tag:blogger.com,1999:blog-61578704650120384112024-03-14T01:04:46.772-07:00Adventures into the UnknownUnknownnoreply@blogger.comBlogger4125tag:blogger.com,1999:blog-6157870465012038411.post-5241842708116076252016-03-27T20:38:00.007-07:002016-05-09T15:56:48.816-07:00Numbers<h2>
<span style="color: #674ea7; font-family: "verdana" , sans-serif;">Figuring out Numbers</span></h2>
<span style="font-family: "verdana" , sans-serif;">So I was having a play, again, this time with a further look into composites and primes.</span><br />
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<span style="font-family: "verdana" , sans-serif;">I am curious to play with representations of digits in manners that allow visualization of distributions of prime numbers and non-prime numbers... not naively in any form or shape of faith in finding something someone else hasn't of course, just because it's fun, for me, to look, play, learn and figure things out by myself...</span><br />
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<span style="font-family: "verdana" , sans-serif;">So what did I get to - from this point on, it may well seem absolutely obvious, already known/common fact - but this is playful musings, explored with no math guides or prior knowledge applied, this is number theory basics a la Chris.</span><br />
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<span style="color: #674ea7; font-family: "verdana" , sans-serif;">'Types' of Numbers...</span></h3>
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<span style="font-family: "verdana" , sans-serif;">My number playing revolves around positive integers... of which a number is categorized in this exercise, as composite or prime.<br /><br />With this, a couple of terms are thrown around which you may be familiar with:</span><br />
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<a href="https://www.google.co.uk/webhp?hl=en#hl=en&q=what+is+a+factor+of+a+number" style="font-family: verdana, sans-serif;" target="_blank">Factor</a></h4>
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<span style="font-family: "verdana" , sans-serif; font-size: x-small;"><span style="background-color: white; color: #222222; line-height: 19.2px;">A </span><b style="background-color: white; color: #222222; line-height: 19.2px;">factor</b><span style="background-color: white; color: #222222; line-height: 19.2px;"> is a whole </span><b style="background-color: white; color: #222222; line-height: 19.2px;">number</b><span style="background-color: white; color: #222222; line-height: 19.2px;"> which divides exactly into a whole </span><b style="background-color: white; color: #222222; line-height: 19.2px;">number</b><span style="background-color: white; color: #222222; line-height: 19.2px;">, leaving no remainder. For example, 13 is a </span><b style="background-color: white; color: #222222; line-height: 19.2px;">factor</b><span style="background-color: white; color: #222222; line-height: 19.2px;"> of 52 because 13 divides exactly into 52 (52 ÷ 13 = 4 leaving no remainder). </span></span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-small;"><span style="background-color: white; color: #222222; line-height: 19.2px;">The complete list of </span><b style="background-color: white; color: #222222; line-height: 19.2px;">factors</b><span style="background-color: white; color: #222222; line-height: 19.2px;"> of 52 is: 1, 2, 4, 13, 26, and 52 (all these divide exactly into 52).</span></span><br />
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<a href="https://www.google.co.uk/webhp?hl=en#hl=en&q=what+is+a+prime+number" style="font-family: verdana, sans-serif;" target="_blank">Prime Number</a></h4>
<span style="font-family: "verdana" , sans-serif; font-size: x-small;"><span style="background-color: white; color: #222222; line-height: 19.2px;">A </span><b style="background-color: white; color: #222222; line-height: 19.2px;">prime number</b><span style="background-color: white; color: #222222; line-height: 19.2px;"> is a whole </span><b style="background-color: white; color: #222222; line-height: 19.2px;">number</b><span style="background-color: white; color: #222222; line-height: 19.2px;"> greater than 1, whose only two whole-</span><b style="background-color: white; color: #222222; line-height: 19.2px;">number</b><span style="background-color: white; color: #222222; line-height: 19.2px;"> factors are 1 and itself. </span></span></div>
