history of numeration systems

tally

Numeration Systems and Psychology

Looking at different historical numeration systems demonstrates how language and grouping systems profoundly effect human thinking, perception and function, and how the the system you ‘naturally’ use to perceive the universe isn’t the only way.

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In some Western Hemisphere high rise buildings there are no thirteenth floors. Well, there are thirteenth floors, but the floors are labeled 10, 11, 12, 14, 15 to give the superficial appearance of having no thirteenth floors. The building owners know many have a superstition against the numeral thirteen and it’s easier to rent an apartment or office if it’s called ‘fourteen.’

In Korea and Japan where four is considered unlucky as it’s the sign of death, some buildings ‘omit’ the fourth floor.

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Our base-10 numeral system

The common modern human counting system— the one you and I use– is based on 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, and many popular groupings are divisible by ten: 10, 20, 100, 300, 10,000, century, decade, top 10 lists, golden anniversary, etc.

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 and it seemed only natural to base a counting system on the 10 digits.

While the base-10 is a good system and has served us well, ten as the base was a somewhat arbitrary choice. 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). These different base systems would work. 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.

Quick comparison: counting with base-10 versus base-8

The above pictures compare 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.

8v10

This comparison picture shows how assorted designs (top row) are counted with the base-10 and with the base-8 systems. 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 10, amongst other numeral symbols, is not an absolute. It depends on what base is being used.

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Another example of counting with different bases

 base-5  base-8  base-9  base-10  symbols
 0  0  0  0  $
 1  1  1  1  #
 2  2  2  2 @
 3  3  3  3  !
 4  4  4  4  %
 10  5  5  5  ^
 11  6  6  6  &
 12  7  7  7  *
 13  10  8  8  )
 14  11  10  9  _
 20  12  11  10  +
 21  13  12  11  =
 22  14  13  12  –
 23  15  14  13  <
 24  16  15  14  >
 30  17  16  15  ?
 31  20  17  16  “
 32  21  18  17  ;
 33  22  20  18  ‘

The following table illustrates how you can count symbols (far right column) using the base-10, base-9, base-8 and base-5 systems. If you wish, the symbols can represent physical objects like fruit or cars or plants. In this table the symbols are constant, while the numeral systems create different numeral labels for the symbols (or fruit or cars or plants). For those who consider ‘13’ unlucky, notice that each counting system labels a different symbol as being 13.

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This counting stuff is not 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.

Today’s computer scientists use 2, 8 and 16-base systems. For some mathematical work base-12 is more convenient than base-10. For this base-12 system they usually use the normal 0,1,2,3,4,5,6,7,8,9 numerals and add the letters a and b to make twelve (0,1,2,3,4,5,6,7,8,9,a,b). It goes without saying that these mathematicians, often university professors and researchers, are using this system to perform higher levels of calculations than you or I perform in our daily lives. They aren’t counting change at the grocery store.

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 (60 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 should give 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 have been 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 unique symbol.

Ancient Egyptian numerals for 1,000 (lily flower) and one million (man with raised arms)

tally

Tallying is an ancient basic counting system many of us use. The practical problem with this system is that numbers like 500 and 10,000 require a whole lotta tally marks. 500 requires 500 tally marks. Over history, numeral systems have changed and evolved to correct inconveniences like this. Notice we use the tally system only for simple tasks, like keeping score in a ping pong game and marking days.

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A kid’s counting system: Eeny meeny miny moe

Kids have long used counting rhymes to decide who is it. The below common rhyme does the equivalent of counting to twenty, with the last word being the twentieth word.

Eeny, meeny, miny, moe / Catch a tiger by the toe If he hollers let him go / Eeny, meeny, miny, moe

There are a few interesting things about this eeny meeny counting system. First, it is quasi base-20, not our normal base-10. Second, words are used as numerals, or as the practical equivalent of numerals. Kids could count to 20 for the same practical result, but they chose to use words. Third, while lucky 7, 10 and unlucky 13 have popular importance compared to other numerals in our base-10 system, the seventh, tenth and thirteenth words in the rhyme do not.

This is an example where a different counting system changes what numbers are perceived as important. Most kids who count with this rhyme aren’t even aware which are the seventh, tenth and thirteenth words.

