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.
* * * *
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.
* * * *
Our base10 numeral system
The common modern human counting system the one you and I use
is based on ten, and in mathematics is called base10.
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 base10 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 base10 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 base10 system. Math professors, nuclear physicists
and tax accountants could make their calculations using a 9 or
11base 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 base10 versus base8
The above pictures compares counting with a base10 system based
on the ten digits of the hand (fingers + thumbs), and with a
base8 system based on just the eight fingers (thumbs not used).
Notice that the base8 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 base10 and with the base8 systems. As base8 omits
the two symbols 8 and 9, '10' comes sooner when counting in base8.
As you can see, the real value of 10, amongst other numbers,
is not an absolute. It depends on what base is being used.
* * * *
Another example of counting with different bases
The below table illustrates how you can count symbols (far right
column) using the base10, base9, base8 and base5 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.
base5 
base8 
base9 
base10 
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 
' 

* * * *
This counting stuff is not idle abstraction. Civilizations
have used and use different numeral systems.
The Yuki Indians of California used a base8 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 base20 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 16base systems.
For some mathematical work, base12 is more convenient than base10.
For this base12 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 base60 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 base20, not our normal base10. 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
base10 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 base10, but here is a base20 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
base10 system to a non base10 system (like say the Yuki's base8
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
base10 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 10base
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.
* * * *
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.
* * * *
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 base9 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.
* * * *
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 base10 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 base8 or base11 system seems strange
and even unnatural to most people because they were raised on
base10.
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
postits 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 base10 or a base8 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.
* * * *
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.
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