The Development of Numbers The Development of Numbers next Taking the Fear out of Math © Math As A...

The Development of Numbers

The Development of Numbers

next Taking the Fearout of Math

© Math As A Second Language All Rights Reserved

hieroglyphics

tally marks

Roman numerals

The Fall of Tally MarksThe Fall of Tally Marks

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and

The Rise of Roman Numerals

The Rise of Roman Numerals


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 andEgyptians 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© Math As A Second Language All Rights Reserved

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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Each X represents ten. Therefore, two X’s represents twenty; and since each I

represents one, we see that XXIII represents twenty three.

X

Hence, XXIII represents 20 + 3 or 23.

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X I I I

10 + 10 1 + 1 + 1+ or 23

With X and I as defined previously, what number is named by XXIII?


next 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.

1Remember that we judge progress by what the new concept replaces not by what it was later replaced by.

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Notice also the new level of abstraction.

Note

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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nextFor 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.

Notes

However, it would then have represented two 5’s and three 1’s or 13. 3

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.

notes

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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.

They replaced ten X’s by the letter C.

X X X X X X X X X X = C

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The letters were not chosen at random. The letter ‘C’ is the first letter of the Latin

word centum that means a hundred.


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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In the Real World

C C C C C C C C C C = MThey replaced ten C’s by the letter M.

nextnextnext 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.

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.


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

mills in a dollar4.

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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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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.

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M M M C XX I I I I

3,000 100 20 4+ + + = 3,124

What number is named by the Roman numeral MMMCXXIIII?


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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!

Note


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

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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.


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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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C is the Roman numeral for 100, and there are 3 C’s.

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C C C I X

300 9+ = 309

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.

Find the number represented by the modern Roman numeral CCCIX.


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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.

hieroglyphicshieroglyphics

tally markstally marks

Roman NumeralsRoman Numerals

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