The Earliest Counting Systems

Numeration systems, which include the many different systems used by different cultures throughout history, is a mind-examining area to study. Studying the different numeration systems, you learn that they are simply languages. Some ancient cultures, such as the Hebrews and Greeks, use their alphabet symbols as their numeral symbols.

The following is a brief overview of the origins of numerations systems. It is an excerpt from my book An Introduction to Ancient Counting Sytems.

The Beginnings of Mathematics: Number sense

Number sense is not counting, but a vague neurological ability to identify changes in a small collection. Humans and several other species of animals have number sense. If you make changes in amount to small groups the animal with number sense can notice. The animal can identify changes in amount even when unable to tell exactly how much has changed. It can tell that there is more or fewer than before.

Many mammals and birds have small numbers of young, say three or four. If you take away one or two of the young, the animal will notice. One reason not to take a baby or egg from a bird's nest is the mother may notice and leave the entire nest.

Crows are famously intelligent and numerous stories have been told how the crows can, up to a number, tell how many people have entered and left a barn inside where the bird has its nest. The crow doesn't want to enter the barn unless it is free of people, so waits outside until it is empty.

crows are well known for their intelligence and counting abilities

People have tested crows’ counting abilities by having people enter then leave in different combinations of people. Up to a number, the crow can tell when all the people have left. However, it is not certain whether the bird was counting or recognized the individual people. It is known that crows can recognize human faces.

Humans do not have an especially great number sense. What separates us from the other animals is we have developed counting systems. Even rudimentary systems of counting on our fingers help record numbers and identify changes in numbers. This ability to count and mark numbers, and later multiply and divide, has been an integral part of human's success story on earth.  Number sense is innate, while numbers may be cultural.

Early human counting

In the earliest days, humans did not have a great need to count things, other than in vague number sense ways of “little” or “lots.” As they started trading and owning more things, it became important to count. A shepherd needed a way to count his grazing sheep to make sure he had gathered them all before returning home.

The earliest counting systems used one-for-one correspondence. The shepherd may have carried with him a bag of pebbles, one pebble for each of his sheep. At the end of the day, he could match each pebble to each sheep to make sure he had them all. Related marking systems included tying knots in a rope and cutting notches on a stick. Tally marks and counting on one's fingers are one-to-one correspondence systems we use today.

As humans developed language they began assigning symbols and words to numbers. The physical hand with its five digits may be used to symbolize five, one finger may symbolize one, two fingers symbolize two, a twig with two branches may symbolize two. These symbols can be written down as pictures on paper, forming written numeral symbols. The Ancient Egyptian numerals were pictures of things: a finger, a frog, a flower.

As you see, different people can assign different, personal symbols for numbers. People can have similar counting systems but with different symbols. If asked to make their own systems for fun, one person might use the hand for five, another might use an oak leaf, another could use a written symbol of a star. One person might use colors for symbols, another might use shapes, another physical objects. 

Make up your own numeral system, and write down the following numbers. Use whatever symbols you wish for 1 to 9, then write down the following number in your symbols

12

235

The following is mine:

Many historical counting systems are similar to ours except they use their own symbols.

GROUPING 

As we've already seen, grouping is a common and convenient part of counting. When counting large numbers, grouping is essential. Anyone who has counted a large pile of pennies, quarters, nails or buttons knows that establishing groupings is important to counting and later re-calculating what you have. 

If you are counting a pile of buttons, you might count in groups of ten, twenty or twenty-five. This way if you lose count, or want to go back and calculate the whole amount, you don't have to start from the beginning. You can count a pile of buttons, coins or pebbles using different groupings. People can and do count a pile using different groupings, all coming up with the correct final number.

OUR COMMON GROUPING: BASE-10

The common modern human counting system— the one you and I use-- is based on groupings of ten, and is referred to as base-10. It uses 10 different numeral symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) to represent all numbers. Many popular groupings are divisible by ten: 10, 20, 100, 300, 10 000, century, decade, top 10 lists, golden anniversary.

Our base-10 system is based on the number of digits on a human’s hands: eight fingers and two thumbs. As with today, many ancient humans found fingers and thumbs convenient for counting.

While the base-10 is a good system and has served us well, ten as the base was an arbitrary choice. Remember that the piles of buttons and pennies can be counted in piles other than ten. Our numeral system could have been based on 3, 8, 9, 11, 12, 20 or other number. Instead of basing it on the total digits on a pair of hands, it could have been based on the points of an oak leaf (9), the sides of a box (6), the fingers on a pair of hands (8). Some might work as well or better than our base-10 system. Nuclear physicists and tax accountants could make their calculations using a 9 or 11-base system. Once you got used to the new system, you could count toothpicks and apples just as accurately as you do now.

Figure 2. Quick comparison: counting with base-10 versus base-8

Figure 3. Compares counting with a base-10 system based on the ten digits of the hands (fingers + thumbs), and with a base-8 system based on just the eight fingers (thumbs not used). Notice that the base-8 system, not using the thumbs, is missing two numeral symbols: 8 and 9.

Figure 3 shows how assorted designs (top row) are counted with the base-10  and with the base-8 systems.  Base-10 not only means nine unique symbols 0-9 but also using powers of 10, while base 8 means powers of 8. When using modern notation: 123 = 1x10² + 2x10 + 3 in base 10 but 1x8² + 2x8 + 3 in base 8. 

As base-8 omits the two symbols 8 and 9, ‘10’ comes sooner when counting in base-8. In one numeration system, the cat is ‘9’ and in the other is ’11.’ As you can see, the real value of the symbol 10, amongst other numeral symbols, is not an absolute. It depends on what base is being used. If we continued ‘unlucky 13’ would be assigned to different objects.

This counting stuff is not an idle abstraction. Civilizations have used and use different numeral systems. The Yuki Indians of California used a base-8 numeral system. Instead of basing their system on the digits on their hands, they based it on the spaces between the digits. The Ancient Mayans used a base-20 system, as they counted with the digits on their hands and feet. They lived in a hot climate where people didn’t wear closed-toe shoes.

Our normal lives show the vestiges of ancient numeral systems. We sometimes count with Ancient Roman numerals (Super Bowl XXIV, King Richard III), letters (chapter 4a, chapter 4b, chapter 4c... Notice how this combines two different systems, standard numerals with letters), and tally marks. We group loaves of bread, inches and ounces by the dozen, and mark time in groups of sixty. Sixty seconds per minute, 60 minutes per hour. Counting inches and ounces by twelve comes from the Ancient Romans. Our organization of time in groups of 60 comes from the Sumerians, an ancient civilization that used a base- 60 system.

The traditional counting of bread into groups of twelve has practical convenience. At the market, a dozen loaves can be divided into whole loaves by two, three or four. Ten loaves can only be divided by two into whole loaves. Sellers and customers prefer the grouping that gives more whole loaf options, not wanting a loaf to be torn apart. This gives you an idea why feet and yards are divisible by twelve, and there were twelve pence in a shilling: You get more ‘whole’ fractions out of twelve than you do ten. 

These are just some examples of other numeral systems, as there have been a wide and varied number over history. This not only includes systems with different bases but with different kinds and numbers of numeral symbols. In Ancient Eastern countries, physical rods were used to represent numbers. The number, position, direction and color of the rod represented a number. In Ancient Egypt, pictures, known as hieroglyphics, were used to represent numbers. One thousand was written as a lily, and 10,000 as a tadpole. The Ancient Hebrews had a similar system to ours, except they used 27 different symbols to our ten. For the Hebrews, numbers 20, 30, 40, etc each got its own symbol.