Transcript Sketch 1 - Your Space

Sketch 1
Keeping Count
Writing Whole Numbers
Early Counting
• Tally Bones
• Bodily Mathematics
• Finger Counting
• Number Words & Body Parts
Egyptian Mathematics
Hieroglyphics (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.
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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:
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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