Romans
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It has long been recognized that there was no ancient “Roman Mathematics”.
There were original mathematicians who worked in the Roman empire, primarily in
Alexandria, but all of them were part of the continuing Greek tradition.
The Roman imperial government believed that mathematical research was not an
important national interest and therefore did not support it.
Few Greek scholars were invited to teach mathematics to the children of the elite. Soon,
no one in Rome could even understand, let alone extend, mathematical works, such as
Euclid’s.
The Greek tradition did continue for several centuries under the Roman government in
Egypt, particularly because of the Alexandrian Library, but soon after its destruction in
the late 4th century, Greek mathematics ceased to be.
It is believed that around 200 B.C. the Romans developed a system of numeration.
Today, we call this system “Roman Numerals”.
The dominance of Europe by the Roman Empire from about the first century B.C. to the
fifth century A.D. made Roman numeration the commonly accepted European way of
writing numbers for centuries afterwards, even into the Renaissance (14th-17th centuries).
Roman Numerals
1
I
5
V
10
X
50
L
100
C
500
D
1000
M
5000
10,000
50,000
100,000
500,000
1,000,000
V
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L
C
D
M
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Note, a bar was used for numbers larger than 1000 to mean multiplication by 1000.
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Double bars were used to represent multiplication by 1000 twice. So, for example, L =
50,000,000.
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In this system, the value of the numeral was originally obtained by adding the value of
each symbol.
Later, Roman numerals also included subtraction, for efficiency purposes.
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was used with numeral written with 4’s and 9’s – that is, 4, 9, 40, 90, 400, 900, and so on.
In other words, only symbols representing powers of ten could be subtracted, and they
only may be paired with the next two larger values. This was to avoid ambiguity.
By this method, not more than three adjacent copies of the same basic symbol are needed
in any numeral.
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Try translating the following Roman numerals to our number system:
MMCDLXXXIV
D M V CCLVII
DCCXXIV
"L MMCDLVII
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Translate the following into Roman numerals:
2007
101,900
2,893,429
12,400,359
Fractions
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The Romans didn't have a standard way to write fractions using their numerals.
Instead, they just wrote out the word for the fraction: for example, two-sevenths was
"duae septimae" and three-eighths was "tres octavae."
The Romans did most of their practical calculations with fractions by using the uncia.
The uncia started out as 1/12 of the as, a unit of weight (the word uncia is related to our
word "ounce"), but it soon came to mean 1/12 of anything.
You can add up twelfths to make halves, thirds, or quarters, so the uncia was fairly
versatile. When they wanted smaller fractions, the Romans usually cut the uncia into
smaller parts.
The system is very similar to measuring length in inches and fractions of the inch: you
might not measure an object's length exactly, but you can still come very close.
There were Roman and medieval symbols for multiples of the uncia. However, uncia
symbols were never standardized, and not everybody used them. Some late medieval
writers even substituted the modern fraction bar.
Computation
Consider the following addition problem: 23 + 58. In Roman numerals, that is XXIII + LVIII.
• Begin by writing the two numbers next to each other: XXIII LVIII. Next, rearrange the
letters so that the numerals are in descending order: LXXVIIIIII.
• Now we have six I’s, so we can rewrite them as VI: LXXVVI. The two Vs are the same
as an X, so we simplify again and get LXXXI, or 81, as our final answer.
Now let’s try another addition problem: 14 + 17, or XIV + XVII.
• Notice that the I in XIV is being subtracted, so this problem is going to be a little more
complicated.
• We begin the way we did before, by writing the numbers side by side: XIV XVII.
• The subtracted I in XIV cancels out another I, so we cross them both out: X I V XVI I .
• Next we put the remaining letters into the right order: XXVVI.
• Simplifying gives us XXXI, or 31.
There are similar methods for subtracting, multiplying, and even dividing Roman numerals.
However, they can be frustrating, and for a good reason. Even the Romans did not use them.
When they wanted to do complicated arithmetic problems, the Romans used a special counting
board or an abacus.
Add the following numbers:
XLIII + XXVIII
CMXIV + DLV
Subtract the following numbers:
LXV – VII
CMXIV – DLV