Numeration Systems | Dr. Carlene Arceo

Numerations Systems

Dr. Carlene Arceo

UP Diliman

I am Carlene Arceo from the Institute of Mathematics of the University of the Philippines Diliman. Today, I would like to share with you what I know about numeration systems.

What are numeration systems? 

A numeration system, or numeral system, is a set of symbols used to represent numbers. Implicit in that definition is a distinction made between numbers and numerals. Many of us think that numbers and numerals are one and the same. They are not. A number is a concept, an idea, an abstraction, whose purpose is to measure and quantify. A numeral is a symbol for a number. 

Five is a number. As an idea, a concept, an abstraction, it can take on different meanings for different people. I like the number five because it is significant in many ways to me and my husband. I like how its Hindu-Arabic numeral looks and how it is uniquely drawn, ending with a bold upward stroke.  I like how its Roman numeral represents a “V”, just like a checkmark. Do you have any guess why it’s a “V”? The most natural way of counting is done using our fingers, right? And we have five fingers. Now, try raising your hand to show these five fingers. Can you see the “V”? It is said that the “V” comes from tracing the top of the thumb to the base of the palm, to the top of the little finger. 

Okay, so I like the number five for certain reasons. Imagine me teaching University of the Philippines students and declaring that I just love the number five. You could probably hear sharp intakes of breath or see eyes opening wider, jaws dropping, beads of sweat starting. Because in UP, a grade of 5 means you failed the subject. They must be thinking “this teacher likes giving failing grades!” 

How about the number four? This is still a dreaded number in UP because it’s a conditional grade, saying that you [didn’t] exactly pass yet—You have to prove your mastery by taking another exam or re-taking the course. However, a grade of 4 is definitely not dreaded in universities like De LaSalle where 4 is the highest grade you can get in a subject. So, Maroons would run away from a 4 as swiftly as a Green Archer would run towards it . 

So, different meanings for different people, but do I write differently from my UP students? No, we all write it in basically the same way—the Hindu-Arabic way. Quantitatively, it stands for the same amount. Conceptually, it can send different messages. Do UP students write 4 differently from the way De LaSalle students do? Of course not, it’s written in the same Hindu-Arabic way. The numeral used is the same, although it delivers contrasting messages and its sight evokes opposite emotions. 

I hope my examples have sufficiently explained the difference between a number and the numeral. The former is a concept, the latter is a symbol. 

Now, let’s get to the “meat” of this talk. What brought about the numerals? Note that each civilization is expected to have a numeration system. What needs of civilizations were met by numeration systems? More than the basic need to count, which could just be done orally, was a need to record, compare and communicate. For at least these reasons, a written representation was needed. How the collection of written representations or numerals works is a mark of a civilization’s level of sophistication. 

To the question of how the numerals were chosen to represent the respective numbers, there are several answers. You would find civilizations who used what was commonly seen around them—for example, fingers, a hand, a shell, a stick, a stone, a flower—or what could easily be drawn like a scratch on a wall, a single brush stroke of ink, whether vertical or horizontal. Or even what was divine, like faces or symbols of gods and whatever else they may revere. Prehistorically, Tali marks on rocks, stone walls and animal bones are evidence to earliest man’s need to record their counting on some medium. 

At this point, let’s look at some major ancient civilizations and their numeration systems. They are very interesting, and speak volumes about the culture and environment of a civilization. As we “travel” to these civilizations through their numeration systems, I suggest that, at the back of our minds, we do a running comparison with the numeration system that we are using now—the Hindu-Arabic system. We will eventually close our talk with some explanation of how we ended up using practically everywhere on the globe, the Hindu Arabic system, okay?

So let’s start. 

The Egyptian Numeration System.

Formal Egyptian numerals made use of pictures or what are called “hieroglyphs”. They believed [that] this was the language of their gods. Their 1 is a common single vertical stroke. Their 10 looks like an inverted “U” and it is said to represent a heel bone of cattle or oxen. Their 100 is said to be a coil of rope. Their 1,000 a lotus flower. Their 10,000 a bent finger. What I find more interesting, though, is their 100,000 which is said to be a tadpole that looks like it’s already transforming into a frog. To me, it illustrates growth; it declares that this is a major number, yet there is more. So this brings us to their 1,000,000. It is represented by a man who is kneeling with arms raised then opened wide. It is said that the man is supposed to be their god of eternity or infinity. Fittingly, this is their last and biggest numeral.

How did the Egyptian system work? Let’s limit our answer to how they represent their numbers. They simply repeated the numerals which were in powers of 10. For the number thirty, you’d need three inverted “U”. For three thousand, three lotus flowers. To write the year now, 2018, you’d need two lotus flowers, a heel bone and eight vertical strokes in no particular order. This is called simple grouping. Numerals are written in repetition and the final quantity is determined by counting the number of repetitions.

The Babylonia Number System.

The Babylonians used [the] so-called Cuneiform writing. Cuneiform means “wedge-shape”, sort of angular [shape]. These wedges were formed by using a stylus on a wet clay tablet. These tablets are then dried and hardened under the sun. One well-known Babylonian tablet, called the Plimpton 322, contains examples of these wedge-shaped numerals. They are neatly arranged in columns and rows. 

