by: Stefanos Gialamas, Ph.D., DeVry University
Oakbrook Terrace, IL 60181
Spectrum, The Illinois Mathematics Teacher Journal, Summer 2002
The purpose of this article is to briefly introduce three numerical systems used in ancient Greece and to engage the reader in activities using these systems to solve problems and finally develop his/her own numerical system. Therefore this historical journey with the activities will provide the reader with an appreciation for and an understanding of the arduous path mathematics has traveled during the centuries.
The Herodianic System of Numerical Notation
The Herodianic system was named for Herodianus, the Byzantine grammarian who introduced the system to his contemporaries in 200 A.D. Variations of this system were used in areas in which the Athenian influence was present. The Herodianic system was used as early as 600 B.C., during the reign of Solon, and continued to be utilized for several centuries. Archaeological evidence suggests that these variations on the Herodianic system of numerical notation were in use as late as 40 B.C., during the time of Cicero.
The Herodianic system was largely based on the use of abbreviations . Perhaps the most interesting aspect of this system is how it was created. In this system the number 5 and the first five powers of 10 (100 = 1, 101 =10, 102= 100, 10 3 =1000, 104 =10,000) received special symbols; all other numbers were formed by the creation of compound symbols. The following were the 10 essential symbols of the Herodianic system.
The introduction of deductive reasoning to mathematical thinking resulted in the creation of compound symbols. These symbols were created by the concurrent use of fundamental symbols in a combination format, hence
In abstract form there were no symbols for fractions and complex operations. The desired operations were described by using the language. As negative numbers were considered “forbidden,” there was no need for signed numbers.
The Boeotian System of Numerical Notation
Outside of Attica, in the part of Greece called Boeotia, a variation of the Herodianic system was in use. This system was developed as a result of the increasingly hostile relations between the various city states of ancient Greece. Essentially, the Herodianic principles were applied to the creation of fundamental symbols; these symbols where then compounded . Those fundamental symbols were:
The similarities between the Attic and Boeotian systems unfortunately doomed both systems as the ancient Greek society became increasingly sophisticated.
The Alphabetic Numerical System
I believe that the Greeks, more than any other people of antiquity, possessed an unbounded love of knowledge for its own sake. Their philosophers traveled throughout the known world in order to gather and benefit from all the wisdom that other nations, with longer histories, had accumulated over the centuries. One culture with which the Greeks were very familiar as the Phoenician. It seemed quite natural that the Greeks should create their own alphabet by borrowing, modifying, improving, and extending the Phoenician alphabet. By 600 B.C., the Greek alphabet had twenty-seven letters (3 of which are now obsolete). This alphabet was quickly transformed into what is known as the Ionic numeration system. The use of letters as numerals was an application unique to the ancient Greeks. The use of the letters of the alphabet as numerals was original with Greeks; they did not derive it from the Phoenicians, who never used their alphabet for numerical purposes but had separate signs for numbers (3). In fact, the earliest archaeological evidence of the use of numerals in conjunction with an alphabet is a Halicarnassion inscription dated at approximately 450 B.C.
The alphabetic numerical system was specifically created to fulfill the increasing need for sophisticated mathematical functions. The twenty-seven basic symbols are listed in a table on the following page.
To distinguish between numbers and letters, a horizontal line was drawn over the numbers. For instance, the number 3,542 = 3,000 + 500 + 40 + 2 can be written in this system as, γ φ μ β.
Despite the sophistication of the alphabetic system, there was no uniform notation of fractions; therefore, most Greek writers expressed fractional values in words. When fractions were expressed in symbols, they were generally denoted by first writing the numerator and marking it with an accent; the denominator was then marked with two accents and written twice . For example, the fraction ½ can be written as α’ β’’ β’’. In spite of the seemingly weak alphabetic system, many fundamental elements of modem mathematics were developed within the confines of this seemingly inadequate system.
Activity 1: The Farmer
You’re an ancient Greek fanner who grew a crop of 450 bushels of com this year, which is 40 bushels of com more than you produced last year and twice as many bushels of barley than you produced this year. You hope to increase production of both com and barley next year by 35 bushels each.
Problem 1: Using Herodianic symbols, record the number of bushels of barley produced this year and the amount you’d like to produce next year.
Problem 2: Using Alphabetic symbols, record this year’s crop of both barley and com.
Problem 3: If each bushel of barley costs 40 drachmas (Greek currency) and each bushel of corn costs 30 drachmas, find your total crop value for this year in both the Herodianic and Alphabetic systems’ notation.
Activity 2: Develop Your Own Numerical System
Step 1: Decide which base you’ll use for your numerical system. For this example the base is 3. The powers of the base are 30=1, 31=3, 32=9, 33=27, 34=81, …
Step 2: Choose a form of notation to represent these powers. For our example, we chose:
Step 3: Determine whether the symbol will follow (or proceed) the numerical value. In this example the symbols precede the numerical value.
Step 4: Define the procedure for addition. We chose juxtaposition for our example.
*Students might also want to define procedures for denoting multiplication, fractions, and even a positional numerical system.
Sir Thomas Heath in his book, A History of Greek Mathematics Volume I, writes, “Most people, when they think of the
Greek genius, naturally call to mind its master pieces in literature and art with their notes of beauty, truth, freedom and humanism. But the Greek, with his insatiable desire to know the true meaning of everything in the universe and to be able to give a rational explanation of it, was just as irresistible driven to natural science, mathematics and exact reasoning in general or logic”. The Greeks were able to accomplish all of the above using numerical systems which were not easy to use.
Nevertheless the limitations of their numerical systems did not prevent them from contributing to the development of mathematical thought.
I would like to express my appreciation to Ms. Susan Busch, Executive Assistant at the Academic Affairs Department of DeVry University, for her assistance in the completion of this article, and to my son Panayiotis Gialamas for his valuable recommendations on improving the quality of this article.
Eves, Howard, An Introduction to the History of Mathematics, Holt, Rinehart, and Winston, 1969.
Gialamas, Stefanos, Zero the Exceptional Number, Consortium of Mathematics and its Applications, Spring 1990.
Heath, Sir Thomas, A History of Greek Mathematics, Vol. 1, Dover, 1981.
Answers to Activity 1
The value of the com crop is 450 x 30 =13,500.
The value of the barley crop is 225 x 40 =9,000.
If There Were No Rectangles
If we didn’t have rectangles, long wide or thin,
Take a moment to imagine what a state we’d be in.
Rectangles are plentiful. We use them a lot.
Two sides are longer, two sides are not.
No bricks to make buildings. No tape and no labels.
Rugs on floors, dresser drawers, dining room tables.
You’d need new ideas for new courts for most games,
And new shapes for flags, boxes, mirrors, and frames.
Say goodbye to tickets and playing card decks.
Credit cards, business cards, personal checks.
Without the rectangle, how would we cope?
There’d be no stairs to climb, just a slippery slope!
No reading or writing in books with no pages.
Expressing ourselves would be like the Stone Ages.
To send a kind note to make someone feel better,
Would take a new envelope, stamp, and a letter.
Different windows and doors for all of our houses,
And new shapes for the traps for all those smart mice.
So let’s keep all the rectangles, books, beds, and tape.
So our world can avoid being bent out of shape.
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