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. 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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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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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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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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