Can you guess what kind of information this ancient matrix contains? Would you have imagined that each row contains a Pythagorean triple? Well, that’s what those rows are. More than ten rows—so, that’s more than 10 sets of Pythagorean triples already known to the Babylonians thousands of years before common era, and with no imaginable contact yet with the ancient Greeks. It seems, then, that the Babylonians had their own trigonometry already. 

Notice also that they only had two numerals: an upright symbol for 1 that looks like a capital Y or T and an angular symbol for 10 that looks like the symbol for an angle or “less than”. 

How did the Babylonian system work with just these two? 

It was sort of simple in that they repeated the numerals but they had the beginnings of place value, or a positional system, where, like the Hindus and Arabs, the tens are written to the left of the ones or units. This is done until the quantity 59 after which 60 reverts to the symbol for 1. This means that the Babylonian system is a sexigestimal or base 60 system. They gave us the measures for time—60 seconds in a minute, 60 minutes in an hour—and for angles—360 degrees for a full circle.

Going back to the numerals, the immediate loophole, though, is how to read the wedge for 1 if it stands alone. Does it represent 1 or 60? Or maybe another power of 60 for that matter? 

The Mayan Numeration System.

So, let’s move to another continent: South America, where there were ancient civilizations like the Mayans. Like the Egyptians who had a simpler everyday version of their formal hieroglyphs, the Mayans had two ways of writing their numerals. They had an icon-based “God’s language” too, where they uses faces of divinities to represent numbers. Imagine having to draw these faces! But then, they had a very simple counterpart, where they use what look like dots and thick dashes. 1 is represented by a big dot and this can be repeated up to four times, after which, 5 is represented by a thick dash. This goes on until 19, which is represented by three thick dashes topped by four big dots arranged in a row. And then, 20 reverts back to the single dot. 

This brings us to the conclusion that the Mayan system is a vigesimal or base 20 system. In contrast to the Babylonian system, however, they have a numeral for zero which acts as a placeholder. The numeral takes the form of an open empty shell—very self-explanatory. 

So how does this system work? It is almost Babylonian; there are only two main numerals—“ones” and “fives”—and it reverts to the “ones” when the based number 20 is reached. The Babylonian numerals are read from left to right, while the Mayan positional system reads from top to bottom. Whereas there is confusion in the Babylonian case because of the absence of a placeholder, there is no confusion in the Mayan case because they have a placeholder. Where the Babylonian 60 is represented merely by a 1 and nothing else, the Mayan 20 is represented by a 1—a big dot—with a shell beneath it. This clearly shows that there is a positional system in place. The dot on top takes the place value for twenties. The shell below takes the place value for ones which can run from 1 to 90. Therefore, 21 would be written as two big dots—one on top of the other, 25 would be written with a big dot with a thick dash underneath, 30 would be written as a big dot with two thick dashes underneath, and 40 would be written as two big dots aligned horizontally with a shell underneath. Very systematic!

The Chinese Numeration System.

Let’s change continents again and go to Asia—The Chinese system. They say the numerals correspond to the spoken version of the numbers, and thus, take on the corresponding characters for the spoken syllables. Although they trace the origins of their numerals to counting rods, they have evolved to unique symbols which can be beautifully drawn in brush strokes. 

The early rods were very simplistic. A single horizontal stroke for 1 which can be drawn repeatedly, one on top of another until 4, much like the Mayan big dots. After which, a change is adopted at the number five. 5 is written as a vertical stroke and additional “ones” would be written horizontally underneath it until 9. Thus, they had a base 10 system, but, similarly, had no clear place valuation. You can imagine how unwieldy the system can get, as the numbers become bigger and bigger. For example, do the two vertical rods mean 10 or 55? Their more well-known characters used a multiplicative grouping system which remedies the confusion in place valuation, especially with the addition of a numeral for zero. 

There are Chinese characters for powers of 10 which are inserted in a row of numerals to indicate place value. For example, to write 75, one draws a character for 7, followed by the character for 10, and ends with the character for 5, from left to right. The 7-and-10 pair means that they are seven tens. The year 2018 is written using the characters for—from left to right—2 then 1000, 0 then 100, 1 then 10, and then, 8. The units digit does not have to be multiplied or appended by the character or numeral for 1. 

This system is also used in Japan and Korea.

The Greek Numeration System. 

Now, let’s move to Europe. We must discuss the Greek system, for they gave us the prefixes centi-, deci-, deca-, kilo-, and so on. 

The Ionic or Alexandrian numerals were based on the Greek alphabet. The unit numerals 1 to 9 took on the first nine Greek letters, alpha to theta. The tens numerals 10, 20, 30 up to 90 took on the next nine Greek letters starting with iota. The hundreds numerals 100, 200, 300 up to 900 took on the next 9 letters starting with rho. And so on. Then, they had the myriad, M, for 10,000. After which, they used a variety of symbols and some form of exponentiation for bigger numbers.

