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Errrrr... Well, I have a function which is defined this way: (x^2+y^2)/(x+y) when x+y!=0 and 0 when x+y=0. I am asked to discuss the continuity of the function, and, for this purpose, I'll have to calculate a limit, isn't it?
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Question: 1 (C) A, B and C decides to open an account of RM 10000 for three years without withdrawal. A keeps the amount at an interest rate of 2.5% per annum for a duration of 1 month renewable at the end of each month.
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993:. As Lambiam has also remarked, the function is clearly discontinuous at those points. Finally, it's not nice to lash out at people only because you have not succeeded in describing the problem correctly. --
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I don't see a spot where inserting an equals sign makes the first part more plausible, but perhaps you are thinking of changing x+y=0 to x=y=0. With the same mod applied to my answer, it would equally stand.
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It seems that image C is illustrating the effects of scaling a polynomial by a constant factor. Any ideas about what the others represent, and whether they could be of any use if uploaded to the
Commons?
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I've repeated this a thousand times, so I don't think one thousand and one will make any difference. Your idea will prove the nonexistence of the limit at the origin, not along the straight line y=-x. --
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Without regard to dimension, we can take a continuous function and replace the value at one point with something arbitary. Thus we should not assume that a value and a limit coincide.
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1026:(re-tab and reply to everybody). Ok. Maybe I'm misreading this, but it seems both Taraborn and Lambiam are getting snippy and making assumptions. I call a small time-out.
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that should maybe be converted to SVG and moved to the
Commons (perhaps in the opposite order), if only we can figure out what they are meant to illustrate. One such image (
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approaching it. I wrote above: "Take L to be any value other than zero, and let (x,y) approach zero along the parabolic curve y = –x + 2x/L." This settles the question. --
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using sigma notation, but since there's no sigma in the RHS of the equation in the question, I'm a bit puzzled as to how to proceed. Thanks for any help you can give.
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You need to find the amount in the account at the end of 3 years. You can carry on working it out month by month for 36 months - or you can look for a shortcut.
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I have completed the question this far: 102.5/100*10000=10250 But I do not understand the phrase "for a duration of 1 month renewable at the end of each month".
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I don't have any reason to think you're more of an idiot than the rest of us; I also don't know what you are aware of. So forgive me if I state familiar facts.
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Then integrate from 0 to π. The cos parts of the integral are all 0 (can you see why ?) and you are just left with 3π/8. Now generalise this to any value of
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You wrote: "I am asked to discuss the continuity of the function". Obviously, the function is not continuous at those points (x,y) for which x+y = 0, except
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The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the
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studying what happens when x+y APPROACHES 0 isn't that hard to understand either. I guess you need a course in basic vector calculus urgently. --
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635:. I think that's sufficient for the limit to be defined, but it's all idle speculation without the original poster's clarifying the problem.
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I think the point of the question is that the interest is paid and compounded monthly. At the end of one month the amount in the account is
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Of course the function f(x,y) = (x^2 + y^2)/(x+y) is discontinuous at x=-y. It seems the function
Taraborn is asking about actually is.
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a bit, will you? If you're not a native
English speaker you may perhaps not realize how rude and offensive your use of the interjection
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I'm thinking that there's an equals sign missing from the first part of the problem. That's the only way the problem might make sense.
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711:(-y) (remember the condition x+y=0) and got infinity, but I'm by no means sure of that, that's why I'm asking for help. Thanks. --
149:-intercepts of a cubic polynomial, is already on the Commons, but the other ones, shown below, are somewhat more of a mystery.
1174:{\displaystyle g(x,y)={\begin{cases}{\frac {x^{2}+y^{2}}{x+y}},&{\mbox{if }}y\neq -x\\0,&{\mbox{if }}y=-x\end{cases}}}
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One possible reason (I haven't worked it out) that there might be no "sigma" in the right side is the following: recall that
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Is that so? Take L to be any value other than zero, and let (x,y) approach zero along the parabolic curve y = –x + 2x/L. --
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This sum now earns interest during the second month, so the amount in the account at the end of the second month is
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However, I'm not sure how to proceed with the expansion in the case of a variable, 2n, rather that a constant.
