Sketch 1

Keeping Count

Writing Whole Numbers

Early Counting

• Tally Bones

• Bodily Mathematics

• Finger Counting

• Number Words & Body Parts

Egyptian MathematicsHieroglyphics (2700 BC)

Hieratic Script

Egyptian Mathematics

• What is important about Egyptian mathematics?

• Ancient Egyptian Mathematics

• Egyptian Fractions

• Egyptian Numbers

Egyptian Mathematics

• Materials for writing– Papyrus made of reeds– Leather– Cloth like cotton or linen– Stone

Egyptian Mathematics

• Rhind Papyrus– In the middle of the Hyksos period, a scribe

named Ahmose wrote a treatise on Egyptian mathematics

• Ahmose Papyrus or Rhind Papyrus (after the man who purchased it)

• Most comprehensive Egyptian document that still exists as a complete work

Egyptian Mathematics

• Rhind Papyrus– 18 ft by 13 inches– Probably written around 1650 BC– Ahmose claimed his work was based on the

work of an older manuscript that was never found

• May or may not have been Ahmose’s work• To make it more likely that a person’s work would

be copied in the future, one might claim it was the work of a great thinker of the past

Egyptian Mathematics

• Complete 387 + 245 using Egyptian hieroglyphics.

• Complete 437 – 159 using Egyptian hieroglyphics.

Egyptian Mathematics

• Multiplication– Multiplication and division were the first

mathematical operations described in the Rhind papyrus

– Not explained, but work was shown– Multiplication = repeated addition– Doubling and halving

Egyptian Mathematics

• Multiplication– Example: 27 x 34

Babylonian Mathematics

• Area between the Tigris and Euphrates rivers—modern day Iraq

• In about 3000 BC the Sumerians developed as a group of city states close to the Persian Gulf– Ur was the best know city state– Each city state was its own political entity,

which made it easy to conquer these

Babylonian Mathematics

• Eight different civilizations inhabited this area over the years of 3000 BC to about 300 BC when Alexander the Great died and the regions around the Fertile Crescent were ruled by general Seleucus.– Sumerians– Akkadians– Amorites– Hittites– Assyrians– Chaldeans– Persians– Seleucids

• We often inaccurately refer to all of this as “Babylonia.”

Babylonian Mathematics

• The Sumerians were the ones who invented the method of writing known as cuneiform or “wedge-shaped” by pressing a stylus into wet clay. They baked the tablets to preserve them.– More than ½ million clay tablets exist today.

• There was trade in Mesopotamia and the need for irrigation. Thus, much of their mathematics dealt with the digging of canals. The lack of isolation in Mesopotamia, vs Egypt, affected their mathematics.

Babylonian Mathematics

• Notation and Computation:– Two symbols, one for units and one for tens– Base 60 positional system—sexagesimal– What we will discuss are the symbols used

during just one time period. There were changing all the time and looked different at different times and in different places in Mesopotamia.

• Babylonian Mathematics

Babylonian Mathematics

• Write the following numbers in Babylonian, base 60, system.– 22– 84– 62– 614

• Write 3, 42, 31; in our number system, base 10.

Babylonian Mathematics

• What are some problems with the Babylonian number system?

Maya Mathematics• Two symbols, a dot for one and a line or bar for five (p. 67)• Maya Mathematics• Arranged vertically instead of horizontally and placed the place

value amounts in each group• The groups were:

– 1-19– 20s– 18 x 20– 18 x 202

– 18 x 203

– etc• Used a zero for an empty place value. • Due to their lack of contact with Europe, their numeration system

had no influence on European numeration.• Maya PowerPoint.

Maya Mathematics

• Try writing 43,274 using Mayan symbols

Roman Number System

• Roman numerals

• Used subtraction

• Larger numbers get a bar overtop of the number– For example, a V with a bar over it would be 5

x 1000 = 5000– A V with two bars over it would be 5 x 1000 x

1000 = 5,000,000

Greek Number System

• Greek Number Systems

• More on Greek Number Systems

• Used their Greek alphabet plus two/three other symbols (450 BC)– Nine symbols for the units, nine symbols for

the tens, and nine for the hundreds– Special mark used for numbers over 1000

Greek Number System

• Greek Number Systems• More on Greek Number Systems

– Used uppercase letters and then lowercase letters– Since they also used alphabet for words, they put a

bar over alphabet letters used to represent numbers– Used a large M, myriad or myrioi, for 10000 and then

put correct symbol over the M for a larger number.• For example, an M with the symbol for 4 over the M would be

40000.

Greek Number System

• 98,375 would be:

– written in lowercase Greek alphabet

– written in uppercase Greek alphabet

Current Number System

• Hindu Arabic Number System• Invented by the Hindus sometime before 600 AD

and refined over time• Picked up by Arabs during Islamic expansion

into India in 7th and 8th centuries • Europeans took it from the Arabs• Basic symbols are 0-9 and called digits• Roman number system existed for a long time

– Worried about changing 2 to 20 (theft)– Hard to compute with Roman numerals—they used

an abacus