Humans often say they can’t conceptualize numbers in anything but the normal base-10, but here is a base-20 words counting system that we have all used. Granted this counting system is simplistic in the extreme, used for one and only one purpose— to count to twenty (moe). You wouldn’t want to try and use it to calculate your taxes.

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Numerals and human psychology

Humans form psychological attachments and biases for the numeration systems they use. Having grown up using a particular system, and seeing all those around them using the same, many people assume their numeration is absolute and eternal. Before reading this chapter, you may not have known or thought about the existence of other systems. Your base-10 system was all you knew, the prism which you saw the universe. 10, 100 and 1000— popular products of your base-10 system— are numbers you are attracted to. Thinking in base-8 or base-7 is foreign.

It’s telling to look at how humans change their perception from system to system, and how a change of numeration system changes peoples’ perceptions of things. The perception is not just about the numeration system itself, but the things the numeration system is used to count— objects, time, ideas.

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As the earlier tables showed, a different base numeral system doesn’t change the accuracy of our calculations or the physical objects we calculate. However, if we retroactively changed our base-10 system to a non base-10 system (like say the Yuki’s base-8 system) we would change how humans perceive and react to objects and concepts.

As with the high rise buildings and the superstitious renters, the historical changes would be caused in large part by human perceptions of the numerals themselves rather the things the numerals represent. No matter what the Mexico City building owner calls the thirteenth floor, it is the same floor. If he changes the label on the elevator directory from ‘13’ to ‘9988’ or to ‘789’ or to ‘Q,’ it is the same floor with the same walls, ceiling and windows and distance above the sidewalk. The numerologist apartment seekers aren’t reacting to the floor but to the symbol ’13.’ It should not surprise that a change to the symbols, such as caused by the changing to a new counting system, will change their reaction to the floors, along with many other things.

With a large lot of stones lined up on a table, changing the numeral system has no direct effect on the amount or physical nature of the stones. With a new counting system, the stones would be the same stones, but many to most would be assigned different numeral names. While the stones are the same stones no matter what we call them, human perceptions of the stones change as the stones’ numeral names change. Under our popular base-10 system, humans consider certain numerals to be special, including 10, 100, 1000 and 13, and react accordingly to objects labeled with these names. With the new numeral representations, humans’ perception and treatment of the stones will change. If before a person avoided a stone because it was unlucky 13, in the new system a different stone would be called 13. If in the old system the stone labeled ‘100’ was singled out as special, in the new system ‘100’ would represent a different stone.

If a human is asked to count and group the stones, the grouping will change with the different counting system. In the base-10 system, it’s likely the person would make piles of 10 or 25 stones or similar standard. In an 8 or 9 base system, the number and size of the piles would be different. To someone standing across the room, the rock design would be different. Her aesthetic reaction to the formation would be different.

This shows that your numeration system isn’t just an objective observation system, but helps form how you perceive objects. Under a different system, you would perceive things differently.

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Changing numeral systems, changing history

As a numeration system changes how we perceive, organize and react to things, a retroactive change to the numeral systems would change human history. The amount and type of change can be debated, but today’s history books would read different. With a change to the standard numeration system, time would remain the same but human marking of time would change. The decade, century and millennium equivalents would be celebrated at different times. No Y2K excitement at the same time as we had. Special milestones, like current marriage 10th or 25th anniversaries, would be at different times. People who now receive 30 years of service awards might receive equivalent awards but after a different duration.

Think of all those sports championships decided in the last moments, including the improbable upsets and bloop endings. If the events took place at different times and under different numeral influenced conditions some of the outcomes would be different. If an Olympic sprint is decided by a fraction of a second, it’s unlikely the first to last place order would be identical if it took place the day before with the runners in switched lanes and running a different length race. The changes to marking of time and distance would likely result in different gold, silver and bronze medal winners over the years. If a horse race was a tie, it is unlikely the same horses would tie if the race had been run earlier or later in the day or on a different day over a different length race. Realize that the change to the numeration system would likely change the standard race distances, even if the changes were just slight.

Think of all the razor close political elections. If the elections took place at a different time, even if just a day earlier or later, it’s possible some would have different outcomes. A few of the outcomes could have been for President, Prime Minister, judge or other socially influencing position. Think of all those close historic battles that may or may not have had a different outcome if started at different times, using different size platoons and regiments and Generals who made decisions using different number biases. Napoleon Bonaparte was superstitious of 13 and made his government, social and military plans accordingly. Think of the influential or not yet influential people who died at relatively young ages in accidents, from Albert Camus to General Patton to Buddy Holly. James Dean died in a sports car crash at age 25. Would he have crashed if he started his drive at an earlier or later time? Popular perception of the actor no doubt would be quite different if we watched him grow old and bald.