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This is set in the context of de Moivre's theorem, and preceding questions have taken forms such as "Express
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Hi, all. I've got a bit stuck on an exercise in my A2 further maths book and I'm hoping someone can help.
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The likely point of the exercise is for you to discover and explore these facts in a specific case. --
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Can somebody help me with this limit? (x^2+y^2)/(x+y) when x+y=0 and (x^2+y^2)/(x+y) when (x,y)-: -->
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I moved the tabs over for these next two messages as they are not related to my above post.
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the x,y plane that either cross or "ends up" at the point, like say y=t^3,x=t or the
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Taraborn, discussing continuity includes the possibility that there is discontinuity.
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Another important difference can be illustrated with polynomials. A polynomial in
360:(0,0). Is the second case already analyzed with the first one (x+y=0)? Thanks. --
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glue any surfaces together. The key is proving this with absolute certainty.
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What is the name of this formula by
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807:. I think I'm wasting my time here, well, forget about my question. --
371:(0,0)?" I assume that you know the "multidimensional" definition of
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E is the shifting of (x-1)x(x+1) through the transformation x -: -->
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0, or more accurately, when we are approaching a point on the line
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Lambiam provided a solution to problem point at the origin. OK.
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is used in the question - what do you get if you integrate an
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I was thinking of doing something along the lines of defining
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Welcome to the
Knowledge Mathematics Reference Desk Archives
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336:... and at the end of month 3 the amount in the account is
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language" :). Thank you for your help and your time. --
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hope for is the limit does not depend on the approach.
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110:Ugly bag of mostly water
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1584:
1582:
1581:
1576:
1572:
1567:
1566:
1565:
1547:
1546:
1533:
1517:
1516:
1500:
1498:
1497:
1492:
1484:
1483:
1465:
1463:
1462:
1457:
1455:
1453:
1452:
1451:
1433:
1432:
1419:
1399:
1382:
1381:
1368:
1363:
1345:
1343:
1342:
1337:
1329:
1328:
1239:
1196:
1180:
1178:
1177:
1172:
1170:
1169:
1151:
1147:
1120:
1116:
1108:
1106:
1095:
1094:
1093:
1081:
1080:
1070:
992:
990:
989:
984:
960:
958:
957:
952:
934:
932:
931:
926:
897:
862:
822:
665:
634:
632:
631:
626:
609:
601:
600:
588:
587:
553:
507:
505:
504:
499:
475:
473:
472:
467:
415:
380:
315:
296:
200:
198:
186:
184:
172:
170:
158:
75:
38:Mathematics desk
34:
2535:
2534:
2530:
2529:
2528:
2526:
2525:
2524:
2419:
2410:
2402:
2401:
2349:
2330:
2329:
2323:
2288:
2269:
2268:
2262:
2261:
2257:
2241:
2232:
2224:
2223:
2189:
2167:
2139:
2117:
2098:
2089:
2081:
2080:
2036:
2020:
2019:
2013:
2012:
1993:
1988:
1987:
1916:
1915:
1876:
1848:
1837:
1836:
1783:
1767:
1766:
1760:
1759:
1754:
1753:
1723:
1718:
1717:
1709:in the answer.