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<span style="color: #222222; font-family: "verdana" , sans-serif;"><span style="line-height: 19.2px;">I particularly liked this visual example of what a prime number is:</span></span></div>
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<span style="font-family: "verdana" , sans-serif;"><a href="https://upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Prime_rectangles.svg/450px-Prime_rectangles.svg.png" target="_blank"><img border="0" height="270" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfIPX42rMp2l5nir-bT-f1Tvb9teDcKVlKzHIB1mMMEHaEn2PasbJ-p9HsP2hivlRX-7hW0lcPukY_hsAUwhS74D9RimXtC_RG4xxnurZIJm0Q33z2oIWU5rM59xHofx91ROc7jhKSYJs/s320/primes+as+shape+arrangement.png" width="320" /></a><span id="goog_947860871"></span><span id="goog_947860872"></span><a href="https://www.blogger.com/"></a></span></div>
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<span style="color: #222222; font-family: "verdana" , sans-serif; font-size: x-small;"><span style="line-height: 19.2px;"><a href="http://ponderering-things.blogspot.co.uk/2016/03/factors-expressed-geometrically.html" target="_blank">[Which in fact, led me to explore a bit further with this manner, which may aid in providing an additional level of comprehension]</a></span></span></div>
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<a href="https://www.google.co.uk/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=what%20is%20a%20composite%20number" style="font-family: verdana, sans-serif;" target="_blank">Composite Number</a></h4>
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<span style="font-family: "verdana" , sans-serif; font-size: x-small;"><span style="background-color: white; color: #222222; line-height: 19.2px;">A </span><b style="background-color: white; color: #222222; line-height: 19.2px;">composite number</b><span style="background-color: white; color: #222222; line-height: 19.2px;"> is a positive integer which is not prime (i.e., which has factors other than 1 and itself).</span></span></div>
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<span style="color: #674ea7; font-family: "verdana" , sans-serif;">Sourcing, Manipulation, Confirmation!</span></h3>
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<span style="font-family: "verdana" , sans-serif;">Manually formulating a table for all factors of a number seemed like an unnecessary task to proceed with, seeing as what I wanted to delve into being already somewhat of a large task. </span></div>
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<span style="font-family: "verdana" , sans-serif;">So I scoured the net for a table as such, <a href="https://en.wikipedia.org/wiki/Table_of_divisors" target="_blank">and this is what I found.</a> </span></div>
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<span style="font-family: "verdana" , sans-serif;">Which for a quick display of the content, it looks something like this:</span></div>
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<span style="font-family: "verdana" , sans-serif;"><a href="https://en.wikipedia.org/wiki/Table_of_divisors" target="_blank"><img border="0" height="196" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj41eo0bTHbf4PFt8cF0Na2FRdIIdtsoNoD6tHrJuedZQNrh-2h-tAAOJfrZSCSjh4U8yZIzVvS7XGQVlUmxXP31kmj_SBi8kZ3Z1LxdddsG49XWlfhKQWddIY3oWwSCuzrDerP1wEj44c/s640/table+of+divisors+example+1-9.JPG" width="640" /></a></span></div>
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<ul style="background-color: white; color: #252525; line-height: 22.4px; list-style-image: url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20version%3D%221.1%22%20width%3D%225%22%20height%3D%2213%22%3E%0A%3Ccircle%20cx%3D%222.5%22%20cy%3D%229.5%22%20r%3D%222.5%22%20fill%3D%22%2300528c%22%2F%3E%0A%3C%2Fsvg%3E%0A"); margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;"><span style="font-family: "verdana" , sans-serif; font-size: xx-small;"><a href="https://en.wikipedia.org/wiki/Divisor_function" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; text-decoration: none;" title="Divisor function"><i>d</i>(<i>n</i>)</a> is the number of positive divisors of <i><b>n</b></i>, including 1 and <i><b>n</b></i> itself</span></li>
