A Look at Numeral Systems and Numeral Biases
by David Rudd Cycleback

(c) cycleback 2003, 2005 all rights reserved


























































































































































































































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 to give the superficial appearance of having no thirteenth floors. The building owners know that 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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This page is a is a brief introduction to numeration systems and how humans form psychological biases for familiar numerals and numeral systems. As you will see, human thought and even physical function is greatly influenced by which numeral system is used.

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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 in mathematics is called base-10. It uses 10 unique 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 hand: 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 largely arbitrary choice. Our numeral system could have been based on 3, 8, 9, 11, 12, 20 or another 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. Math professors, 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.

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

The above pictures compares counting with a base-10 system based on the ten digits of the hand (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.

The above 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. As you can see, the real value of 10, amongst other numbers, is not an absolute. It depends on what base is being used.

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Another example of counting with different bases
The below 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 whether fruit or cars or plants. In this table the symbols are constant, while the different numeral systems create different numeral names for the symbols. For those who consider '13' unlucky, notice that each counting system labels a different symbol as being 13.

 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  '  

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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 the hands and feet. They lived in a hot climate where people didn't wear closed 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 (Superbowl 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 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 a practical convenience. At the market, a dozen loaves can be divided by two, three or four into whole loaves. Ten loaves can only be divided by two into by whole loaves. Many bread sellers and customers prefer the grouping that gives them 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. 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 rods represented a number. Black rods reprsented positive numbers and red rods represented negative. 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)


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 lot of 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.

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 the 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 do your taxes or even keep score at a basketball game.

Psychological Reactions & Actions to Numeral Systems
As the earlier table 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 human history. The degree of change can be debated, but today's history books would read different.

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 '789' or '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 objects.

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, human's 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 10-base 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 20 feet away, the rock design would be different. Her aesthetic reaction to the formation would be different.

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Changing numeral systems, changing history

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 items. 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 would likely result in different Gold, Silver and Bronze medal winners. 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. 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? Our perspective of the actor would be 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 fourteenth has to be added 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 numeric 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 ground, 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 the 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 ($9.99) looks distinctly different than a four digit number ($10.00), literally being shorter 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 like 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. Numerals seem so tangible, so concrete. 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 (Spanish for cat) it's the same animal, and whether you call a number 5, five or V (Roman), 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. Mathematicians are as influenced by aesthetics as the rest of us.

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Sounds good

Many Chinese traditionally 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 numerals and words are pronounced differently in different areas, a numeral's perceived goodness and badness depends on where you are. If you took a train ride across the country, at some stops 6 would be considered good and at other stops it would be considered bad. glass


(c) david rudd cycleback, all rights reserved