Chapter 2 Reading and Writing Arithmetic Presented by Lucas Mellinger MAT 400

Activity Write a solution to the following equation without using any arithmetic symbols: Write a solution to the following equation without using any arithmetic symbols: 1½x + 4 = 10 1½x + 4 = 10

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.” “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.” - A’h-mose Moscow Mathematical Papyrus

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

Math in Ancient Egypt Two papyri containing collections of mathematical problems with their solutions written by the scribe A’h-mose in 1650 BCE: Two papyri containing collections of mathematical problems with their solutions written by the scribe A’h-mose in 1650 BCE: Rhind Mathematical Papyrus Rhind Mathematical Papyrus Moscow Mathematical Papyrus Moscow Mathematical Papyrus

Math in Ancient Egypt Addition and Subtraction: Addition and Subtraction: Hieroglyphic Grouping Hieroglyphic Grouping Multiplication and Division: Multiplication and Division: Doubling process expressed in a table Doubling process expressed in a table Example: multiply 12 by 13 Example: multiply 12 by 13 ‘ ‘448 ‘896

Activity Express the following without using any arithmetic symbols: Express the following without using any arithmetic symbols: (5+6) – 7 = 4

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

Possible Solutions 1470’s Written shorthand is the earliest form of arithmetic symbols. et is “and” in Latin

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”

Possible Solutions 1494 Symbols used in Luca Pacioli’s Summa de Aritmetica. Widely used throughout Europe

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.

Possible Solutions 1629 Widely used as subtraction in the 17 th and 18 th centuries. (5+6) 7 = 4

Possible Solutions 1631 Clavis Mathematicae by William Oughtred used colons to emphasis a grouping of terms. Also in the same year were first used as “less than” and “greater than.”

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

History of Arithmetic Symbols Early 1700’s – 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. Early 1700’s – 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.

Other Important Dates 9 th and 10 th Centuries: Indian manuscripts first began placing quantities to be multiplied next to one another. 9 th and 10 th 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. 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.” 1417: Manuscript uses the plus symbol as the abbreviation for the Latin “et.” 1600’s: X symbol appears for multiplication in Europe 1600’s: X symbol appears for multiplication in Europe 1698: Leibniz introduces the raised dot for multiplication to avoid confusion between X and the variable x 1698: Leibniz introduces the raised dot for multiplication to avoid confusion between X and the variable x

References Berlinghoff and Gouvea Berlinghoff and Gouvea Cajori, Florian. A History of Mathematical Notations. Dover Publications, New York, Cajori, Florian. A History of Mathematical Notations. Dover Publications, New York, Katz, Victor J., A History of Mathematics, Pearson/ Addison Wesley, 2004 Katz, Victor J., A History of Mathematics, Pearson/ Addison Wesley, 2004 Parkinson, Claire L., Breakthroughs: a chronology of great achievements in science and mathematics, G. K. Hall, Boston, Parkinson, Claire L., Breakthroughs: a chronology of great achievements in science and mathematics, G. K. Hall, Boston, 1985.

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

1600’sX symbol appears for multiplication in Europe and German manuscripts begin using * for multiplication. 1600’sX symbol appears for multiplication in Europe and German manuscripts begin using * for multiplication. 1629Our current division sign appears as a minus sign. 1629Our current division sign appears as a minus sign. 1631Colons used to group terms in a mathematical equation. 1631Colons used to group terms in a mathematical equation. 1637Descartes uses odd new symbol for equality. 1637Descartes uses odd new symbol for equality. 1698Leibniz introduces raised dot a symbol for multiplication. 1698Leibniz introduces raised dot a symbol for multiplication. 1700’sAcceptance of our current mathematical symbols. 1700’sAcceptance of our current mathematical symbols. Timeline for the progression of arithmetic symbols