In a Roman state of mind. If someone mentions Ancient Rome, what are some of the things you think of?

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2 In a Roman state of mind If someone mentions Ancient Rome, what are some of the things you think of?

3 Properties of Roman Numerals Values of symbols: I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1000 groups of 5 were added by the Etruscans to shorten the amount of symbols needed to represent a quantity Positional value of symbol depends on its position in the string (XI does not equal IX ) Additive symbol of lower value on right is added to value of symbol on its left (VI = 5+1=6) Subtractive symbol of lower value on left is subtracted from value of symbol on its right (IV = 5-1 =4) Multiplicative a bar over a symbol multiplies it by 1,000

4 Intro to Roman Numerals Activity: Take a few minutes and write down answers to the following addition problems: 1) MLXXXII + MDCCXIV 2) LXII + CDVIII I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1000

5 Arithmetic with Roman Numerals Carly DeSalvo

6 Objectives: Warm up activity What is an abacus? History What can an abacus do? Some applications Examples of arithmetic on a counting board Intro of Hindu numerals Abacists vs. Algorists Conclusions Done!

7 What is an abacus? abacus comes from the Greek word abax which is interpreted as flat table or board A manual computing device consisting of counters arranged in columns or rows (different cultures have their own version) -Counters below the center divider are units -Counters above the center divider are fives -Columns represent place value increasing from right to left abacus applet

8 Some History Evidence of abacus in ancient Greece/Rome: In 7 th century B.C. Solon, the great law-giver of Athens, compared a tyrant s favorite to a counter whose worth depended entirely on the whim of the person who pushed it from one column to a another. the Salamis Tablet (date unknown) the only preserved counting board of ancient Greece tool for reckoning money

9 What can an abacus do? Perform computations - addition, subtraction, multiplication, division, square roots Simplify calculations for number systems that do not rely on place value (example: Roman numerals) place value - the position of the number represents a power of the base (for base 10: 246 is 2*10^2 + 4*10^1 + 6*10^0) Roman numerals: C = 100, CCC = = 300 position does not relate to a power of a base number Remove ambiguity in positional number systems without a written zero (example: six hundred two & sixty-two both look like 6 2 ) abacus applet

10 Uses in ancient Rome Reckoning of money Teach arithmetic in schools abacus applet Counting board type of abacus pebbles in columns Throughout the middle ages some form of an abacus was used in schools, monasteries, royal treasuries, in the offices of town officials, and in the counting rooms of merchants

11 Arithmetic with Roman Numerals Multiplication Rules: Basic shift shift the multiplicand pattern until the units position falls under the multiplier character Etruscan shift same as basic shift, but the tally is written twice, plus once more in the column to the right Negative rule 1) negative is represented with a primed T and located in its normal column 2) partial product of an unprimed multiplier character is shifted and written without change 3) partial product of a primed multiplier character is shifted and written with primes added to the unprimed T s and with the primes removed from the primed T s

12 Impact on mathematical concepts: Counting board makes addition and subtraction simple and quick just add up the counters! No need for a zero because an empty column represents no value Multiplication able to perform without knowledge of multiplication tables but time consuming and inefficient compared to today s operations.

13 Introduction to Hindu numerals Gerbert (later became Pope Sylvester II) Learned about Hindu numerals in Spain in 967 brought the 9 Hindu numerals to Rome a few years later (without zero) Replaced counters of abacus with pieces of horn called apices, each carved with a different Hindu numeral believed to have made calculations more tedious because to add had to replace with 1 and 4 apices instead of just grouping 14 counters in a single column He had good intentions, just did not fully understand the concept of computing with Hindu numerals

14 Hindu Numerals continued 1240 Johannes de Sacrobosco introduced Western Europe to Hindu numerals, including zero, and their use in arithmetic computation his work became known as Algorismus Leonardo of Pisa (Fibonacci) wrote Liber Abaci in 1202 his book embodied all the numerical knowledge of his time interpreted with Hindu numerals. - purpose was to teach the Italians the Hindu number system and its operations City Council of Florence outlawed use of Hindu numerals in accounting records had to use Roman numerals and write values out in words (like modern day checks!) Protection of fraud

15 Abacists vs. algorists Abacists preferred computations on the abacus Algorists preferred pen and paper computations using Hindu numerals Controversy: Italians did not want to accept this new system. They did not realize how much easier calculations would be. Paper was expensive and they found the calculations more tedious than the traditional abacus process. Finally in the 16 th century when cheaper, disposable paper was introduced, the Italians fully adopted the method of the algorists.

16 Conclusions Abacus was the primary tool used for performing calculations for at least twenty centuries, and probably more. We no longer involve use of an abacus in our mathematical computations, but we should understand that it was a major stepping stone in the conception of place value and our numeral system.

17 The End!