tag:blogger.com,1999:blog-2433841880619171855.comments2020-08-10T22:33:48.786+01:00Pat'sBlogUnknownnoreply@blogger.comBlogger1183125tag:blogger.com,1999:blog-2433841880619171855.post-38555935176598426882020-08-10T22:33:48.786+01:002020-08-10T22:33:48.786+01:00You've got me curious. What are the other univ...You've got me curious. What are the other universal symbols?Denise Gaskinshttps://www.blogger.com/profile/11928843626113889088noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-23594061534246682222020-07-31T06:32:40.683+01:002020-07-31T06:32:40.683+01:00Delighted that I found your site, fantastic info. ...Delighted that I found your site, fantastic info. I will bookmark and try to visit more frequently.<br /><br /><i><b><a href="https://www.amazon.com/Flat-Earth-Map-Stationary-Ferguson/dp/B01GSRWAWO" rel="nofollow">Flat Earth Map</a></b></i>Humaun Kabirhttps://www.blogger.com/profile/07115372749293027201noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-9036311176093421562020-07-29T23:15:47.323+01:002020-07-29T23:15:47.323+01:00This solution is obviously incorrect for two reaso...This solution is obviously incorrect for two reasons. First, it expresses the area of the segment of a circle in terms of rational operations and the square root of given integral measures, rather than as a transcendental number. Second, the height of the region in question is about 1.156, the difference of the two computed heights, so the area must be less than 10 times that, not more than 30 times that height.<br />The area of the larger sector can be computed (using radian mode) as <br />6^2*arcsin(5/6) - 5*SQRT(11) = 18.881, and for the smaller segment,<br />6^2*arcsin(4/6) - 4*SQRT(20) = 8.382, which is quite close to what is shown in the solution.<br />The difference, about 10.499, is slightly less than 10 times the height, as we would expect.Fred Gloffhttps://www.blogger.com/profile/08649019237954824008noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-11717227207808893872020-07-28T07:26:55.550+01:002020-07-28T07:26:55.550+01:00Thank you for taking the time to read and respond ...Thank you for taking the time to read and respond to my comment, Pat. Knowing where you’re coming from, I do regret my tone and should not have assumed that you were a historian of mathematics. <br /><br />Under my assumption, I didn’t understand why your website did not cover mathematical contributions of Eastern and other societies, given how monumental and influential they were to Western societies. A simple Google on “Chinese mathematical contributions” opens a labyrinth of interesting facts and domino effects to Western theories. <br /><br />Of course, America also opened her door and lifted the Chinese during their stale growth, elevating Eastern mathematical advancements to this day. <br /><br />I understand now that you were not intentionally snubbing other culture’s mathematical contributions. Thank you for your clarification. <br /><br />Good talk. Vamphttps://www.blogger.com/profile/15219986004852127996noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-84906925782720709822020-07-25T20:02:03.950+01:002020-07-25T20:02:03.950+01:00I was so annoyed at the distorted appearance of th...I was so annoyed at the distorted appearance of the table that I decided to try it another way, that should be much more readable. I also changed the 0.0 to just 0, because unlike the other 9 values in the last two columns, it is not the result of rounding off.<br />0 -----> 23.5 --------> 0<br />1 -----> 64.5 ------> 64.5<br />2 -----> 88.5 -----> 177.1<br />3 -----> 80.9 -----> 242.8<br />4 -----> 55.4 -----> 221.6<br />But there is more (Isn’t there always?). Now that we have an easy way of getting the mean number of matched birthday groups (call it M), we can use that value and a normal approximation to the discrete distribution to get a handle on the range of frequencies to expect. Using SQRT(M) as an overestimate of the standard deviation, compute M-2*SQRT(M) and M+2*SQRT(M) and round inward to the next integer to create a +/-2 standard deviation interval, enclosing about 95.5% of the cases. Using this rule of thumb for the case of quadruple birthdates, with M=55.4, the two endpoints are 40.5 and 70.3 (nearest tenth is enough), so about 96% of the time the number of quadruple birthdates would fall in the 41 to 70 interval. Because we have overestimated the SD (more pronounced with large