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History and mystery of zero

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History and mystery of zero

  1. 1. The History and Mystery of Zero Mark Darby Ken Doherty
  2. 2. Topics  Why We like Math, Do you?  History  Religion  Players, Cultures, Contributors  A Few Equations Along the Way!  If You Can Divide By Zero, You Can Do Anything!  Zero Today – All ok? 2
  3. 3. References  Zero The Biography of a Dangerous Idea Charles Seife  The Nothing That Is Robert Kaplan 3
  4. 4. Math Myths  I am not good at ___________ [fill-in the blank: counting, multiplying, etc.]  To do Math, you have to be born that way.  Math is boring, it does not involve creativity. 4
  5. 5. Intro- Example 1  Johann Carl Friedrich Gauss - mathematician and scientist (1777 – 1855)  Story of his punishment as a child 1  100  101 1  2  3    98  99  100  ? 2  99  101 Answer :101 50  5, 050 Generally: the sum of numbers 1+2+  +n   n  1  n 2 5
  6. 6. Engineering and Math  Solve equations Scientific laws Engineering principles  Predict Breaking point of a material Number of customer orders next month  Optimize Minimize cost, maximize profit of manufacturing 6
  7. 7. Mathematicians vs. Engineers – Example 2  You are 2 steps away from ___________ [fill-in the blank: beautiful woman, handsome man, $1,000].  But you may only approach according to the following rule: Each step must be ½ of the previous step.  Should you try? 7
  8. 8. Example 2, cont’d  To solve this problem, we need to know the answer to 1 1 1 1       ? (infinite number of terms) 2 4 16  Does it have an answer?  Can we calculate the answer? 8
  9. 9. Numbers…in the beginning Used to count or tally  30,000 year old wolf bone with carved notches (discovered 1930’s). Groups of 5 – why?  Ishango bone, Congo (20,000 - 25,000 years old). Groups of 28 or 29. Why? 9
  10. 10. Ishango Bone  Would have been reflective of phases of the moon & women’s menstrual cycle.  Women – The first mathematicians? 10
  11. 11. Early History – No Need for Zero  Why worry about 0 bushels, 0 buffalo?  Counting, geometric significance only.  Also, scary and/or mind boggling  Zero ↔ Nothingness No such thing as nothing in the Greek universe (300 BC)  Don’t want to think about it! But: there were problems… 11
  12. 12. Calendars B.C. A.D. …, -4, -3, -2, -1, 1, 2, 3, 4,…  Zero is missing  Consider a child born on Jan 1, 4 BC  On Jan 1 in 2 AD, child is 5  But would calculate age 6 (2- -4) without zero! 12
  13. 13. Any Better in 2000?  When should we have celebrated the new millennium?  It was celebrated on Dec 31, 1999.  2000 years after 1 AD would make the date Dec 31, 2000/Jan 1, 2001! 13
  14. 14. Representation of Numbers  Egyptians (5,000 years ago) – pictures, symbols  Greeks (600 B.C.) – Use of letters (e.g., M for 1,000) Messy for larger numbers – 87 required 15 symbols)  Babylonians (1,800 B.C.) – 1 thru 60 (base 60) Didn’t need zero for their “abacus”, but had problem with writing numbers - could not distinguish between 61, & 3,601. 14
  15. 15. Abacus used for calculations by the Romans 15
  16. 16. Arabic Numbering (Base 10) [Should be called Indian Numbering!] 1' s 1 2 3 4 5 6 7 8 9 10 ' s 10 20 30 40 50 60 70 80 90 100 ' s 100 200 300 400 500 600 700 800 900 Consider the number 107 1' s 1 2 3 4 5 6 7 8 9 7 1 10 ' s 10 20 30 40 50 60 70 80 90 0 10 100 ' s 100 200 300 400 500 600 700 800 900 1100 0 as a place holder 16
  17. 17. Myans (200- B.C. – 250 A.D.) Did have zero! 17
  18. 18. Zeno – Paradox of Achilles (490 BC) Achilles runs 1 foot / sec Tortoise runs ½ foot sec After 1 sec, Achilles has caught up to where tortoise was But tortoise has moved up 1/2 foot In next ½ sec, Achilles makes up the ½ foot But tortoise has moved up 1/4 foot Achilles never catches the tortoise! Obviously not true but why? 18
  19. 19. Remember Example from Earlier? 1 1 1 1 1       n    ? (infinite number of terms) 2 4 16 2  Series approaches a limit  Each (individual) term gets closer to 0 19
  20. 20. Some (creative) Math! 1 1 1 1 S  1     n  2 4 16 2 1 multiply by 2 1 1 1 1 1 S      n  2 2 4 16 2 1 Subtract S from S 2 1 S  1, or S  2 2is the limit! 2 20
  21. 21. Or, Estimate/Guess with Excel 21
  22. 22. Influence of India (5th century AD)  Hinduism embraced duality  Similar to Yin Yang of Far East  Good / Evil  Creation and Destruction  Accepting of original nothingness (infinite)  Numbers became distinct from geometry  Abstraction  Zero the number (not just a place holder)  Rules of zero (what are they?)  Negative numbers 22
  23. 23. Religious Aspects  Christianity influenced by Aristotelian view  Stationary earth  Planets moved by each other  God is prime mover  No void or infinity What is conflict?  Islam  Embraced the void (creation came from the void)  Muslim scholars (Al-Khowarizmi, “Al-jabr” 800 AD) 23
  24. 24. Alegbra with Zero  If a X b = 0, Then A or B must be zero, Or, they both are zero; one of the keys to algebra as we know it today.  a ÷ b not defined if b = 0 24
  25. 25. Zero and infinity - 1 ÷ 0? 1 1 1 1  10 0.1  1  10 0 (a bigger and bigger number!) 0.0 01 a lim  ? (a is postive number) Answer : Infinity "in the limit" x 0 x We cheat (a bit) when we say a ÷ 0 = ∞ 25
  26. 26. Zero and infinity - 1 ÷ ∞? 1  0.1 10 1  0.01 100  1  0.00...01 100...0 a lim ? Answer : 0 "in the limit" x  x We cheat (a bit) when we say a ÷ ∞ = 0 26
  27. 27. Zero and Infinity 0 ∞ 27
  28. 28. Vanishing (Zero) Point in Art.   28
  29. 29. Leonardo da Vinci was one of the first to use a vanishing point in his art.  In one of his books about painting, he warned “let no one who is not a mathematician read my works.” 29
  30. 30. Zero Today - Double entry book keeping Must Balance: Difference = 0 30
  31. 31. Zero and Infinity Today  Routine use in  Mathematics (e.g., Calculus)  Science  Engineering  All problems resolved? 31
  32. 32. A Little More Math…Where’s The Problem? a  b 1 b  ab 2 a2  a2 a 2  b 2  a 2  ab (a  b)(a  b)  a (a  b)  a  b  a  b  0 But we started with b  1! What happened? 32
  33. 33. USS Yorktown (1997) 33
  34. 34. Thanks for Your Attention  Questions? 34
  35. 35. Extra 35
  36. 36. 36
  37. 37. Descartes 1596 1650 37
  38. 38. Still a confounder for me. 38

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