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The Elegance of Mathematics

The Elegance of Mathematics. Calculus for the 7th Grade. Keynotes at California Mathcounts 2003 Dr. Zhiqin Lu University of California, Irvine. 03/22/2003. The title is from Brain Greene’s best selling book The Elegant Universe

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The Elegance of Mathematics

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  1. The Elegance of Mathematics Calculus for the 7th Grade Keynotes at California Mathcounts 2003 Dr. Zhiqin Lu University of California, Irvine 03/22/2003

  2. The title is from Brain Greene’s best selling book The Elegant Universe To professor Greene, the universe is elegant. To us, mathematics is elegant. Brain Greene’s best selling book

  3. Let’s begin with a problem Find the value of the following:

  4. Idea?

  5. Then we have

  6. What is the observation here? In order to find the value of We write

  7. Then we have Theorem This is the idea of Calculus!

  8. Another Example Back-to-the-envelope calculation

  9. One more Example 1+2+…+99+100=? Carl Friedrich Gauss solved this problem when he was six! (but probably he didn’t use our method.) Can you do it? I am sure.

  10. What is Calculus? Calculus=Differentiation+Integration Differentiation (or subtraction) : Integration (or addition) : The fundamental theorem of calculus states the duality between differentiation and integration. Theorem If , then If a is the differentiationof b, Then b is the integration of a.

  11. Go to Infinity

  12. Theorem (Fundamental theorem of Calculus)

  13. The idea I have just showed you can be used to prove theFundamental Theorem of Calculus

  14. Of course, we need to make the above setting precise. However, the basic duality between differentiation and integration roots in the very simple duality property between subtraction and addition. That is the elegance of mathematics. You may be able to discover your own theorem by enjoying and paying attention to the mathematics you are learning now.

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