1682:
1681:
1641:
1622:
1621:
1615:
1595:
1594:
1551:
1535:
1534:
1528:
1527:
1508:
1503:
1502:
1475:
1470:
1469:
1443:
1421:
1420:
1400:
1370:
1350:
1349:
1317:
1312:
1311:
1302:
1191:Yes, it is. --
1165:
1164:
1143:
1134:
1133:
1112:
1096:
1085:
1072:
1071:
1062:
1035:
1034:
966:
965:
937:
936:
908:
907:
742:alone, such as
592:
579:
510:
509:
478:
477:
437:
436:
357:
297:
286:
271:
204:
201:
196:
190:
187:
182:
176:
173:
168:
162:
159:
135:
120:Euler's formula
106:
104:name of formula
101:
30:
29:
28:
12:
11:
5:
2533:
2531:
2523:
2522:
2501:
2500:
2499:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2437:
2433:
2426:
2422:
2418:
2413:
2409:
2399:
2387:
2383:
2380:
2376:
2371:
2365:
2362:
2359:
2356:
2352:
2348:
2343:
2340:
2337:
2333:
2326:
2322:
2319:
2315:
2310:
2304:
2301:
2298:
2295:
2291:
2287:
2282:
2279:
2276:
2272:
2265:
2260:
2255:
2248:
2244:
2240:
2235:
2231:
2221:
2210:
2205:
2202:
2199:
2196:
2192:
2188:
2183:
2180:
2177:
2174:
2170:
2166:
2163:
2160:
2157:
2152:
2149:
2146:
2142:
2138:
2135:
2130:
2127:
2124:
2120:
2116:
2112:
2105:
2101:
2097:
2092:
2088:
2078:
2065:
2060:
2055:
2049:
2046:
2043:
2039:
2035:
2030:
2027:
2023:
2016:
2011:
2008:
2005:
2000:
1996:
1977:
1976:
1953:
1950:
1947:
1944:
1939:
1934:
1931:
1928:
1924:
1912:
1894:
1891:
1888:
1883:
1879:
1875:
1872:
1869:
1866:
1863:
1860:
1855:
1851:
1847:
1844:
1835:Remember that
1812:
1807:
1802:
1796:
1793:
1790:
1786:
1782:
1777:
1774:
1770:
1763:
1741:
1738:
1733:
1730:
1726:
1698:
1695:
1692:
1689:
1668:
1663:
1657:
1654:
1651:
1648:
1644:
1640:
1635:
1632:
1629:
1625:
1618:
1614:
1611:
1608:
1605:
1602:
1580:
1575:
1570:
1564:
1561:
1558:
1554:
1550:
1545:
1542:
1538:
1531:
1526:
1523:
1520:
1515:
1511:
1490:
1487:
1482:
1478:
1450:
1446:
1442:
1439:
1436:
1431:
1428:
1424:
1418:
1415:
1412:
1409:
1406:
1403:
1397:
1394:
1391:
1388:
1385:
1380:
1377:
1373:
1367:
1362:
1358:
1335:
1332:
1327:
1324:
1320:
1310:By expressing
1301:
1298:
1297:
1296:
1237:
1236:
1210:
1209:
1205:
1204:
1203:
1202:
1181:
1168:
1163:
1160:
1157:
1154:
1144:
1142:
1139:
1136:
1135:
1132:
1129:
1126:
1123:
1113:
1111:
1105:
1102:
1099:
1092:
1088:
1084:
1079:
1075:
1068:
1067:
1065:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1031:
1024:
1023:
1022:
1021:
1020:
1019:
1018:
1017:
1016:
1015:
1014:
1013:
1012:
1011:
1010:
1009:
1008:
1007:
1006:
1005:
995:Meni Rosenfeld
982:
979:
976:
973:
950:
947:
944:
924:
921:
918:
915:
795:
794:
793:
792:
791:
790:
781:
780:
779:
736:
732:
721:
720:
719:
718:
702:
701:
680:
679:
678:
677:
676:
675:
674:
673:
672:
671:
649:
648:
647:
646:
645:
644:
643:
642:
624:
621:
618:
615:
612:
608:
604:
599:
595:
591:
586:
582:
578:
575:
572:
569:
566:
563:
560:
557:
552:
549:
546:
543:
540:
537:
534:
531:
528:
525:
522:
518:
497:
494:
491:
488:
485:
465:
462:
459:
456:
453:
450:
447:
444:
426:
425:
424:
423:
422:
421:
402:
401:
400:
399:
387:
386:
356:
353:
352:
351:
342:
341:
340:
334:
333:
332:
326:
325:
324:
285:
282:
270:
268:Bring Radicals
265:
264:
263:
262:
261:
260:
259:
244:
243:
242:
241:
206:
205:
202:
193:
191:
188:
179:
177:
174:
165:
163:
160:
153:
134:
131:
130:
129:
105:
102:
100:
97:
95:
91:
90:
82:
81:
71:
70:
64:
48:
41:
40:
31:
15:
14:
13:
10:
9:
6:
4:
3:
2:
2532:
2521:
2518:
2514:
2510:
2506:
2502:
2483:
2480:
2474:
2471:
2465:
2462:
2459:
2456:
2450:
2447:
2441:
2438:
2431:
2424:
2420:
2416:
2411:
2407:
2400:
2385:
2381:
2378:
2374:
2369:
2363:
2360:
2357:
2354:
2350:
2346:
2341:
2338:
2335:
2331:
2324:
2320:
2317:
2313:
2308:
2302:
2299:
2296:
2293:
2289:
2285:
2280:
2277:
2274:
2270:
2263:
2258:
2253:
2246:
2242:
2238:
2233:
2229:
2222:
2203:
2200:
2197:
2194:
2190:
2186:
2181:
2178:
2175:
2172:
2168:
2164:
2161:
2158:
2155:
2150:
2147:
2144:
2140:
2136:
2133:
2128:
2125:
2122:
2118:
2110:
2103:
2099:
2095:
2090:
2086:
2079:
2063:
2058:
2053:
2047:
2044:
2041:
2037:
2033:
2028:
2025:
2021:
2014:
2009:
2006:
2003:
1998:
1994:
1986:
1985:
1983:
1979:
1978:
1975:
1972:
1967:
1951:
1948:
1945:
1942:
1937:
1932:
1929:
1926:
1922:
1913:
1911:
1908:
1889:
1881:
1877:
1873:
1870:
1867:
1861:
1853:
1849:
1845:
1842:
1834:
1833:
1832:
1831:
1828:
1827:Seth Bresnett
1810:
1805:
1800:
1794:
1791:
1788:
1784:
1780:
1775:
1772:
1768:
1761:
1739:
1736:
1731:
1728:
1724:
1713:
1710:
1696:
1693:
1690:
1687:
1666:
1661:
1655:
1652:
1649:
1646:
1642:
1638:
1633:
1630:
1627:
1623:
1616:
1612:
1609:
1606:
1603:
1600:
1578:
1573:
1568:
1562:
1559:
1556:
1552:
1548:
1543:
1540:
1536:
1529:
1524:
1521:
1518:
1513:
1509:
1488:
1485:
1480:
1476:
1466:
1448:
1440:
1437:
1429:
1426:
1422:
1416:
1413:
1407:
1404:
1395:
1392:
1389:
1386:
1383:
1378:
1375:
1371:
1365:
1360:
1356:
1347:
1333:
1330:
1325:
1322:
1318:
1308:
1305:
1299:
1295:
1292:
1287:
1282:
1281:
1280:
1279:
1275:
1271:
1270:
1265:
1261:
1260:Golden spiral
1255:
1253:
1249:
1245:
1244:
1235:
1232:
1227:
1226:
1225:
1224:
1220:
1216:
1215:
1206:
1201:
1198:
1194:
1190:
1189:
1188:
1187:
1185:
1161:
1158:
1155:
1152:
1140:
1137:
1130:
1127:
1124:
1121:
1109:
1103:
1100:
1097:
1090:
1086:
1082:
1077:
1073:
1063:
1058:
1052:
1049:
1046:
1040:
1032:
1029:
1028:
1027:
1004:
1000:
996:
980:
977:
974:
971:
963:
948:
945:
942:
922:
919:
916:
913:
904:
903:
902:
899:
895:
891:
887:
882:
878:
877:
876:
873:
869:
868:
867:
864:
860:
856:
852:
848:
844:
841:
840:
839:
836:
829:
828:
827:
824:
820:
815:
814:
813:
810:
806:
803:
802:
801:
800:
799:
798:
797:
796:
789:
786:
782:
777:
773:
769:
765:
761:
757:
753:
749:
745:
741:
737:
733:
730:
729:
727:
726:
725:
724:
723:
722:
717:
714:
709:
706:
705:
704:
703:
700:
697:
692:
691:
690:
689:
686:
670:
667:
663:
659:
658:
657:
656:
655:
654:
653:
652:
651:
650:
641:
638:
619:
616:
613:
606:
597:
593:
589:
584:
580:
573:
567:
564:
561:
555:
547:
544:
541:
529:
526:
523:
508:then finding
495:
492:
489:
486:
483:
463:
460:
454:
451:
448:
442:
434:
433:
432:
431:
430:
429:
428:
427:
420:
417:
413:
408:
407:
406:
405:
404:
403:
398:
395:
391:
390:
389:
388:
385:
382:
378:
374:
369:
368:
367:
366:
363:
354:
350:
347:
343:
338:
337:
335:
330:
329:
327:
322:
321:
319:
318:
317:
313:
309:
305:
304:60.50.173.111
301:
293:
290:
283:
281:
280:
277:
269:
266:
258:
255:
250:
249:
248:
247:
246:
245:
240:
237:
233:
232:
231:
228:
223:
222:
221:
220:
216:
212:
199:
192:
185:
178:
171:
164:
157:
152:
150:
148:
144:
140:
132:
128:
125:
121:
117:
116:
115:
114:
111:
103:
98:
96:
88:
84:
83:
80:
77:
76:
68:
61:
57:
53:
47:
42:
39:
35:
27:
23:
19:
2512:
2508:
2504:
1981:
1965:
1907:Donald Hosek
1714:
1711:
1467:
1348:
1309:
1306:
1303:
1268:
1263:
1256:
1242:
1238:
1213:
1211:
1184:the function
1183:
1025:
961:
885:
880:
854:
850:
846:
804:
775:
771:
767:
763:
759:
755:
751:
747:
743:
739:
707:
696:Donald Hosek
681:
637:Donald Hosek
394:Donald Hosek
358:
294:
291:
287:
276:Black Carrot
272:
254:Black Carrot
207:
146:
136:
107:
94:
78:
1291:Rainwarrior
888:0. It is a
298:—Preceding
26:Mathematics
962:approaches
886:everywhere
754:, such as
2517:Gandalf61
843:Cool down
346:Gandalf61
274:content.
50:<<
1971:Tesseran
1286:gradient
1231:Taraborn
881:possibly
872:Taraborn
835:Taraborn
809:Taraborn
713:Taraborn
685:Taraborn
362:Taraborn
312:contribs
300:unsigned
236:Capuchin
24: |
22:Archives
20: |
1193:Lambiam
894:Lambiam
859:Lambiam
847:Goddamn
819:Lambiam
805:GODDAMN
662:Lambiam
412:Lambiam
377:Lambiam
139:Hamedog
89:pages.
99:July 6
67:July 7
46:July 5
785:KSmrq
764:ratio
476:when
211:Bkell
69:: -->
63:: -->
62:: -->
44:<
16:<
1984:=2:
1269:Root
1264:name
1243:Root
1214:Root
999:talk
849:is.