values of M), a shorter interval (43 to 68) is actually sufficient, but for simplicity, I don’t think this is a big concern. <br />When M is small (say below 7) the normal approximation does not work quite as well, so we need a little adjustment. Let’s use M=5.3 as an example. The computed endpoints are at 0.7 and 9.9, but the interval from 1 to 9 is not quite enough. Because the 9.9 is so close to 10 (within 0.2), we need to include that frequency as well. This is not an issue for the left-hand endpoint. <br />Don’t use this method on the expected frequencies in the third column. For the case of the number of people that are in a group of 4 matching birthdates (avg. = 221.6), take the interval we previously found and multiply it by 4, getting a 164 to 280 interval. Can you explain why?<br />FredFred Gloffhttps://www.blogger.com/profile/08649019237954824008noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-78604484880617867282020-07-24T20:07:06.870+01:002020-07-24T20:07:06.870+01:00My recent comment should have included the followi...My recent comment should have included the following:<br />But we are not done yet. By using the binomial distribution with parameters 1000 and 1/365, we can extend the exploration in my first paragraph to obtain the entire frequency distribution for the single, double, triple, etc. birthdates. The table below includes the first five rows, rounded as shown. The first column gives number of matched birthdates, the second gives the mean number of times a matched group of that size would occur, and the third gives the mean number of people that would fit into that group size.<br />0 23.5 0.0<br />1 64.5 64.5<br />2 88.5 177.1<br />3 80.9 242.8<br />4 55.4 221.6<br />Notice that the most frequent group type is the matched pairs, with matched triples close behind. On the other hand, individuals are most likely to belong to a matched triple, with matched quadruples next in line. I would encourage interested readers to look at a larger portion of the table, using either a spreadsheet or a calculator with good list-handling capabilities. One feature to observe is the sums in the 2nd and 3rd columns. The 2nd column adds to 365 (no surprise since it is 365 times a probability distribution), and the 3rd totals 1000—confirmation that every possible birthdate is counted exactly once, and every person is accounted for exactly once. I first noticed this many years ago when dealing with an analogous situation, using MUCH larger numbers, and still consider it one of many great examples of the beauty of mathematics.<br />Fred<br />P.S.: This system will not permit the spacing between columns that I have input. I hope you can interpret the table in spite of this.Fred Gloffhttps://www.blogger.com/profile/08649019237954824008noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-16094698752896835452020-07-24T05:16:53.715+01:002020-07-24T05:16:53.715+01:00So close and yet so far! You were genuinely on the...So close and yet so far! You were genuinely on the right track in the paragraph beginning with “I reasoned that the probability . . .”, but you skipped one step before using the Poisson distribution. You had correctly computed the probability of a particular birthdate not occurring among the 1000 people. But that applies to each of 365 birthdates. So we should multiply that probability by 365, thus finding that the expected value of the number of birthdates NOT occurring is 23.486…. Notice that we have used an exact technique to obtain this result. If this expected frequency had turned out to be merely 4 or 5, the probability of no missed birthdates would be “pretty low.” When it is larger than 23, that probability must be extremely small. If you want to get a handle on how small, now is the time to reach for the Poisson distribution, or since we are talking about getting zero occurrences, just compute e^(-23.486), getting about 6E-11 (Now we are approximating). Notice that carrying out these computations on a basic scientific calculator would take little more than 10 seconds. <br />But we are not done yet. Even if one is not familiar with the Poisson distribution, it should seem apparent that the probability distribution for the number of missed birthdays would be skewed to the right (many more options above 23.5 than below), so the mode would be smaller than 23.5. In the Poisson distribution, it is right at 23, with a probability of about 