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teaching:topics:calculus:sequences [2020/08/07 12:31]
simon
teaching:topics:calculus:sequences [2022/07/26 13:43] (current)
simon
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 Can we possibly get a sensible, finite, answer when we try to add up an infinite number of positive numbers all greater than zero?  Can we possibly get a sensible, finite, answer when we try to add up an infinite number of positive numbers all greater than zero? 
  
-This is a very ancient problem in mathematics ​... and the different ways of resolving (or banishing) it have been profoundly important to mathematics,​ physics and wider ideas of philosophy.+This is a very ancient problem in mathematics ​--- the different ways of resolving (or banishing) it have been profoundly important to mathematics,​ physics and wider ideas of philosophy. And to the history of the development of modern science, technology and industry along with modern economies and societies.
 == == == ==
 <wrap #​paradox>​Consider a simple question like this:</​wrap>​ <wrap #​paradox>​Consider a simple question like this:</​wrap>​
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 Making up new language, new notation, to work with particular ideas and problems is at the heart of mathematics. Making up new language, new notation, to work with particular ideas and problems is at the heart of mathematics.
  
-===Once ​apon a time=== +===Once ​upon a time=== 
-mathematicians were trying to make some sense of calculating continuous motions ... in particular the paths taken by planets, cannonballs,​ stones dropped from the Leaning Tower of Pisa and balls rolling on a sloping platform. This was 400 years ago (in the 1600s), just over 2000 years after Xeno famously talked about the [[#​paradox]] above and the many challenges involved with playing around with infinites and infintessimals.+mathematicians were trying to make some sense of calculating continuous motions ... in particular the paths taken by planets, cannonballs, lamps 
 +swinging on long chains hanging in cathedrals, stones dropped from the 
 +Leaning Tower of Pisa and balls rolling on a sloping platform. This was 400 years ago (in the 1600s), just over 2000 years after Xeno famously talked about the [[#​paradox]] above and the many challenges involved with playing around with infinites and  
 +[[teaching:​topics:​calculus:​introduction#​infinitessimals]].
  
 It took the life's work of a few generations of mathematicians to develop the ideas, mathematics and physics and it culminated in Liebniz and Newton bringing it all together ... and setting up the next couple of centuries of science and technology. It took the life's work of a few generations of mathematicians to develop the ideas, mathematics and physics and it culminated in Liebniz and Newton bringing it all together ... and setting up the next couple of centuries of science and technology.
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 They were inventing the Calculus you are just about to spend quite a lot of effort to understand yourself! They were inventing the Calculus you are just about to spend quite a lot of effort to understand yourself!
 They discovered that motion could be understood through the relationship between what was eventually called They discovered that motion could be understood through the relationship between what was eventually called
-the '​second derivative'​ of position with respect to time, and the position ​and masses involved. This is quite a big leap.+the '​second derivative'​ of position with respect to time, and the positions ​and masses involved. This is quite a big leap.
 Quite a lot of our school curriculum is learning about the mathematics they developed in that time. Quite a lot of our school curriculum is learning about the mathematics they developed in that time.
-  ​+== ==  ​
 They were laying the foundations of classical mechanics and the science on which we have built modern technology, industry and society. We have dug much deeper since, and found the universe actually behaves in very weird and wonderful ways indeed. Today'​s mathematicians and scientists are still spending their lives exploring and explaining it, there are still questions to be answered as challenging now as these questions were then. They were laying the foundations of classical mechanics and the science on which we have built modern technology, industry and society. We have dug much deeper since, and found the universe actually behaves in very weird and wonderful ways indeed. Today'​s mathematicians and scientists are still spending their lives exploring and explaining it, there are still questions to be answered as challenging now as these questions were then.
  
-Meanwhile others were arguing that dealing with these infinite sums and infinitessimal numbers was bad mathematics ​... that mathematics should not tread any paths that lead to paradoxes, that mathematics is some kind of absolute truth and there was no room for such dangerous and messy ideas.+Meanwhile others were arguing that dealing with these infinite sums and infinitessimal numbers was bad mathematics ​--- that mathematics should not tread any paths that lead to paradoxes, that mathematics is some kind of absolute truth and there was no room for such dangerous and messy ideas.
  
 The next generations of mathematicians worked to find more rigorous and careful ways to define (and extend!) these first versions of calculus. Especially ways to nail down those troublesome infinites and infinitessimals,​ including the idea of **limits** that we will look at next. The next generations of mathematicians worked to find more rigorous and careful ways to define (and extend!) these first versions of calculus. Especially ways to nail down those troublesome infinites and infinitessimals,​ including the idea of **limits** that we will look at next.
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 We understand mathematics very differently now ... but that is another very long (and very interesting) story. We understand mathematics very differently now ... but that is another very long (and very interesting) story.
  
-As for physics ​... when it is or is not realistic to keep dividing quanties into smaller and smaller parts+As for physics ​--- when it is or is not realistic to keep dividing quanties into smaller and smaller parts
 and whether we can consider space and time continuous quantities is again the big question and the answers and whether we can consider space and time continuous quantities is again the big question and the answers
 are sought in quite beautiful and new mathematics. Classical mechanics only takes us some of the way. are sought in quite beautiful and new mathematics. Classical mechanics only takes us some of the way.
  
 <WRAP right>​**=>​[[limits#​limits]]**</​WRAP>​ <WRAP right>​**=>​[[limits#​limits]]**</​WRAP>​
teaching/topics/calculus/sequences.1596767481.txt.gz · Last modified: 2020/08/07 12:31 by simon