Arithmetic with Roman Numerals

In a Roman state of mind…
• If someone mentions Ancient Rome, what are some
of the things you think of?
Properties of Roman Numerals
Values of symbols:
• I=1
V=5
• X = 10 L = 50
• C = 100 D = 500
• M = 1000
•
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)
ƒ groups of 5 were added
by the Etruscans to
shorten the amount of
symbols needed to
represent a quantity
•
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
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
Arithmetic with Roman Numerals
Carly DeSalvo
Objectives:
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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!
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
Some History…
• Evidence of abacus in ancient
Greece/Rome:
– In 7th 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
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 = 100+100+100 = 300 Æ 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
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
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
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.
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 9 + 5 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
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.
• 1299 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
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 16th century when cheaper, disposable paper was
introduced, the Italians fully adopted the method of the
algorists.
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.
The End!