750:and
308:talk
215:talk
56:July
2513:odd
2463:cos
2439:cos
1995:cos
1964:if
1725:cos
1688:cos
1601:cos
1510:cos
1477:cos
1372:cos
1319:cos
1274:one
1248:one
1219:one
1148:if
1117:if
517:lim
410:--
60:Aug
52:Jun
2475:θ
2466:
2451:θ
2442:
2364:θ
2355:−
2342:θ
2303:θ
2294:−
2281:θ
2204:θ
2195:−
2182:θ
2173:−
2151:θ
2129:θ
2048:θ
2042:−
2029:θ
2007:θ
2004:
1923:∑
1874:∫
1871:∑
1846:∑
1843:∫
1795:θ
1789:−
1776:θ
1740:θ
1737:
1697:θ
1691:
1656:θ
1647:−
1634:θ
1610:θ
1604:
1563:θ
1557:−
1544:θ
1522:θ
1519:
1489:θ
1486:
1417:π
1393:θ
1387:θ
1384:
1366:π
1357:∫
1334:θ
1331:
1276:)
1250:)
1221:)
1159:−
1128:−
1125:≠
1001:)
978:−
920:−
851:If
770:/(
683:--
536:→
314:)
310:•
217:)
203:E
189:D
175:C
58:|
54:|
2509:n
2505:n
2487:)
2484:3
2481:+
2478:)
2472:2
2469:(
2460:4
2457:+
2454:)
2448:4
2445:(
2436:(
2432:)
2425:3
2421:2
2417:1
2412:(
2408:=
2386:)
2382:3
2379:+
2375:)
2370:2
2361:i
2358:2
2351:e
2347:+
2339:i
2336:2
2332:e
2325:(
2321:4
2318:+
2314:)
2309:2
2300:i
2297:4
2290:e
2286:+
2278:i
2275:4
2271:e
2264:(
2259:(
2254:)
2247:3
2243:2
2239:1
2234:(
2230:=
2209:)
2201:i
2198:4
2191:e
2187:+
2179:i
2176:2
2169:e
2165:4
2162:+
2159:6
2156:+
2148:i
2145:2
2141:e
2137:4
2134:+
2126:i
2123:4
2119:e
2115:(
2111:)
2104:4
2100:2
2096:1
2091:(
2087:=
2064:4
2059:)
2054:2
2045:i
2038:e
2034:+
2026:i
2022:e
2015:(
2010:=
1999:4
1982:n
1966:C
1952:C
1949:n
1946:=
1943:C
1938:n
1933:1
1930:=
1927:i
1893:)
1890:x
1887:(
1882:n
1878:f
1868:=
1865:)
1862:x
1859:(
1854:n
1850:f
1811:n
1806:)
1801:2
1792:i
1785:e
1781:+
1773:i
1769:e
1762:(
1732:n
1729:2
1694:2
1667:)
1662:2
1653:n
1650:i
1643:e
1639:+
1631:n
1628:i
1624:e
1617:(
1613:=
1607:n
1579:n
1574:)
1569:2
1560:i
1553:e
1549:+
1541:i
1537:e
1530:(
1525:=
1514:n
1481:4
1449:2
1445:)
1441:!
1438:n
1435:(
1430:n
1427:2
1423:2
1414:!
1411:)
1408:n
1405:2
1402:(
1396:=
1390:d
1379:n
1376:2
1361:0
1326:n
1323:2
1272:(
1246:(
1217:(
1162:x
1156:=
1153:y
1141:,
1138:0
1131:x
1122:y
1110:,
1104:y
1101:+
1098:x
1091:2
1087:y
1083:+
1078:2
1074:x
1064:{
1059:=
1056:)
1053:y
1050:,
1047:x
1044:(
1041:g
997:(
981:x
975:=
972:y
949:y
946:+
943:x
923:x
917:=
914:y
776:y
774:+
772:x
768:y
760:y
758:+
756:x
752:y
748:x
744:x
740:x
623:)
620:y
617:+
614:x
611:(
607:/
603:)
598:2
594:y
590:+
585:2
581:x
577:(
574:=
571:)
568:y
565:,
562:x
559:(
556:f
551:)
548:0
545:,
542:0
539:(
533:)
530:y
527:,
524:x
521:(
496:0
493:=
490:y
487:+
484:x
464:0
461:=
458:)
455:y
452:,
449:x
446:(
443:f
306:(
213:(
209:—
197:E
183:D
169:C
161:B
147:x
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