8.3%. So, as Jon Ingram found by the matrix method, the most likely outcome is 342 distinct birthdays. How do our probabilities compare with his? The first is high (but the discrepancy is hardly meaningful) and the second is low. For those approximations, I actually prefer to use a binomial distribution. With “birthday problems”, using an n-value about half the maximum number of possibilities (364/2 here) seems to work fairly well. With n=182, the probabilities come out around 1E-11 and 8.8%, clearly closer to Jon’s results. Sometimes being willing to accept approximate answers can produce useful results while saving a tremendous amount of effort in both reasoning and computation.<br />In the previous day’s post, Pat showed that the average group size needed to cover all 365 possible birthdates is about 2154. Using that in place of 1000 produces an expected frequency of .990 (pretty close to 1, which is typical when dealing with waiting time random variables) and a probability of about 37% (not far from e^-1). With a group size of 2284, we get an expected frequency of 0.693 (near ln(1/2)), with a probability very close to 50%. Thus the median waiting time is approximately 2284.<br />FredFred Gloffhttps://www.blogger.com/profile/08649019237954824008noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-18921911847658447882020-07-23T23:00:33.026+01:002020-07-23T23:00:33.026+01:00Yes, this response is delayed, but I have just rec...Yes, this response is delayed, but I have just recently started looking at some of Pat’s earliest blogs and came across this one. Regarding the question raised in paragraph 4, we can use the formula<br />P(exactly one matched pair) = 365*C(n,2)*P(364,n-2)/365^n, where n is the group size. For n=23, it is about 36.4%, in good agreement with the simulation results. For the case of “either two separate matches, or three of a kind”, I get about 12.3%, but the method is more involved.<br />1) My best idea is to use the formula P(no match) =P(D,n)/D^n, where D is the number of options (pseudo-days) and find the smallest n that produces a value <= 0.5. The table of values feature or tracing on a graph are two quick ways to implement this on many common calculators. For D=400, n=24, and for D=52 (assuming 52 equally likely weeks, with no 365th day), n=9. But a warning: on some devices (such as the TI-83), trying it with 1000 produces an overflow error. In this case we call fall back on the traditional recursive approach.<br />4) Using either the formula in #1 above or the recursive approach, the maximum probability (about 3.23%) occurs for the 20th person. But all positions from 17 to 23 have probabilities within 0.1% of the max, so the exact position is not really that big a deal.Fred Gloffhttps://www.blogger.com/profile/08649019237954824008noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-51498473906755181622020-07-22T16:33:12.885+01:002020-07-22T16:33:12.885+01:00This is one amazing piece of article. Helped a lot...This is one amazing piece of article. 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Perhaps as outsiders they "saw more of the game".Korhommehttps://www.blogger.com/profile/02290764661952746389noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-17795823730779631002020-07-04T10:03:46.679+01:002020-07-04T10:03:46.679+01:00I found this article which is related to my intere...I found this article which is related to my interest. The way you covered the knowledge about the subject and the <a href="https://www.samglobaluniversity.ac.in/" rel="nofollow"> best university in bhopal<br /> </a> was worth to read, it undoubtedly cleared my vision and thoughts towards B <a href="https://www.samglobaluniversity.ac.in/" rel="nofollow"> about best private university in bhopal </a>. Your writing skills and the way you portrayed the examples are very impressive. The knowledge about <a href="https://www.samglobaluniversity.ac.in//" rel="nofollow"> universities in bhopal </a> is well covered. Thank you for putting this highly informative article on the internet which is clearing the vision about Top Private University in Bhopal and who are making an impact in the Education sector by building such amazing campuses.