What is the difficulty with this system? Although it is indeed decimal and easily depicts small and big numbers, it requires users to memorize more than thirty six characters. 

Logical thinkers that they were, there was another Greek system involving the Attic numerals. They were not so alphabetical, as they were acrophonic. That means [the] numbers were based on the first letters of the symbols used to represent them. 10 got delta. 100 got hekaton. 1,000 got chilioi. This eventually emerged in the metric units of measure—or a prefix of deca– meant the factor of 10, a prefix of hecto- meant the factor of 100, and the prefix of kilo- meant the factor of 1000. These Attic numbers were the precursor of the Roman system of numeration. 

The Roman Numeration System. 

So, as history tells us, the Greeks became dominated by the Romans and the Romans met the Attic numerals. Eventually the Romans came to seven numerals they would call their own. The capital I for 1, capital V for 5, capital X for 10, capital L for 50, capital C for a hundred, capital D for 500, and capital M for a thousand. You can see here the beginnings of C for century and M for millenium. 

Let us take note at this point that, like the Mayans, the decimal base includes a sub based of 5. Why? You guessed correctly if you concluded that the most natural counting implement for a man is his set of fingers. So, although there are ten fingers by hand, there are 5 fingers each, thus, the interim base of 5 before the main decimal base of 10. Remember how we said that the V resembles a hand drawn from the thumb to the wrist, to the pinky? If you join two such V’s at their vertex, whether this way or that way, what do you get? An X, which is appropriately representative of 10 because two V’s, or two fives, make 10.

May I insert a question? If we had eight fingers, what do you think would be the base of most numeration systems? What if we had thirteen fingers? Just something to think about. 

Alright, back to the Roman numerals. 

How do these seven numerals work? They follow some place valuation, in that they are written from left to right, beginning with the biggest value, moving down to the units. They also had a unique way of writing 4, and 9, and 40, and 90, and so on, involving some subtraction of numerals. Without going into any computational details, imagine doing addition, subtraction, multiplication and division using Roman numerals. I can imagine the expressions on your faces, coupled with vigorous head shaking. Why would you refuse? What makes it difficult? What makes it super complicated? I know that you are basing your reactions on the way we perform operations now and how relatively easy the Hindu-Arabic decimal system allows us to do so. 

Here is the catch: no matter how advanced the Roman civilization may have been, there was a big, even huge gap in their numeration system—they had no zero. So, the story goes that the Roman numeration system was taken over by the Hindu-Arabic system in around the first millenium, and the Roman numerals, after thousands of years, were relegated to ceremonial, formal, aesthetic symbol use. They were not used anymore for computation. 

The Hindu-Arabic Numeration System.

And so, we come to our finale, the winner in terms of being chosen as the universal numeration system across almost every nation on the globe. It has ten numerals, it has a zero, it is a decimal system and it writes straightforwardly from left to right in descending powers of 10. Most of all, it allows us to fluidly perform calculations because of zero. 

How did the zero come to be in our system? The Indians had it, through their mysticism that embraced both nothingness and infinity. The nomadic Arabs, in their forays East, came across the Indian numerals and calculations called Modus Indus or the Indian Way. The Arabs adapted them, taking this new knowledge with them in their journeys through deserts from one oasis to another. Then, the Arabs did trade with North Africans who, then, did trade with Southern European. This is the short version of how zero could have traveled from the Indian subcontinent to Europe, particularly Italy. 

It is the Italian mathematician Leonardo Pisano—Leonardo of Pisa, better known as Fibonacci—who is credited for introducing to the west the Hindu-Arabic numerals and calculations, paving the way for the displacement of the Roman numerals. He did this through his 13th century book entitled Liber Abaci or the Book of Calculations. Leonardo was the son of an Italian official, and he was able to spend time from childhood to travelling on business in North Africa. He brought what he learned during these travels back to Pisa via the Liber Abaci. 

Needless to say, it was met with much resistance. You can imagine the scenario— mighty authoritative Romans being told that their system was not the best and that the heretofore obscure Indian method performed much, much better. But as many stories go, the underdog won. Acceptance of zero spread little by little across Europe. It took a long time but the underdog won. A few centuries later, zero emerged in other European countries through other mathematical landmarks. 

The Frenchman René Descartes’ Cartesian plane is a grid peg on a central point called the origin, represented by the ordered pair 0,0. Close to those years, Calculus was developed by the Englishman Isaac Newton and the German Gottfried Leibniz, pegging it on the concept of a limit where an already small value is taken closer and closer to zero. And as Europe discovered the world, it spread zero and the Hindu-Arabic system along with it.

We have come to the end of our talk. Instead of proverbially starting from zero our story ends with zero, and happily so. Now, I hope the number zero doesn’t evoke negative thoughts. I can tell you now that zero is my other favorite number aside from 5. Its story, the one that I just told you, tells you why and just like its Hindu-Arabic numeral our story has come full circle. 

Again, this is Carlene Arceo. I hope you had as much fun as I did. Thank you very much for listening.