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.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.