The math lie - are Real numbers Real

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Should we be so quick to use real numbers? Are they "Real" at all?

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The math lie - are Real numbers Real

  1. 1. <ul><li>The Math Lie </li></ul><ul><li>Are real numbers real? </li></ul>
  2. 2. <ul><ul><li>What are real numbers? </li></ul></ul><ul><ul><li>Let’s take a ride down history’s lane </li></ul></ul>
  3. 3. <ul><li>In the beginning, there was 1 </li></ul>
  4. 4. <ul><li>1 gave birth to 2 (1+1), 3 (1+1+1), … to infinity </li></ul><ul><li>And we got what is called “The Natural Numbers” (N), or positive integers </li></ul>
  5. 5. <ul><li>They’re call “Natural” for a reason </li></ul><ul><li>They’re very intuitive, and have been around for as long as there have been man </li></ul>
  6. 6. <ul><li>God gave us the integers, all else is the work of man </li></ul><ul><li>L. Kronecker </li></ul>“ ”
  7. 7. <ul><li>But, they aren’t perfect </li></ul><ul><li>You can formulate questions with natural numbers, that have no answer IN natural numbers </li></ul>
  8. 8. <ul><li>1+? = 1 </li></ul>
  9. 9. <ul><li>So the zero came into being </li></ul><ul><li>And we get what is called N+ </li></ul><ul><li>N+ = 0,1,2… </li></ul>
  10. 10. <ul><li>But there were still unanswerable questions </li></ul><ul><li>1+? = 0 </li></ul>
  11. 11. <ul><li>So once again, new numbers were added - negative numbers </li></ul><ul><li>-x was define to be the number that answers </li></ul><ul><li>X+? = 0 </li></ul>
  12. 12. <ul><li>This gave us the Z, the integers </li></ul><ul><li>But unanswerable questions kept popping up </li></ul>
  13. 13. <ul><li>Though we had to look beyond addition to find them </li></ul><ul><li>3*? = 1 </li></ul>
  14. 14. <ul><li>So we gave birth to the “Rational” numbers - Noted Q </li></ul><ul><li>Rational = represent a ratio between two integers </li></ul>
  15. 15. <ul><li>The rational numbers had to fit the known image of the world </li></ul><ul><li>And indeed 3 (integer) = (rational) </li></ul>
  16. 16. <ul><li>Rational numbers seem to solve any question they can represent </li></ul><ul><li>A little hard work, but no conceptual problem </li></ul>
  17. 17. <ul><li>But, life can’t be that good </li></ul><ul><li>Around 400BC, the Greeks have found that isn’t rational </li></ul>
  18. 18. <ul><li>Which means that this question has no answer in the rational domain </li></ul><ul><li>?*? = 2 </li></ul>
  19. 19. <ul><li>So, a new set of numbers were introduced - irrational-numbers, </li></ul><ul><li>Irrational just means it can’t be expressed as a ratio, but we’ll see it turns out to have a deeper meaning </li></ul>
  20. 20. <ul><li>Actually, the trip doesn’t end here, because there’re still unanswerable questions </li></ul><ul><li>?*? = -1 </li></ul><ul><li>But let’s focus on the irrationals </li></ul>
  21. 21. <ul><li>Rational + Irrational numbers </li></ul><ul><li>= </li></ul><ul><li>Real numbers </li></ul>
  22. 22. <ul><li>They are casually presented to every junior-high pupil as a natural extension of rational numbers </li></ul><ul><li>No question is asked </li></ul>
  23. 23. <ul><li>As it turns out, the irrational numbers are the really important ones </li></ul><ul><li>e, π, cos, sin, … </li></ul><ul><li>And there are infinitely more irrational numbers then rational ones </li></ul>
  24. 24. <ul><li>So, Real numbers are pretty much the corner stone of modern math </li></ul><ul><li>And we all use then quite regularly </li></ul>
  25. 25. <ul><li>Good thing they work </li></ul>
  26. 26. <ul><li>Do they? </li></ul><ul><li>Shouldn’t we stop and ask? </li></ul>
  27. 29. <ul><li>That’s pretty impressive </li></ul><ul><li>Any specific rational problem can be solved and narrowed down to a answer </li></ul><ul><li>Lets what the “Real” numbers can do </li></ul>
  28. 32. <ul><li>Something’s fishy… </li></ul><ul><li>It seems we’re quite helpless in the face of real numbers </li></ul>
  29. 35. <ul><li>Now, that’s just cheating </li></ul>
  30. 36. <ul><li>It’s not that it never works… </li></ul><ul><li>But it almost never does </li></ul>
  31. 37. <ul><li>Wait, does it ever work? </li></ul><ul><li>Is , really? </li></ul>
  32. 38. <ul><li>What is anyway? </li></ul><ul><li>Is it 1.141? </li></ul><ul><li>No, that’s only the beginning… </li></ul>
  33. 39. <ul><li>Is an unending beast </li></ul><ul><li>No one has ever held in its hand </li></ul><ul><li>And no one ever will </li></ul>
  34. 40. <ul><li>So what do we mean by ? </li></ul><ul><li>If we’ll ever be able to hold an unending beast, and multiply it by another unending beast (a process which will never end), we will get the exact result 2 </li></ul>
  35. 41. <ul><li>Anyone convinced? </li></ul>
  36. 42. <ul><li>But maybe the problem lays with us </li></ul><ul><li>We’re idiots </li></ul><ul><li>The real smart professors must have the answer… </li></ul>
  37. 43. <ul><li>Well, actually… </li></ul>
  38. 44. <ul><li>The whole thing is pretty young </li></ul><ul><li>A wide consent on the nature of real numbers was only achieved around 1920+, when Cantor’s theory of sets grew popular </li></ul>
  39. 45. <ul><li>This means that for ~2300 years, people had no idea what they’re talking about </li></ul><ul><li>Not even the smart professors </li></ul>
  40. 46. <ul><li>(It doesn’t mean that they know what they’re talking about NOW. It just means most of them agree on the nonsense they’re saying) </li></ul>
  41. 47. <ul><li>Real numbers are extremely unnatural </li></ul><ul><li>You have to go through 2 months of university calculus before you’re (kinda) convinced that </li></ul>
  42. 48. <ul><li>But still, if you draw a 1X1 square, the diagonal just IS . </li></ul><ul><li>And if you want to know a area of a circle, π is the only way to go </li></ul>
  43. 49. <ul><li>These are not some invented pure mathematical entities </li></ul><ul><li>They’re really “out there” in nature </li></ul>
  44. 50. <ul><li>We can’t help but deal with them </li></ul><ul><li>And that’s OK </li></ul>
  45. 51. <ul><li>But let’s not kid ourselves </li></ul><ul><li>Real numbers are not real </li></ul><ul><li>And shouldn’t be presented casually </li></ul>
  46. 52. <ul><li>They are irrational, in the fullest sense of the word </li></ul>
  47. 53. <ul><li>And we should be asking hard questions about them </li></ul>
  48. 54. <ul><li>So, start filling the blanks </li></ul>
  49. 55. <ul><li>_______? </li></ul>

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