<br />vinuhttps://www.blogger.com/profile/15443106238177530710noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-79517187274750870872020-07-04T10:03:13.683+01:002020-07-04T10:03:13.683+01:00I found this article which is related to my intere...I found this article which is related to my interest. The way you covered the knowledge about the subject and the <a href="https://www.samglobaluniversity.ac.in/" rel="nofollow"> best university in bhopal<br /> </a> was worth to read, it undoubtedly cleared my vision and thoughts towards B <a href="https://www.samglobaluniversity.ac.in/" rel="nofollow"> about best private university in bhopal </a>. Your writing skills and the way you portrayed the examples are very impressive. The knowledge about <a href="https://www.samglobaluniversity.ac.in//" rel="nofollow"> universities in bhopal </a> is well covered. Thank you for putting this highly informative article on the internet which is clearing the vision about Top Private University in Bhopal and who are making an impact in the Education sector by building such amazing campuses.<br />vinuhttps://www.blogger.com/profile/15443106238177530710noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-90744375451133975872020-07-04T05:53:29.633+01:002020-07-04T05:53:29.633+01:00Hi pat
You may not know but jarret's symbol is...Hi pat<br />You may not know but jarret's symbol is still used to this day in Arabic mathematics the L for factorial<br />We never use the exclamation mark(!) Abdelrahmanhttps://www.blogger.com/profile/07877098074485317120noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-9503843068976428152020-07-03T14:27:23.913+01:002020-07-03T14:27:23.913+01:00† ††
O ^.^
|< ° ° ^.^
/\ ...† <b>†</b>†<br /><br />O ^.^<br /> |< ° ° ^.^<br />/\ ° ° <o <o<br /><a href="http://matthewrevelation.blogspot.com" rel="nofollow"><br />Matthew 13 [4] And when he sowed, some seeds fell by the way side, and the fowls came and devoured them up:<br /><br />Revelation 13 [5] And there was given unto him a mouth speaking great things and blasphemies; and power was given unto him to continue forty and two months.<br /></a><br /> >>>>>>>>>>>>>>>>>>>>>>>>>>><br /><a href="http://matthew-revelation.blogspot.com" rel="nofollow"> <b> The end is near !</b></a><br /><<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<br /><br /><a href="http://billionairesonparade.blogspot.com/2018/10/2018.html" rel="nofollow">Research your favorite <i><b>Billionaires !</b></i></a><br /><br /> <a href="mailto:scratchwiththechickens@gmail.com?subject=possible%20benefactor&body=Are%20you%20still%20available%3F%21" rel="nofollow">scratchwiththechickens@gmail.com <br><br>DON'T DO IT </a><br /><br /> !<br />Unknown Soldierhttps://www.blogger.com/profile/00796210982427639682noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-28414148876849488062020-07-03T14:23:05.063+01:002020-07-03T14:23:05.063+01:00† ††
O ^.^
|< ° ° ^.^
/\ ...† <b>†</b>†<br /><br />O ^.^<br /> |< ° ° ^.^<br />/\ ° ° <o <o<br /><a href="http://matthewrevelation.blogspot.com" rel="nofollow"><br />Matthew 13 [4] And when he sowed, some seeds fell by the way side, and the fowls came and devoured them up:<br /><br />Revelation 13 [5] And there was given unto him a mouth speaking great things and blasphemies; and power was given unto him to continue forty and two months.<br /></a><br /> >>>>>>>>>>>>>>>>>>>>>>>>>>><br /><a href="http://matthew-revelation.blogspot.com" rel="nofollow"> <b> The end is near !</b></a><br /><<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<br /><br /><a href="http://billionairesonparade.blogspot.com/2018/10/2018.html" rel="nofollow">Research your favorite <i><b>Billionaires !</b></i></a><br /><br /> <a href="mailto:scratchwiththechickens@gmail.com?subject=possible%20benefactor&body=Are%20you%20still%20available%3F%21" rel="nofollow">scratchwiththechickens@gmail.com <br><br>DON'T DO IT </a><br /><br /> !<br />Unknown Soldierhttps://www.blogger.com/profile/00796210982427639682noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-31330042440450803002020-06-23T13:39:28.560+01:002020-06-23T13:39:28.560+01:00Vamp, yes,I disagree. I just reread the Wikipedia...Vamp, yes,I disagree. I just reread the Wikipedia article and it says"The abacus, also called a counting frame, is a calculating tool that was in use in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the written Hindu–Persian numeral system. The exact origin of the abacus is still unknown. ". <br />I might also point out that my note is not a history of the abacus, but of its use in American Education. Nor do I, or have I, ever claimed to be an "expert". I'm a retired math teacher with an interest in the history of math and math education.