The powerful nineteenth century Irish Leader Charles Stewart Parnell would not sign a legislative bill that had thirteen clauses. A clause had to be added or subtracted before it could become law. Irish law would have been different under a different numeral system.

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United States consumer prices would likely be affected by a different numeral system, if just marginally. Again, this would be due to human psychological perceptions of numerals.

Even though most current US sellers and buyers think nothing of one penny, often tossing it in the garbage or on the sidewalk, sellers regularly price things at $9.99 instead of $10, and $19.99 instead of $20. Check the newspaper ads. This pricing is purely aesthetic, intending to play on consumers biases towards numerals.

The shallowness of this 1 cent game is illustrated when it is used by stores that have a ‘give a penny, take a penny’ tray, and that it is used in many states with different sales tax rates. Most people psychologically affected by $9.99 pricing at home are also affected by $9.99 pricing when traveling by car across the country. That the daily change in sale tax charge dwarfs the one cent between $9.99 and $10, illustrates the traveler’s irrationalness.

Under a base-9 numeral system that omits the numeral ‘9,’ $9.99 and $19.99 would no longer exist, and the visually appealing “one cent below big number” pricing would land elsewhere. In a 9 digit system, it’s likely that there would be many $8.88 and $18.88 pricings in newspaper ads, and the same types of travelers would be attracted to $8.88 and $18.88 prices as they go state to state even though the taxes change state to state.

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There are a variety of intertwined reasons behind irrational biases towards numerals and numeral systems.

One reason is people form psychological attachments towards a system, its symbols and the standard groupings of objects made from the system. A three digit numeral price ($9.99) looks distinctly different than a four digit numeral price ($10.00), literally being shorter. One hundred stones grouped into 10 groups of 10 each will look different than 11 groups of 9 stones each with one left over. It’s the same amount of stones, but their physical designs look different. There’s an aesthetic aspect to how humans view symbols and groupings.

Closely related reasons are tradition and habit. If you have used our base-10 system all your life, it’s as natural to you as your native spoken language. In fact words such as nine, ten and decade are part of your daily vocabulary. If everyone you know uses this numeral system, the idea of using a different system may not have even crossed your mind before now. The idea of calculating using a base-8 or base- 11 system seems strange and even unnatural to most people because they were raised on base-10.

Another reason behind irrational biases towards numerals is the seeming, if nonexistent, absoluteness of the familiar numerals. While the true nature of time, supernatural, war, love and the cosmos are shrouded in mystery, the numerals traditionally used in representing these things seem tangible, concrete. Unlike philosophical abstractions, numerals can be written down and typed into the calculator. Even little kids can count numerals on their fingers. That folks like Isaac Newton and Albert Einstein used these same numerals seem to numerologists to indicate the numerals’ potency. Though, if asked, both scientists would agree they could have used other numeral systems to do their work, and there was nothing uniquely special about the system they adopted.

Numerals are used only as convenient notations, proverbial post-its to label objects. They have no absolute, inborn connection to the things they represent. Whether you call the animal cat or gato it’s the same animal, and whether you call a number 5, five or V, it’s the same number. Whether you count a grove of trees with a base-10 or a base- 8 system, they are the same trees. If you count and label the trees a,b,c,d,e,f,g, they are still the same trees. Numerologists incorrectly assign an absolute meaning and identity to the numerals that doesn’t exist.

Even in academia, mathematicians considered to be too enamored with the beauty of numbers at the expense of practical use are sometimes derogatorily called numerologists by applied scientists like engineers. Mathematicians are as influenced by aesthetics as the rest of us.

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Sounds Good

Many Chinese judge numbers as good or bad by what words they sound closest to. As their pronunciation of 3 sounds closest to their word for ‘live,’ 3 is considered good. Their pronunciation of 4 sounds close to their word for ‘not,’ so is often considered negative.

China is a huge country with many dialects. As numbers and words are pronounced differently in different areas, a number’s perceived goodness and badness depends on where you are. For example, 6 is considered good in some places and bad in others.

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