<li style="margin-bottom: 0.1em;"><span style="font-family: "verdana" , sans-serif; font-size: xx-small;"><a href="https://en.wikipedia.org/wiki/Divisor_function" style="background: none; color: #0b0080; text-decoration: none;" title="Divisor function">σ(<i>n</i>)</a> is the sum of all the positive divisors of <i><b>n</b></i>, including 1 and <i><b>n</b></i> itself</span></li>
<li style="margin-bottom: 0.1em;"><span style="font-family: "verdana" , sans-serif; font-size: xx-small;"><a href="https://en.wikipedia.org/wiki/Divisor_function" style="background: none; color: #0b0080; text-decoration: none;" title="Divisor function"><i>s</i>(<i>n</i>)</a> is the sum of the <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Proper_divisor" style="background: none; color: #0b0080; text-decoration: none;" title="Proper divisor">proper divisors</a> of <i><b>n</b></i>, which does not include <i>n</i> itself; that is, <b><i>s</i>(<i>n</i>) = σ(<i>n</i>) − <i>n</i></b></span></li>
<li style="margin-bottom: 0.1em;"><span style="font-family: "verdana" , sans-serif; font-size: xx-small;">a <a href="https://en.wikipedia.org/wiki/Perfect_number" style="background: none; color: #0b0080; text-decoration: none;" title="Perfect number">perfect number</a> equals the sum of its proper divisors; that is, <b><i>s</i>(<i>n</i>) = <i>n</i></b></span></li>
<li style="margin-bottom: 0.1em;"><span style="font-family: "verdana" , sans-serif; font-size: xx-small;">a <a href="https://en.wikipedia.org/wiki/Deficient_number" style="background: none; color: #0b0080; text-decoration: none;" title="Deficient number">deficient number</a> is greater than the sum of its proper divisors; that is, <b><i>s</i>(<i>n</i>) < <i>n</i></b></span></li>
<li style="margin-bottom: 0.1em;"><span style="font-family: "verdana" , sans-serif; font-size: xx-small;">an <a href="https://en.wikipedia.org/wiki/Abundant_number" style="background: none; color: #0b0080; text-decoration: none;" title="Abundant number">abundant number</a> is less than the sum of its proper divisors; that is, <b><i>s</i>(<i>n</i>) > <i>n</i></b></span></li>
<li style="margin-bottom: 0.1em;"><span style="font-family: "verdana" , sans-serif; font-size: xx-small;">a <a href="https://en.wikipedia.org/wiki/Prime_number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; text-decoration: none;" title="Prime number">prime number</a> has only 1 and itself as divisors; that is, <b><i>d</i>(<i>n</i>) = 2.</b> Prime numbers are always deficient as <b><i>s</i>(<i>n</i>)=1</b></span></li>
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<span style="font-family: "verdana" , sans-serif;">Taking this data and loading it into Excel, allows me to manipulate and produce variations in data representation for the fun of spotting characteristics, similarities/groupings and/or patterns... but what it did lead to, actually saw me producing the results manually with the application of formula in Excel - so in fact, the table, in the end, was just a confirmation base for my calculations and output.</span></div>
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<span style="font-family: "verdana" , sans-serif;">The below shows the divisors for <i style="font-weight: bold;">n</i> from the source table; to the right, in the grid and with colors - is the output of a formula, that divides <i style="font-weight: bold;">n </i>by all whole integers presented in the top row increasing by 1 across column by column - but only is the product of the division displayed, if the value is a whole integer.</span></div>
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<span style="font-family: "verdana" , sans-serif;">The diagonal line is the boundary of this output as that is the indicator of where <i style="font-weight: bold;">n</i> is divided by <i style="font-weight: bold;">n</i>. <br />Any whole integer in that top row, as a divisor of <i style="font-weight: bold;">n</i>, will not be whole if it is <b><i>></i></b><i style="font-weight: bold;">n</i>, and so, will not display a populated number.</span></div>