<br />Thank you for sharing your comments, and opinion. Sorry you felt you had to be abusive to try to teach me something. Most of what I share here came from comments and information shared by far more learned folks than I, and I've tried to acknowledge each of them. You may know lots, but nothing you wrote seemed to enhance my edu ation about either math,or math educationPat's Bloghttps://www.blogger.com/profile/15234744401613958081noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-31610975203012979832020-06-18T09:21:03.490+01:002020-06-18T09:21:03.490+01:00OMG!!! Did you really blog about the abacus witho...OMG!!! Did you really blog about the abacus without mentioning the Chinese? White-washing history much? Sad that Wikipedia is more historically accurate than an “expert.” <br /><br />The Romans and Greeks didn’t invent the abacus, they used boards and they were not the first, even. There’s a difference between beads rolling along a board versus wired beads (which is what an abacus truly is and it’s powerful because it’s much faster than marks/beads on boards). The Russian’s version came later than the Chinese and think where they are geographically and take a gander at how they were influenced. <br /><br />It’s okay if you disagree, but at least your readers should know your biased Supremacy. <br /><br />I’m sure you’ll delete my comment, but that only proves you’re a racist. Vamphttps://www.blogger.com/profile/13209312356692660473noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-82713078306745023032020-06-10T10:17:28.572+01:002020-06-10T10:17:28.572+01:00It is very nice blog so that every one can underst...It is very nice blog so that every one can understand the importance of abacus.<br />Abacus are becoming widely popular as the demand for brain development takes a plunge upward. Abacus is not just a tool for calculation anymore; it has become an instrument which offers numerous cognitive benefits. UCMAS provides Abacus math program for children aged 4-13 years where child can develop some amazing abacus math skills so that they can make the most out of it in their live.<br /><br /><br />https://www.ucmas-usa.com/piyuhttps://www.blogger.com/profile/08339946239944156724noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-34547634635840149322020-05-13T17:47:50.338+01:002020-05-13T17:47:50.338+01:00D. Crompt,
Don't worry about being a little b...D. Crompt, <br />Don't worry about being a little behind, I'm a couple of years behind, and I write this stuff. To paraphrase Einstein, "Don't worry about your problems with punctuality, I assure you mine are much worse." <br />Pat BPat's Bloghttps://www.blogger.com/profile/15234744401613958081noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-34186281528556393762020-05-09T23:00:50.277+01:002020-05-09T23:00:50.277+01:00That was interesting as s***!! I know im 7 yrs beh...That was interesting as s***!! I know im 7 yrs behind on this post but thanks for thatD. Crompthttps://www.blogger.com/profile/12494678401805028051noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-1333330813793560842020-04-25T06:35:15.787+01:002020-04-25T06:35:15.787+01:00So, conclusion drawn is that, if i i ha an AP: a1,...So, conclusion drawn is that, if i i ha an AP: a1, a2, ... an, and I want to find the sum of (a1)²+(a2)²+...+(an)²=(a1)(an)(n)+[d²(n-1)(n)(2n-1)]/6. Is this it?Starhttps://www.blogger.com/profile/04072109681135551303noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-35752560057025662332020-04-21T14:37:26.669+01:002020-04-21T14:37:26.669+01:00Wow!!!, you have done so much research on division...Wow!!!, you have done so much research on division. Hats off to u sir. Anonymoushttps://www.blogger.com/profile/16979433984600941655noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-47980548720150723752020-04-21T13:28:20.634+01:002020-04-21T13:28:20.634+01:00Patrick, Thank you, I corrected it. I knew his nam...Patrick, Thank you, I corrected it. I knew his name and have written other things about the pair of them. Not sure if I mistyped what I got from Dr Rickey or didnt notice it in his. Glad you enjoy it, and keep the corrections comming. I seem to have the ability to read past a mispelling or grammer error a dozen types and everything looks great. Pat's Bloghttps://www.blogger.com/profile/15234744401613958081noreply@blogger.com