Survey of Math - MAT 140
Page: 1
Now to start with, there is a distinct different between Numbers (quantity) and Numerals (symbols that represent a quantity). Numbers are the abstract way that we understand "how many?" where Numerals are symbols in which we communicate these numbers, usually written but sometimes just the use of hands.
Now a Numeration System is a combination of Symbols to represent the Numbers and Rules on how to use the
Symbols. One of the earliest Numeration Systems was the Egyptian System. They used Symbols that represented
1's, 10's, 100's,..., and 1,000,000's.
Now to discuss the Egyptian method, we will also have to introduce the Hindu-Arabic System. This is the one that you are most familiar with, that is because this is the one that we use. I will call it H-A for short.
H A 1 10 100 1 ; 000 10 ; 000 100 ; 000 1 ; 000 ; 000
Egyptian
Now these symbols are a little different than the books, especially the Frog. The book uses a Tadpole, but I think it looks like the Christian Fish. I wonder if this symbol was used when the Romans were persecuting Christians and they needed a symbol that wouldn't arouse suspicion and used the Egyptian Numeral for 100,000???
To use this system we only need to add enough symbols to "make" the number up, additively.
Example 1 Write the number 235 in Egyptian:
H A 2
Hundreds
3
T ens
5
Ones
Spacing is unnecessary, and so is order, though it is a polite way of writing the Number. They could also be Grouped:
Example 2 What H-A number is this Egyptian number?
Copyright 2007 by Tom Killoran

Survey of Math - MAT 140 Page: 2
There is one "Fish" = 1,000,000
There are 7 "Fingers" = 7 10 ; 000
D
70 ; 000
There are 3 "Plants" = 3 1000
D
3000
There are 5 "Loopies" = 5 100
D
500
There are 9 "Staffs" = 9 1
D
9
So:
1 ; 000 ; 000
70 ; 000
3 ; 000
500
C
::::::::: 9
1 ; 073 ; 509
The next one is one of the more popular ones to teach in Elementary School. Its usefulness is to gure out what time it is on Big Ben, locating pages that are before Page 1, dating old movies, and teaching us a little about where we come from and why we do things the way we do.
The Roman's added a little more to the Egyptian System, by putting in a notation with subtraction. With this trick it would be unnecessary to write 99 with 18 symbols in Egyptian. Also the Symbols were reduced to something any laymen could pen down, and not an artistic drawing.
H A 1 5 10 50 100 500 1000
Roman I V X L C D M
1. One cannot write more than Three Symbols in a row (it causes eyestrain)
2. To add one must write a larger symbol in front of a smaller symbol C L X
D
160
3. To subtract one must write a smaller symbol in front of a larger symbol, BUT it cannot be more than one logical
"place" away from the larger symbol. Example: place away.
C D
D
900 is ok but L D
6D
LC
D
50 is impractical so XC
D
90 would be one logical
450 is not, because the L is two logical places away.
4. To get a number larger than M (since the Roman Empire surely collected taxes in the millions) a product by a
1000 is formed by putting a bar on top, N is 5 1000
D
5000 and N
D
100 1000
D
100 ; 000
This last one is why we are taught wrong in Elementary School. They teach that there is a line above and below
Roman Numerals, when in fact those would be 1000 times larger!!!
X
D
10 and X
D
10 ; 000 :
Copyright 2007 by Tom Killoran

Survey of Math - MAT 140
Example 3 Write the Number 3745 in Roman Numerals:
3000
D
700
D
40
D
5
D
M M M
DCC
X L
V
M M M DCC X LV
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Example 4 Write the number 7 ; 894 in Roman Numerals
7 ; 000
800
4
D
90
D
D
D
V M M
DCCC
XC
I V
V M M DCCC XC I V
Example 5 What number in H-A is this Roman Numeral? M MC DL X X I V
M M
C D
L X X
I V
D
1000
C
1000
D
2000
D
D
D
500 100
50
C
5 1
10
D
C
4
D
10
400
D
70
2474
An ancient system that got extremely close to a modern place value system was the Chinese Numeration System.
They had symbols that represented the 10's place, 100's place, and 1000's place (they had higher places but this book doesn't cover them, much harder to draw!). And they had the digits well covered, except you will notice they had no symbol for zero!
Chinese Numeration System-Multiplicative
H A 1 2 3 4 5 6 7 8 9 10 100 1000
Chinese
The Chinese wrote their Numerals Vertically, with the Largest Value up high...
Copyright 2007 by Tom Killoran

Survey of Math - MAT 140
Example 6 Write the Number 5 ; 792 in Chinese
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Example 7 What H-A number is represented by this Chinese Numeral?
6 .
1000 /
C
4 .
10 /
C
8
D
6048
Another way of doing a simple numeration system was used by the Greeks. This system is in the right direction as our place value system, but it lacks a place value and there are so many Numerals that one is hard pressed to remember them all. The Greek Numeral System uses their alphabet and if that wasn't enough they had to borrow a few symbols from other cultures! This system is long because it not only contains a symbol for 1-9 but also 10-90 and 100-900.
That is a system with 27 symbols! They are in the book so I will not reiterate them here. All I will say that the Fonts of Greek letters differ by a little, just like our letters sometimes look different in different Fonts. So if it looks like it, it probably is. Also there is a small hint: All Greek Numerals are written from Largest to Smallest.
Example 8 Write the Number 847 in Greek:
!
800 !
40
7
To write numbers larger than 900 ,"sampi", the multiplication by a 1000 trick is used. In the Greek system though it is a apostrophe that is used.
0
D
2 1000
D
2000
Example 9 What number is represented by the Greek Numeral?
0 0
80 ; 000
C
88 ; 603
8000
C
600
C
3
Copyright 2007 by Tom Killoran