{"id":40,"date":"2016-11-08T09:36:02","date_gmt":"2016-11-08T16:36:02","guid":{"rendered":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/?page_id=40"},"modified":"2018-11-08T11:00:22","modified_gmt":"2018-11-08T18:00:22","slug":"infinitesimals-the-ghosts-of-departed-quantities","status":"publish","type":"page","link":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/infinitesimals-the-ghosts-of-departed-quantities\/","title":{"rendered":"Infinitesimals: The Ghosts of Departed Quantities"},"content":{"rendered":"

Warmup<\/strong> What\u2019s a $dx$<\/span>?<\/em><\/p>\n

Reflection<\/strong> p. 77 \u201cThanks to the real numbers, we can measure the length of any line, even irrational lines such as the diagonal of the unit square.\u201d
\nWhy do we need permission to measure something? How do the real numbers provide it? Are they needed?<\/em><\/p>\n

How do you measure the area of a rectangle, triangle, generic polyhedron? How about a circle? Why is how to measure things a hard question?<\/em><\/p>\n

Arbitrarily Small Pieces<\/h2>\n

This concept as you use it in Calculus class isn\u2019t about a set of things. You don\u2019t grab hold of them or use them. Instead, when invoking them a scenario is described.
\nWarmup<\/strong> With a partner, take turns assigning real numbers and picking smaller ones than the assigned size.<\/em><\/p>\n

Draw a triangular prism cut from a rectangular parallelepiped.<\/em><\/p>\n

Impossible warmup<\/strong> Try to find the volume of a tetrahedron using geometry and algebra. Try with calculus.<\/em><\/p>\n

Recall<\/strong> Find the volume of a tetrahedron using the geometric series.<\/em>
\nRecall<\/strong> Find and prove the sum of the general geometric series $a+ar+ar^2 +\\ldots$<\/span>.
\n<\/em>Why does Stillwell say it is more cautious to use the limit of partial sums instead of the algebraic trick directly on the infinite series?*<\/p>\n

Knowing the Unknowable<\/h2>\n

What is $\\pi$<\/span> and how do we know it? Using the analogy of the three blind men and the elephant, we know something about $\\pi$<\/span> every time we have a different way to see it. What different ways of seeing it are there?<\/em>
\nArchimedes not only approximated $\\pi$<\/span> (though no one called it that then. When did people start calling it by its modern name? How did this happen?<\/em>) he established a common ratio to the measurement of area and circumference of the circle. Stillwell recounts this and hints at the proof.<\/p>\n

Problem<\/strong> Follow Archimedes and Stillwell. Use Eudoxus\u2019 method of exhaustion to show that the area of a circle is the same as a triangle with height equal to the radius and base equal to the circumference of the circle.<\/em> Hint: Check JTG Chapter 4.
\nRead the entry for [$\\pi$<\/span> on Wikipedia][], especially the history. Take some notes and ask some questions.<\/em>
\n![Progress in computing $\\pi$<\/span> (from Wikipedia)][computing pi]
\nThis progress looks like it gets quick around 1950. Then realize the graph is logarithmic.
\nDiscussion<\/strong> What is the point of all this?<\/em>
\nResearch<\/strong> How many digits does one need for various practical purposes? How many are in use in various places?<\/em>
\nResearch<\/strong> What characterizes computational advances, particularly in the computer age beyond \u201cDuh, computers are fast\u201d?<\/em>
\nResearch<\/strong> What does your calculator do? Wolfram alpha?<\/em>
\nReflect<\/strong> How do you check the answer to a calculation of $\\pi$<\/span> to a quadrillion decimal places?<\/em>
\nResearch<\/strong> What is $\\tau$<\/span> in relation to $\\pi$<\/span> and why does it matter?<\/em>
\nWarmups<\/strong>
\nUse differentials to compute tangents to curves other than the parabola.<\/em>
\nUse the game of suggesting arbitrarily small numbers to see how close $2x+dx$<\/span> is to $2x$<\/span>.<\/em><\/p>\n

Discuss<\/strong> Why are the Marquis L\u2019H\u00f4pital\u2019s arguments \u201cindefensible\u201d as suggested on p. 91?<\/em><\/p>\n

Research<\/strong> Examine Fermat\u2019s concept of adequality from a perspective of scientific or technological change. It is an interesting example because we would today consider it a better approach than what persisted, Fermat was not unknown, and yet adequality was not adopted then.<\/em><\/p>\n

Warmup<\/strong> State briefly the fundamental theorem of calculus.<\/em><\/p>\n

Problem<\/strong> Unpack Stillwell\u2019s argument about the arc length of the circle starting on p. 96. What is an arctan? Where are $y$<\/span> and $x$<\/span>?<\/em><\/p>\n

Computing $\\pi$<\/span><\/h2>\n

We have seen a number of ideas for methods of computing $\\pi$<\/span>. But we have not actually used them to compute $\\pi$<\/span>.
\nProblem<\/strong> Implement these by hand without using a calculator. Then use SAGE to implement them.<\/em>
\nReflect<\/strong> Compare them as technologies. Consider particularly efficiency by hand and computer, ease of implementation from concept, and elegance.<\/em><\/p>\n

Reflect<\/strong> Find a a solid body with finite volume and infinite surface area. What other impossible objects can you find\/create using calculus?<\/em><\/p>\n

Further Topics About Analysis<\/h2>\n

Research<\/strong> Stillwell claims on p. 99 that \u201cBerkeley\u2019s criticism stung, and mathematicians tried to answer it, though for a long time without much success.\u201d Find mathematicians\u2019 responses to Berkeley.<\/em><\/p>\n

Research<\/strong> Find out more about nonstandard analysis.<\/em>
\nResearch<\/strong> Find out more about using spaces of functions as mathematical objects.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"

Warmup What\u2019s a $dx$? Reflection p. 77 \u201cThanks to the real numbers, we can measure the length of any line, even irrational lines such as the diagonal of the unit square.\u201d Why do we need permission to measure something? How do the real numbers provide it? Are they needed? How do you measure the area … Continue reading “Infinitesimals: The Ghosts of Departed Quantities”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":6,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-40","page","type-page","status-publish","hentry"],"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/PbpbST-E","_links":{"self":[{"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/pages\/40","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/comments?post=40"}],"version-history":[{"count":2,"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/pages\/40\/revisions"}],"predecessor-version":[{"id":53,"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/pages\/40\/revisions\/53"}],"wp:attachment":[{"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/media?parent=40"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}