Sifer, Cipher, Zero

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Sifer, Cipher, Zero

• Melinda DeWald
• Kerry Barrett

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• It needed one of those strokes of genius which
  we now take for granted to come up with a way of
  representing numbers that would let you calculate
  gracefully with them and the puzzling zero
• which stood for no number at all was the
  brilliant finishing touch to this invention.
• -Robert Kaplan

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Babylonian Number System

• The number system consists of two different
  symbols.
• It is a base 10 system for the digits up to 59.
• It is a base 60 system for larger numbers.
• By 1600 B.C. Babylonians had a well developed
  place value system.

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1st Major Role of ZeroAs a Place Holder

• Zero originated in the Babylonian system where
  they used an end-of-sentence symbol to make clear
  the number of spaces. (700-300 B.C.)
• In the Hindu number system the zero appeared as a
  small circle to serve as a place-holder. (600
  A.D.)
• The Arabs spread this idea of zero through
  Europe.
• Note Zero as we know it still does not
  exist!

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2nd Major Role of ZeroA Number Itself

• By 800 A.D., the Hindus had begun to recognize
  nothing as something. They began to treat zero
  as a number.
• Mathematicians began investigating zeros
  properties.
• Mahavira stated that a number multiplied by zero
  is zero and zero added or subtracted to or from a
  number results in the original number.
• Bhaskara found that a number divided by zero is
  an infinite quantity.
• Note It does not matter who claimed what. It
  only matters that they were finally using zero.

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Zero as an Abstract Concept

• Before this point people would count by using
  objects to represent numbers.
• Counting animals 1, 2, 3, 4
• People had to think of numbers as an abstract
  concept that remained unchanged regardless of
  what was being counted.
• The Hindus recognition of zero and all numbers as
  abstract concepts paved the way for algebra.

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Zeros Role in Algebra

• The Hindu idea to treat zero as a number took a
  long time to take root in Europe.
• Around 1600 Thomas Harriot and Descartes used
  this concept to change systems of equations as
  Europeans knew them.
• Harriots Principle
• Setting an algebraic equation to zero.
• Find roots of x2 2 3x

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Zero, Descarte, and Geometry

• Descarte was working on coordinate Geometry.
• Harriots Principle allowed Descarte to easily
  determine where the function would cross the
  x-axis.
• This allowed him to approximate roots in
  equations that were not easily factored.

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Zero of a Ring or Field

• By 1700 A.D., mathematicians were commonly
  utilizing zero in their work.
• By the 1800s zero gained prominence in abstract
  algebra.
• It was the basis for the Additive Identity. (i.e.
  the zero of a ring/field)
• It was also the driving force behind a special
  property of an integral domain. (If a product of
  numbers is zero then one of the numbers must be
  zero.)

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All Wrapped Up

• Review writing numbers in Babylonian times versus
  today.
• The zero has simplified our number system.
• Without the zero in our number system we would
  never have made discoveries in algebra, geometry,
  and all other areas of math.

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• Zero makes shadowy appearances only to vanish
  again almost as if mathematicians were searching
  for it yet did not recognize its fundamental
  significance even when they saw it.
• http//www-groups.dcs.st-and.ac.uk/history/HistTo
  pics/Zero.html

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• What did the zero say to the eight??

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Timeline

• 700-300 B.C. The Babylonians used an
  end-of-sentence symbol (say a dot) as a
  placeholder.
• 600 A.D. The Hindus used a small circle as a
  placeholder when writing numbers where zero would
  later appear.
• 800 A.D. The Hindus began treating zero as a
  number.
• 1600 A.D. Thomas Harriot and Descarte treated
  zero as a number in their own work, and in doing
  so revolutionized systems of equations in Europe.
  
• 1700 A.D. Mathematicians were commonly utilizing
  zero in their work.
• 1800 A.D. Zero gained prominence in abstract
  algebra.

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Bibliography

• A History of Zero. Retrieved September 3, 2006,
  from http//www-groups.dcs.st- and.ac.uk/history
  /HistTopics/Zero.html
• Kaplan, Robert (2000). The Nothing That Is A
  Natural History of Zero. Oxford University
  Press.
• Katz, A History of Mathematics, Brief Edition,
  2004.