1. The History and
Mystery of Zero
Mark Darby Ken Doherty

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?
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3. References
 Zero The Biography of a Dangerous Idea
Charles Seife
 The Nothing That Is
Robert Kaplan
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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.
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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
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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
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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?
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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?
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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?
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10. Ishango Bone
 Would have been reflective of phases of the
moon & women’s menstrual cycle.
 Women – The first mathematicians?
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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…
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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!
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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!
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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.
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15. Abacus used for calculations by the Romans
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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
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17. Myans (200- B.C. – 250 A.D.)
Did have zero!
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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?
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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
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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
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21. Or, Estimate/Guess with Excel
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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
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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)
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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
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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 = ∞
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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
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27. Zero and Infinity
0
∞
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28. Vanishing (Zero) Point in Art.
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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.”
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30. Zero Today - Double entry book keeping
Must Balance: Difference = 0
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31. Zero and Infinity Today
 Routine use in
 Mathematics (e.g., Calculus)
 Science
 Engineering
 All problems resolved?
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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?
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33. USS Yorktown (1997)
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34. Thanks for Your Attention
 Questions?
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35. Extra
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36. 36

37. Descartes 1596 1650
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38. Still a confounder for me.
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