Outline Roman Numerals Roman Computation Summary

Introduction to Computing CSCI 1109

Alice E. Fischer

October 25, 2019

Introduction. . . 1/15 Outline Roman Numerals Roman Computation Summary

Roman Numerals

Roman Computation

Summary

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Numbers and Numerals From the dawn of history, people used numbers. I Numbers. not numerals. They had varied representations. I The most primitive cultures did not count far . . . they might have said one, two, three, many. I By 43,000 years ago, people were inscribing notches on bones –examples include one with 29 notches and many with smaller counts. I Numerals were invented later as symbols to represent numbers, the the numeral systems gradually became more sophisticated. I One number might be represented by different numerals, even within the same system. For example, these numerals all represent “one half”: 1/2, 2/4, 3/6 etc.

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History

Rome: ∼750 BC until fall in ∼400 AD I Roman numerals arose from Etruscan numerals and were used during this entire history. I They began in the 9th and 8th century BC I Final decline: started in 300 AD : civil war, plague, apathy, and the rise of Christianity in 2nd Century AD Europe: Roman numerals were used everywhere until replaced by Arabic numerals in the 11th century AD. Today: They are used for ceremonial and decorative purposes, for outlines, page numbers, etc.

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Roman Numerals Humans have 2 hands with five fingers each. Counting in base 10 developed naturally from our hands. I The numeral I represents one finger. I II represents two fingers, and so on. I V represents a whole hand, with all fingers and thumb showing. I X represents two whole hands, with all fingers and thumb showing. I C and M are for Century (100 years) and Millenium (1000 years). I L and D were added later, as simplifications of older number symbols.

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Roman and Chinese Numerals Roman numerals are a combination of base-10 numbers and unary notation. Here is one of the earlier systems, without the shortcuts for 4 and 9: 1 2 3 4 5 6 7 8 9 10 I II III IIII V VI VII VIII VIIII X XI XII XIII XIIII ... XX L C D M 11 12 13 14 ... 20 50 100 500 1000

1 2 3 4 5 6 7 8 9 10

Note the similarities in the symbols for 1, 2, 3, and 10, between Roman and Chinese numerals.

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Strengths of these Systems

I These numbers are simple and natural and versions have been used since the dawn of writing. I Straight lines are easy to make on clay tablets and can be engraved in stone. I Addition and subtraction are easy. I The primary uses were financial accounting and counting soldiers or enemies.

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Weaknesses of these Systems

I How do you represent fractions? I How do you multiply or divide? I It takes a lot of writing to write a number in the thousands! I They do not scale. How do you represent the number of stars in the universe? (Estimated to be about 3 × 1023

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Adding Roman Numerals

Suppose a rich man owns 781 sheep and buys 38 more. How many sheep does he have: I Merge the symbols in the two numbers, then simplify if needed. I DCCLXXXI + XXXVIII = DCCLXXXXXXVIIII = DCCLLXVIIII = DCCCXVIIII = 500+300+10+5+4 = 819 I Scratch paper or a word processor certainly helps!

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Multiply by two.

Compute 81 × 2 I L XXX I × II I Rewrite the number, doubling every symbol in it. I LL XXX XXX II I Simplify. I C LX II I = 162

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Divide by two.

Compute 47 / 2 I XXXXVII / 2 I Starting at the left, I see 4 X’s so I write 2 X’s. If the number of X’s is odd, I write a V instead of the last one. I The first part of the answer is: XX . . . I Now I see one V, so I split it into five I’s to combine with the two that are already there. I I can find half of 7 in my head. The answer is III and I drop the remainder. I So XXXXVII / 2 = XX III = 23

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Full multiplication is more of a challenge. Eventually, the Romans figured out how to multiply any two numbers with a do-able amount of work. The algorithm is fun and is related to binary numbers. To compute a × b : I Make three columns on your paper. I Write the larger number at the top of the left column, the smaller number on the right. I Repeat until the smaller number is 1, write a new line of numbers each time: I If the number on the right is odd, write 1 in the middle column. I Double the number in the left column (on a new row). I Halve the number on the right (on a new row). I Add up all the numbers in the first column that are next to 1’s in the middle column. The sum is your answer.

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Compute 37 * 29 using Arabic Numerals & Roman Method.

Larger Odd? Smaller 37 1 29 74 0 14 148 1 7 296 1 3 592 1 1

Now add 37 + 148 + 296 + 592 = 1073 Using ordinary multiplication, 37 × 29 = 1073 These are equal, so the algorithm works.

Note: the binary representation of 29 is 11101 .

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Compute 37 * 29 using Roman Numerals. Larger Odd? Smaller XXXVII 1 XXVIIII LXX1111 0 XIIII CXXXXVIII 1 VII CCLXXXXVI 1 111 DLXXXXII 1 1

Now add: XXXVII + CXXXXVIII + CCLXXXXVI + DLXXXXII = D CCC LL XXXXX XXXXX XXXXX VVV IIIII III = D CCCC LLL X VV III = D CCCCC L XX III = DDLXXIII = 1073, which is the correct answer.

So it works with Roman numerals also. But BOY is it error prone and a lot of work! Introduction. . . 14/15 Outline Roman Numerals Roman Computation Summary

Summary and Homework

Summary: 1. Vocabulary: numeral, 2. Perform simple arithmetic using Roman numerals. 3. The history of numerals through the middle ages. Homework: Unit 13, Due Oct. 30 Be prepared to talk about your work on November 6.

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