The document provides an overview of the history and development of the concept of zero. It discusses how zero emerged as a distinct numerical concept over centuries, originating from its use as a placeholder in places value systems before being abstracted as an independent number. Key points covered include the influence of Indian and Arabic mathematicians in developing rules for zero and negative numbers, allowing for the development of algebra, as well as how different cultures and religions approached the concepts of nothingness and infinity that zero represents.

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
3

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
5

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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