Transcript Sifer, Cipher, Zero

Sifer, Cipher,
Zero
Melinda DeWald
Kerry Barrett
“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
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.
1st Major Role of Zero:
As 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!
2nd Major Role of Zero:
A 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 zero’s 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.
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.
Zero’s 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.
• Harriot’s Principle
– Setting an algebraic equation to zero.
• Find roots of: x2 + 2 = 3x
Zero, Descarte, and Geometry
• Descarte was working on coordinate
Geometry.
• Harriot’s 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.
Zero of a Ring or Field
• By 1700 A.D., mathematicians were
commonly utilizing zero in their work.
• By the 1800’s 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.)
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.
“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/HistTopics/Zero.html
What did the zero say
to the eight??
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.
Bibliography
• A History of Zero. Retrieved September 3,
2006, from http://www-groups.dcs.stand.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.