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<span style="font-family: "verdana" , sans-serif;">The results of this simple formula, produce the same outputs as what I found on on Wikipedia - allowing me to produce my own counts and sums of divisors of positive whole integers and therefore adding additional columns to flag whether <i style="font-weight: bold;">n</i> is prime or not! <i>(<span style="font-size: x-small;">Prime is identified in this grid, as </span></i><i><span style="font-size: x-small;"><span style="font-weight: bold;">n </span>where the count of divisors = 2 & the sum of divisors = <b>n + 1</b>)</span></i></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgl4WSv9KnkLrAK4F5iuBI0Qr5DDRkRLvoE_BYlP1nfwXFgFsqpQbxDe73YG6nYukhgkmr_PI-ooV0RM0wOZ285BANTk-6NAcjYSwDQevTOM6L62AKQB-QofpBJyMRGxUhLnRkOOj8d8nA/s1600/table+of+divisors+example+1-27+chris+version.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: "verdana" , sans-serif;"><img border="0" height="530" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgl4WSv9KnkLrAK4F5iuBI0Qr5DDRkRLvoE_BYlP1nfwXFgFsqpQbxDe73YG6nYukhgkmr_PI-ooV0RM0wOZ285BANTk-6NAcjYSwDQevTOM6L62AKQB-QofpBJyMRGxUhLnRkOOj8d8nA/s640/table+of+divisors+example+1-27+chris+version.JPG" width="640" /></span></a></div>
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<span style="font-family: "verdana" , sans-serif;"><span style="background-color: white;"><span style="color: #222222;"><span style="line-height: 19.2px;">The coloring may already present some structural reference to the distribution of numbers as divisors of </span></span></span><i style="font-weight: bold;">n</i>.</span></div>
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<span style="color: #cc0000; font-family: "verdana" , sans-serif;">A Stopping Point... <i>for now...</i></span></h3>
<span style="color: #e06666; font-family: "verdana" , sans-serif;">There is more I would like to output on this what I have started to clean up from previous weeks play - however, as per usual, I digressed... and what I produced just now will form a separate post for reference, as I have branched back into an area I already have some background work on, but tidied a portion up for its potential supporting role in analysis of tentative primes... </span></div>
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<span style="color: #e06666; font-family: "verdana" , sans-serif;">Which is to say, the above work leads on to flagging of numbers, based on criteria such as last digit and then mod9 and square root check - to identify definite NON PRIMES; and potential primes... which I had a little curiosity to check whether permutations of base numbers <i>(i.e. 123, 132, 213, 231, 312, 321) </i>can occur where all permutations are prime; whether there is some significance of digit positioning with regards to permutations and prime elements and so on... </span></div>
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<span style="color: #e06666; font-family: "verdana" , sans-serif;">It got to well past 4am so... I want to wrap this part up and come back to it, as this what I posted today, is just the tip of the iceberg of what outputs I would like to share! That's just the way such analysis seems to go, from A to B to C through to Z and then sometimes coming back to A :D</span></div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6157870465012038411.post-59048339368325202382016-03-27T18:14:00.000-07:002016-03-27T18:14:39.913-07:00Factors Expressed Geometrically<span style="font-family: verdana, sans-serif;">This really was one of those side pieces that led of something bigger, as part of how I got to where I got to and somewhat of a guide for those of you interested to see such steps :)</span><br />
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<span style="font-family: "verdana" , sans-serif;">The basis of which, formed from visiting <a href="https://en.wikipedia.org/wiki/Prime_number" target="_blank">Wikipedia's article on Prime Numbers</a> and seeing this below image:</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqgxb4Xyrqhrs6Agy270GNw71-l2bOZjWSWlohdXWZ5vwa4Chkg4Sp7N6HesIObY-ev7LpdohTdFTHgpJBWYYV5dUCe4h7Ciczom4FJpyRU5-Tp6nwQFxXmCCLvyHDxo9AIOH98Rgmv9M/s1600/primes+as+shape+arrangement.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: "verdana" , sans-serif;"><img border="0" height="270" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqgxb4Xyrqhrs6Agy270GNw71-l2bOZjWSWlohdXWZ5vwa4Chkg4Sp7N6HesIObY-ev7LpdohTdFTHgpJBWYYV5dUCe4h7Ciczom4FJpyRU5-Tp6nwQFxXmCCLvyHDxo9AIOH98Rgmv9M/s320/primes+as+shape+arrangement.png" width="320" /></span></a></div>
