Transcript Roman Numerals next

Taking the Fear
out of Math
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The Development
of Numbers
Roman numerals
tally marks
hieroglyphics
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The Fall of
Tally Marks
and
The Rise of
Roman Numerals
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Roman
Numerals
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After tally marks, the next evolutionary
step in counting came from the
Romans, and what we now call Roman
numerals.
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As an historical aside, the Romans and
Egyptians invented quite similar systems
for enumeration.
However, because Roman, not Egyptian,
numerals are taught in the elementary
curriculum (at least in the Western World),
we discuss only the Roman version.
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Realizing that it is hard to keep track of
“too many” tally marks at a time and
wanting to take advantage of the fact that
we are born with ten fingers, when
counting tally marks the Romans decided
to cross them out in groups of ten.
As a “short cut” they came up with the
innovative idea that if you are going to
cross out groups of ten tally marks, why
write the tally marks in the first place?
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As a result, quite cleverly, they decided to
use the “crossing out” symbol by itself to
represent ten tally marks, and since the
“crossing out” symbol looked so much like
the letter ‘X’ in their alphabet; the Romans
eventually introduced the symbol (letter) X
to represent ten tally marks.
=
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X
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In short, the Romans used the symbol
X to replace | | | | | | | | | | .
In a similar way, since a tally mark
looked like the Latin letter ‘I’;
the letter ‘I’ became the symbol (numeral)
for representing a single tally mark.
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With X and I as defined previously,
what number is named by XXIII?
Each X represents ten. Therefore, two
X’s represents twenty; and since each I
represents one, we see that XXIII
represents twenty three.
X X
I I I
10 + 10 + 1 + 1 + 1 or 23
Hence, XXIII represents 20 + 3 or 23.
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While XXIII is more
cumbersome to write than
the numeral 23, it is still
more concise and easier to
visualize than
| | | | | | | | | | | | | | | | | | | | | | |.1
Note
Notice the use of the adjective/noun theme
here. For example, the nouns are I and X
and the adjectives are the number of times
each letter appears.
note
1Remember
that we judge progress by what the new concept replaces not
by what it was later replaced by.
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Note
Notice also the new level
of abstraction.
For example, when we look at XXIII it is
clear that there are two X’s and three I’s,
but unless we are told, there is no way that
we can guess what number is represented
by either X or I.
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Notes
For example, if the Romans
had decided to use
X to represent 5 tally marks 2,
XXIII would still consist of
two X’s and three I’s.
However, it would then have represented
two 5’s and three 1’s or 13. 3
notes
2This
most likely would have happened if the Romans had decided to use the number of
fingers on one hand rather than both hands to keep track of the number of tally marks.
3
What number we elect to have X represent leads to the concept of different number
bases and will be discussed in greater detail in a later presentation.
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Continuing in this way, the Romans
continued to use letters of their alphabet
whenever they exchanged ten of a
particular denomination.
XXXXXXXXXX =C
They replaced ten X’s by the letter C.
The letters were not chosen at random.
The letter ‘C’ is the first letter of the Latin
word centum that means a hundred.
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In the Real World
A century is a hundred years; per cent
means “for each (per) hundred;” there
are one hundred cents in a dollar; and
the leader of an elite group of (usually
100) Roman soldiers was called a
centurion.
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As the numbers they dealt with got
larger, the Romans continued the process
of choosing a new symbol (letter) whenever
they amassed 10 of the previous
denomination.
CCCCCCCCCC=M
They replaced ten C’s by the letter M.
Just as the letter C was not chosen at
random, neither was the letter M. More
specifically, the letter ‘M’ is the first letter of
the Latin word milla that means a thousand.
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A millennium is a thousand years; a meter
is a thousand millimeters; a gram is a
thousand milligrams; and there was once a
coin called a mill (that became obsolete due to
inflation). A mill was a tenth of a cent (just as a
cent is a tenth of a dime and a dime is a tenth
of a dollar). Since there are one hundred
cents in a dollar, there were one thousand
note
mills in a dollar4.
4 In
some states property taxes are assessed at a rate of “per hundred dollars.” Since the
average home is assessed at many thousands of dollars (rather than hundreds of dollars),
a fraction of a cent per hundred dollars can add up to a significant amount of tax money
for a state or municipality. For this reason, even though there is no longer a coin called
a mill, some states still use it in establishing a tax rate. We are all used to reading,
for example, $27.34 as “27 dollars and 34 cents” but not as comfortable with reading
$27.346 as “27 dollars and 346 mils”. This amount is worth more than $ 27.34 but
less than $27.35.
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What number is named by the
Roman numeral MMMCXXIIII?
In terms of our adjective/noun theme, we
see that there are 3 M’s, 1 C, 2 X’s and 4 I’s.
M represents 1,000; C, 100; X, 10; and I,1,
we see that we have 3 thousands,1 hundred,
2 tens, and 4 ones, which in place value
notation would be… 3,124.
M M M C XX I I II
3,000 + 100 + 20 +
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4 = 3,124
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Note
While it’s more cumbersome to write
MMMCXXIIII than to write 3,124; the
representation of this number as a
Roman numeral is still a gigantic
improvement over having to write
3,124 individual tally marks!
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In order to be able to write numbers
more compactly, the Romans were willing to
sacrifice the luxury afforded by “trading in
by tens” in order to invent “in between”
denominations.
For example, they used V to stand for five
(probably because just as the number five is
half of the number ten, the symbol V is half
(in fact, the upper half) of the symbol X.
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The Romans invented the symbols
V to denote five,
L to denote fifty, and
D to denote 500.
Thus, rather than write IIIIIII to denote
seven, the Roman numerals
representation would be written more
concisely as VII, and the number we
would write as 256 would be written as
CCLVI.
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If you have studied Roman numerals,
perhaps you learned to write IV rather than
IIII. In other words, the Romans later
introduced the idea that while VI would
mean 6, IV would means 4.
Thus, they invented what we might call
a “subtractive” concept.
The Romans decided that if a symbol
(numeral) was placed to the left of a symbol
that denoted a greater denomination, it
meant that the smaller denomination
should be subtracted from the
greater denomination.
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Thus, the Roman numeral IIII would
also be written IV, and the numeral
IIIIIIIII (or VIIII) would be written IX.5
The Romans used the symbols IV to
denote 4, IX to denote 9, XL to denote 40,
XC to denote 90, CD to denote 400, and
CM to denote 900.
note
5 Notice
that the idea of incorporating “subtraction” into the process of representing a
number in Roman numerals allows us to write numbers in an abridged form, but it runs
counter to reality. For example, if you have a $10-bill and a $1-bill in your wallet, they
represent $11 regardless of whether the $1-bill is placed on top of the $10-bill or below
it. The point is that the Romans did not do their arithmetic using Roman numerals (they
used the abacus). Roman numerals were used only for such things as counting (or for
numbering pages). In that context a sequence such as IX, X, XI caused no problems.
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Find the number represented by the
modern Roman numeral CCCIX.
C is the Roman numeral for 100, and there
are 3 C’s. However, while I is the Roman
numeral for 1 and X is the Roman numeral
for 10, the fact that I appears to the left of X
means that we subtract 1 from 10.
CCC
300
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I X
+
9
= 309
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However, in the original Roman numerals
(that is, the numerals as they existed before
the subtractive property was introduced)
CCCIX would mean 311.
As a more complicated illustration
MMCIIXIIXM and MMMCXXIIII both represent
3,124. However, it is much easier to read the
number when the symbols are grouped in the
form of MMMCXXIIII.
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What happened next is the subject of
our next lesson.
next plateau
Roman Numerals
tally marks
hieroglyphics
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