Chapter 2 Reading and Writing Arithmetic

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Chapter 2Reading and Writing Arithmetic

• Presented by Lucas Mellinger
• MAT 400

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Activity

• Write a solution to the following equation
  without using any arithmetic symbols
• 1½x 4 10

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One Possible Solution

• Calculate the excess of this 10 over 4. The
  result is 6. You operate on 1½ to find 1. The
  result is 2/3. You take 2/3 of this 6. The result
  is 4. Behold, 4 says it.
• - Ah-mose Moscow Mathematical Papyrus

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Earliest Civilizations and Mathematical
Computations

• Egypt and Mesopotamia
• First dynasty ruled Upper and Lower Egypt in 3100
  BCE.
• Algorithm vs. Theory
• Egyptians used mathematics as a set of
  instruction (architecture, agriculture, commerce,
  etc.).
• Greeks heavily stressed theory in their
  mathematics.

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Math in Ancient Egypt

• Two papyri containing collections of mathematical
  problems with their solutions written by the
  scribe Ah-mose in 1650 BCE
• Rhind Mathematical Papyrus
• Moscow Mathematical Papyrus

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Math in Ancient Egypt

• Addition and Subtraction
• Hieroglyphic Grouping
• Multiplication and Division
• Doubling process expressed in a table
• Example multiply 12 by 13
• 1 12
• 2 24
• 4 48
• 8 96

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Activity

• Express the following without using any
  arithmetic symbols
• (56) 7 4

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Possible Solutions
When 7 is subtracted from the sum of 5 and 6, the
result is 4.

• From the ancient Greeks, through the middle ages,
  until the early Renaissance, people wrote out
  their problems and solutions in words.
• Arithmetic symbols became more consistent with
  the invention of movable-type printing.

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Possible Solutions

• 1470s
• Written shorthand is the earliest form of
  arithmetic symbols.
• et is and in Latin

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Possible Solutions

• 1489
• Plus and minus signs are first used in a
  commercial arithmetic book by Johannes Widman
• was an abbreviation for and
• - denoted a separation
• das ist (German) means that is

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Possible Solutions

• 1494
• Symbols used in Luca Paciolis Summa de
  Aritmetica.
• Widely used throughout Europe

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Possible Solutions

• 1557
• First use of plus, minus, and equality symbols in
  English text book The Whetstone of Witte by
  Robert Recorde.
• No two things can be more equal than parallel
  lines of the same length.

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Possible Solutions
(56) 7 4

• 1629
• Widely used as subtraction in the 17th and 18th
  centuries.

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Possible Solutions

• 1631
• Clavis Mathematicae by William Oughtred used
  colons to emphasis a grouping of terms.
• Also in the same year lt and gt were first used as
  less than and greater than.

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Possible Solutions

• 1637
• Notation was used in Rene Descartes' La
  Geometrie.
• Used in Europe until the early 18th century.

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History of Arithmetic Symbols

• Early 1700s common notation emerged due to the
  influential writings of Leibniz, the Bernoullis,
  and Euler. The notation in their writing has
  been used ever since, and is now the most
  recognized language of the world.

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Other Important Dates

• 9th and 10th Centuries Indian manuscripts first
  began placing quantities to be multiplied next to
  one another.
• 1356 Nicole d Oresme used a figure that looks
  like our current plus and minus signs in a
  manuscript called Algorismus Proportionum.
• 1417 Manuscript uses the plus symbol as the
  abbreviation for the Latin et.
• 1600s X symbol appears for multiplication in
  Europe
• 1698 Leibniz introduces the raised dot for
  multiplication to avoid confusion between X and
  the variable x

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References

• Berlinghoff and Gouvea
• Cajori, Florian. A History of Mathematical
  Notations. Dover Publications, New York, 1993.
  http//members.aol.com/jeff570/mathsym.html
• Katz, Victor J., A History of Mathematics,
  Pearson/ Addison Wesley, 2004
• Parkinson, Claire L., Breakthroughs a chronology
  of great achievements in science and mathematics,
  1200-1930. G. K. Hall, Boston, 1985.

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Timeline for the progression of arithmetic symbols

• 1650 BCE Egyptian papyrus manuscripts containing
  a collection of mathematical problems and
  their solutions.
• BCE thru 1400s Write out problems and solutions
  in words.
• 800 900s Placing quantities to be multiplied
  next to one another.
• 1400s More consistency with mathematical
  terminology with the invention of
  movable-type printing.
• 1356 Possible use of current day plus and minus
  sign in a manuscript by Nicole d Oresme.
• 1417 Manuscript uses plus symbol as
  abbreviation for et.
• 1470s Shorthand symbols commonly used to
  replace Latin words.
• 1489 First use of plus and minus signs in
  print.
• 1494 p and m symbols used widely throughout
  Europe.
• 1557 First use of plus, minus, and equality
  signs in an English text book.

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Timeline for the progression of arithmetic symbols

• 1600s X symbol appears for multiplication in
  Europe and German manuscripts begin using
  for multiplication.
• 1629 Our current division sign appears as a
  minus sign.
• 1631 Colons used to group terms in a
  mathematical equation.
• 1637 Descartes uses odd new symbol for equality.
• 1698 Leibniz introduces raised dot a symbol for
  multiplication.
• 1700s Acceptance of our current mathematical
  symbols.