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<span style="font-family: "verdana" , sans-serif;">This led me to form the below examples for the numbers 11, 12 and 36... which are hopefully self explanatory :)</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiW-r2r8IaBHxcypfk-DW7sB3iGag8PHxe35Oxw5perp0ySlxK4TLAcKeCMun3vQMG0mHtdSPEAec7TYm-ppFkjopD91JmzlIz2Ak_RAA2w5vwunoG_sPmCUNlCwgwpEzZ4CM3IznCsKUM/s1600/factors+of+11.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: "verdana" , sans-serif;"><img border="0" height="289" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiW-r2r8IaBHxcypfk-DW7sB3iGag8PHxe35Oxw5perp0ySlxK4TLAcKeCMun3vQMG0mHtdSPEAec7TYm-ppFkjopD91JmzlIz2Ak_RAA2w5vwunoG_sPmCUNlCwgwpEzZ4CM3IznCsKUM/s640/factors+of+11.JPG" width="640" /></span></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqkaLopkUvGh8fheZWmTA1NMyR8rWuFaPOSpyZBrg9jz7lD35K84BLIruf1769mUbhVHw0f-1Q3KIPmr7_EWdqk6dUOZj-rikhRhLg8-vPG4FLLO5iCbgmF49sef-KBATp5HLh-huWDXY/s1600/factors+of+12.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: "verdana" , sans-serif;"><img border="0" height="306" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqkaLopkUvGh8fheZWmTA1NMyR8rWuFaPOSpyZBrg9jz7lD35K84BLIruf1769mUbhVHw0f-1Q3KIPmr7_EWdqk6dUOZj-rikhRhLg8-vPG4FLLO5iCbgmF49sef-KBATp5HLh-huWDXY/s640/factors+of+12.JPG" width="640" /></span></a></div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6157870465012038411.post-36707635879051121112016-02-24T15:45:00.000-08:002016-02-24T16:04:03.913-08:00Maybe Coming Closer to Tidying up my Collection of Number Musings...<div style="text-align: center;">
<div align="center" style="margin: 0cm 0cm 0.0001pt;">
<span style="font-family: Verdana, sans-serif;">I have
left words of promise to people these past months, to share some insight into
my passion for number analysis; some visions in mind to expand upon, and the
thought of sparking some interest into you... well... it's motivating.<o:p></o:p></span></div>
<div align="center" style="margin: 0cm 0cm 0.0001pt;">
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<span style="font-family: Verdana, sans-serif;">So I had
a gander at what I set up back in 2012 to be part of a presentation slash
workshop; it is a very brief poker at what more could be explored - and the aim
is not to provide answers but rather pester curiosity for further musings.<o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif;">So I include it <a href="https://drive.google.com/open?id=0B4lPR5IeS7IfZzVMOFFQNXlETkE" target="_blank">here</a>, with a <a href="https://plus.google.com/u/0/photos/117708861744151825659/albums/6255007215358202305" target="_blank">page of pictures</a> that I also included into the presentations - images of the paper workings (really </span><span style="font-family: "verdana" , sans-serif;">is the more fun way to play - although Excel for sure gets it's fair share of worthy recognition, as too <a href="http://www.geogebra.org/" target="_blank">GeoGebra </a>for the graphical explorations of 'numbers'.) which can also be found in the shared google drive location in the previous post on this blog.</span></div>
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<span style="font-family: "verdana" , sans-serif;">It may lead to degrees of philosophical recognition as to our perception to/of numbers... once one form of data is transformed into variations of representation manners, colors and shapes... giving form to numbers (or is it numbers giving value to form?) it is hard not to ponder towards energy and mass, space and time, light and sound... forces, motion, spin, quantum, gyro... yet I am just thinking games at this present time.</span></div>
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<span style="font-family: "verdana" , sans-serif;"><img alt="Image result for playing games" height="200" 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" 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<span style="font-family: "verdana" , sans-serif;">My tablet had a rather built up selection of notes I had sketched up with my S pen... said table was stolen recently... lost a lot of musings that were new and fresh during my Asia months, so, to beat discouragement, I am just gently exploring again, which, as I write this and present the little worksheet, factors, primes, composites are being explored, as too cumulative sums, as too further fibonacci related ponderings.</span></div>
