The Development of Numbers The Development of Numbers next Taking the Fear out of Math © Math As A Second Language All Rights Reserved hieroglyphics tally marks Roman numerals

The Fall of Tally Marks The Fall of Tally Marks next and The Rise of Roman Numerals The Rise of Roman Numerals © Math As A Second Language All Rights Reserved

Roman Numerals next © Math As A Second Language All Rights Reserved

After tally marks, the next evolutionary step in counting came from the Romans, and what we now call Roman numerals. next © Math As A Second Language All Rights Reserved

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. next © Math As A Second Language All Rights Reserved

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? next © Math As A Second Language All Rights Reserved

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. next = 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. next © Math As A Second Language All Rights Reserved

next 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 or 23. XIII or 23 With X and I as defined previously, what number is named by XXIII? © Math As A Second Language All Rights Reserved

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. 1 Remember that we judge progress by what the new concept replaces not by what it was later replaced by. note next © Math As A Second Language All Rights Reserved

next 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. next © Math As A Second Language All Rights Reserved

next 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. Notes However, it would then have represented two 5’s and three 1’s or This 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 next 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. next © Math As A Second Language All Rights Reserved

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 next The letters were not chosen at random. The letter ‘C’ is the first letter of the Latin word centum that means a hundred. © Math As A Second Language All Rights Reserved

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. next © Math As A Second Language All Rights Reserved In the Real World

C C C C C C C C C C = M They replaced ten C’s by the letter M. next 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. © Math As A Second Language All Rights Reserved

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 dollar 4. next 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 $ as “27 dollars and 346 mils”. This amount is worth more than $ but less than $ note © Math As A Second Language All Rights Reserved

next 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. MMM C XX II I I 3, = 3,124 What number is named by the Roman numeral MMMCXXIIII? © Math As A Second Language All Rights Reserved

next 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 © Math As A Second Language All Rights Reserved

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. next © Math As A Second Language All Rights Reserved

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. next © Math As A Second Language All Rights Reserved

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. next © Math As A Second Language All Rights Reserved

Thus, the Roman numeral IIII would also be written IV, and the numeral IIIIIIIII (or VIIII) would be written IX. 5 next 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. © Math As A Second Language All Rights Reserved next 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. note

next C is the Roman numeral for 100, and there are 3 C’s. CCCIX = 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. © Math As A Second Language All Rights Reserved

next 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. © Math As A Second Language All Rights Reserved

next What happened next is the subject of our next lesson. hieroglyphics tally marks Roman Numerals next plateau © Math As A Second Language All Rights Reserved