The History and Mystery of Zero

Mark Darby Ken Doherty

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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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References

Zero The Biography of a Dangerous IdeaCharles Seife

The Nothing That IsRobert Kaplan

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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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Intro-

Example 1

Johann Carl Friedrich Gauss - mathematician and scientist (1777 – 1855)

Story of his punishment as a child

1 2 3 98 99 100 ? 1 100 101

2 99 101

1Generally: the sum of numbers 1+2+ +

2n n

n

Answer :101 50 5,050

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Engineering and Math

Solve equationsScientific lawsEngineering principles

Predict Breaking point of a materialNumber of customer orders next month

OptimizeMinimize cost, maximize profit of manufacturing

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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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Example 2, cont’d To solve this problem, we need to know the

answer to

Does it have an answer?

Can we calculate the answer?

1 1 11 ? (infinite number of terms)2 4 16

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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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Ishango Bone

Would have been reflective of phases of the moon & women’s menstrual cycle.

Women –

The first mathematicians?

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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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Calendars

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!

B.C. A.D.

…, -4, -3, -2, -1, 1, 2, 3, 4,…

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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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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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Abacus used for calculations by the Romans

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Arabic Numbering (Base 10) [Should be called Indian Numbering!]

Consider the number 107

1' 1 2 3 4 5 6 7 8 910 ' 10 20 30 40 50 60 70 80 90100 ' 100 200 300 400 500 600 700 800 900

sss

1' 1 2 3 4 5 6 7 8 9 7 110 ' 10 20 30 40 50 60 70 80 90 10100 ' 100 200 300 400 500 600 700 800 900

01 100

sss

0 as a place holder

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Myans

(200-

B.C. –

250 A.D.)

Did have zero!

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Zeno –

Paradox of Achilles (490 BC)

Achilles runs 1 foot / secTortoise 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

footBut tortoise has moved up 1/4 foot

Achilles never catches the tortoise!

Obviously not true but why?

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Remember Example from Earlier?

Series approaches a limit Each (individual) term gets closer to 0

1 1 1 11 ? (infinite number of terms)2 4 16 2n

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Some (creative) Math!

2is th

1 1 1 112 4 16 2

1multiply by 2

1 1 1 1 12 2 4 16 21Subtract from 2

1 1 e , or limit!22

n

n

S

S

S S

S S

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Or, Estimate/Guess with Excel

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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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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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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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Zero and infinity -

1 ÷

0?

1 111 10

0.1

1 10 0 (a bigger and bigger number!)0.0 01

0lim ? (a is postive number)x

ax Answer : Infinity "in the limit"

We cheat (a bit) when we say a ÷

0 = ∞

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Zero and infinity -

1 ÷ ∞?

1 0.110

1 0.01100

1 0.00...01100...0

lim ?x

ax Answer : 0 "in the limit"

We cheat (a bit) when we say a ÷

∞

= 0

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Zero and Infinity

0∞

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Vanishing (Zero) Point in Art.  

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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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Zero Today -

Double entry book keeping

Must Balance: Difference = 0

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Zero and Infinity Today

Routine use in Mathematics (e.g., Calculus) Science Engineering

All problems resolved?

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A Little More Math…Where’s The Problem?

2

2 2

2 2 2

1

( )( ) ( )But we started with 1! What happened?

0

a bb aba aa b a aba b a b a a b a b a b

b

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USS Yorktown (1997)

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Thanks for Your Attention

Questions?

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Extra

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Descartes 1596 1650

38Still a confounder for me.