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<span style="font-family: "verdana" , sans-serif;">Anywho... off you go, have a look at some number play!</span></div>
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<span style="font-family: "verdana" , sans-serif;"><span style="font-size: large;">- <a href="https://drive.google.com/file/d/0B4lPR5IeS7IfZzVMOFFQNXlETkE/view?usp=sharing" target="_blank">Worksheet Download</a> -</span></span><br />
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<tr><td class="tr-caption" style="text-align: center;"><a href="https://plus.google.com/u/0/photos/117708861744151825659/albums/6255007215358202305" target="_blank">Click on Mr Tesla to follow through to the photo album...</a></td></tr>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6157870465012038411.post-68819535408354467312014-02-12T15:17:00.001-08:002014-02-14T12:35:43.475-08:00It's been some time...<span style="font-family: Arial, Helvetica, sans-serif;">Between then and now, time has passed; blogger pages stayed as they were left and maybe will continue like that after today.</span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">Continued investigations into number theory have not answered as many questions as new questions posed!</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">The journey truly is where satisfaction is being found, oblivious to an end aim, just outputting and storing seems to be the game.</span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">Well, I would like to change that and am moving to do something with my collection of 2000+ files - consisting of geogebra, excel, photoshop outputs/templates; photos of hand written notebooks ready for digitization; powerpoints; pdf's... I have been busy enjoying my time yet along the way have managed to store up quite some documentation of learning almost like the Greeks did :)</span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">It needs cleaning up - at least into categories... this is work in progress i.e. watch this space.</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh38FI8rilRjgDuIvXN3HVpQ-btsvyNi0Mp7EeFzw5w1X0yHLdCBB5nUZli0uvl8KjUT3rj3I0RrTs3Cr1KLbXIwR3OJYNfZknRK9YzN7ftBQ7UafHswKEjURjvjEZ9IwANMAXm3zjKrCE/s1600/Folder+Messb.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><span style="font-family: Arial, Helvetica, sans-serif;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh38FI8rilRjgDuIvXN3HVpQ-btsvyNi0Mp7EeFzw5w1X0yHLdCBB5nUZli0uvl8KjUT3rj3I0RrTs3Cr1KLbXIwR3OJYNfZknRK9YzN7ftBQ7UafHswKEjURjvjEZ9IwANMAXm3zjKrCE/s1600/Folder+Messb.png" height="311" width="640" /></span></a></div>
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<span style="font-family: Arial, Helvetica, sans-serif; text-align: start;">Ready items that are just sitting there (ready for eyes, not complete in the sense of ended)....</span></div>
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<span style="font-family: Arial, Helvetica, sans-serif; text-align: start;">Here is a<span style="font-size: large;"> <a href="https://drive.google.com/folderview?id=0B4lPR5IeS7IfSHBiTkdTU3BEWjA&usp=sharing" target="_blank">link</a></span> to googledrive where you can locate some documents I put together - kinda a snippet of the bigger picture in the distance :)</span></div>
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<span style="font-family: Arial, Helvetica, sans-serif; text-align: start;"><br /><br />Kinda thinking that if something I have already done can trigger something for you... cool, I would be interested to hear about developments - imagining that someone out there might see some sense in this and engage me to complete some output together perhaps?<br /><br />I really have got a chunk of stuff 'analyzed' (only word I can really think to apply here) and its in dire need of review and write up - the few doc's I shared above via googledrive, are truly snippets - I don't think I see an end yet... each review leads to new ideas... still... two years on...</span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">Almost two years since this:</span></div>
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<a href="http://www.youtube.com/watch?v=oyJxWQDesh8" target="_blank"><b><span style="font-family: Arial, Helvetica, sans-serif; font-size: large;">http://www.youtube.com/watch?v=oyJxWQDesh8</span></b></a></div>
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<br />Unknownnoreply@blogger.com0