The Base 10 number system we use isn't something that is commonly discussed or talked about in our general
education system; it is mostly taken for granted that our numerical counting system consists of the digits
0-9, rolling over into the next place value 10 when we surpass 9. This method of expression is taught from
the very earliest of ages and continues all the way into the highest of mathematical thinking. But why?
Why is it that the Base 10 Number System has come to be so prolific within our modern society? How do other
number systems work? Are these other number systems used anywhere, and why are they not more prevalent?
All this and more will be explored below.
History, Development, and Background
Early History
All kinds of people have had to find ways of expressing numbers since the earliest time periods. There has
always been a need for people to express an amount, from the earliest uses of counting animals or crops on
a
farm, to counting money and people themselves as groups of people came together to form cities and
civilizations. The earliest methods of counting are thought to have simply been using fingers, as fingers
are
simple and
convenient to use. However, the more that there was for these ancient people needed to count, the more that
using their
hands as
a method for keeping track of amounts simply was not feasible.
The image to the left is the Ishango Bone,
found deep
in Africa. The bone itself is dated as 20,000 years old and is one of the first examples of a tally system
coming into play.
Scientists presume that this bone was used to mark off days on a calendar, and it is one of the earliest
examples
we have
currently of a tally system being used. An older bone, known as the "Lebombo
Bone",
dates back even further to between 44,200 and 43,000 years old. According to "The Universal Book of
Mathematics," the 29 notches
on the Lebambo Bone may have been used as a lunar phase counter. In that case African women may have been
the first mathematicians
because keeping track of menstrual cycles requires a lunar calendar. This makes the Lebambo Bone the
world's oldest known
mathematical artifact. Through the discovery of ancient artifacts like this we can see, even thousands upon
thousands of years in the past, manking
needed a way to express
amounts, and the tally system was a vital first step along that path.
However, as one might expect, the tally system is not perfect when it
comes to accounting for all objects in a given space. Just as counting fingers quickly becomes overwhelming,
using
tally marks becomes unwieldy with larger and larger numbers. Accounting for larger groups of sheep with
tallies could be
just as difficult as counting the sheep. A solution to this problem existed in the Middle East, where small
clay tokens were
used to represent quantities. Different shapes represented different commodities, like oil or sheep, and
notches on the tokens
represented different values. The image to the
right denotes an amount of grain, with the circular notches representing a large measure of grain and the
wedges representing
a small measure of grain. Here, this token would represent four large measures and four small measures of
grain.
Another example here shows a clay token used for oil. The circular
notches represent ten jars, and the wedges represent one. This token specifically would then
represent 33 jars of oil. Tokens like these led the way in facilitating easier methods of inventory
keeping and trade.
Development of Number Systems
As the need to account for the numbers of objects increased, we begin to see the development of the
number systems that we are familiar with today, though each civilization developed their own methods of
counting.
Egyptian Numeral System
Ancient Egyptians had a writing system
(hieroglyphics)
dating from about 3000 BC, and from this writing system, we have found evidence of a number system. Their
number system was
a Base 10 system, similiar to the one we are familiar with. The symbols to the right were used to denote
powers of ten up to
10,000,000. Over time, and with the introduction of papyrus, the number system evolved to a more dynamic
writing system that allowed
for the expression of larger numbers using less symbols. Seen here, more
symbols
were added to denote more numbers,
such as the introduction of symbols for each individual multiple of 100. Numbers like 9,999, which would
originally have take 36 hieroglyphs to express, could now be expressed with 4 symbols.
Babylonian Number System
Meanwhile, the Babylonians used a number system consisting of Base 60,
meaning each number less than 60
needed its own numerical symbol, as seen on the left. As can be seen, clusters of ten were combined together
to more easily facilitate expression. However, the Babylonians did not have a symbol for zero.
Interestingly, the Babylonians
did not have a symbol for zero. For example, to express the number 60, the symbol for one would be used. A
space would be included after the "1,"
leaving both numbers looking almost exactly similiar. Though Base 60 seems
much too difficult to use,
we can see how their number system affects us to this day, such as in the 60 seconds and minutes of angular
measurement, in the 180 degrees of a triangle, and in the 360 degrees of a circle.
Arabic Number Systems
The Arabic Number System is the number system we currently use today and was mainly formed in India around
the 5th-6th century. The numeral system
and the zero concept, developed by the Hindus in India, slowly spread to other surrounding countries that
had commercial and military
activities with India, and the Arabs adopted and modified it. The Arabs adopted and modified it. Even today,
the Arabs call the numerals they use "Raqam
Al-Hind," or the Hindu numeral system. The aspects of the Arabic number system that have had the biggest
impact on modern mathematics
are the introduction of the decimal system and the inclusion of a symbol for zero.
Summary
Through the history of mankind's need to express amounts, we see that the number systems used have had to
become
more advanced in order to facilitate
greater understanding between people. The Arabic Number System we use today came about as a result of
several civilizations coming together to trade
with one another, which required them to have a standardized method of communicating with each other.
This video provides a simple summary of the history of number systems throughout history, should you want to
investigate further.
Explanation of Mathematics
Basics of Place Value
So, what is a number system? Simply put, a number system is a method of representing an amount or value. As
was discussed in the history section above,
one could simply count using one's hands or using individual tally marks, but each has its
limitations. Namely, each method has a very difficult
time expressing large numbers (imagine trying to count 1,000 individual tally marks). Because of this
difficulty, numbers were given positions, which allow us to
express numbers in far simpler terms.
Base Number Systems operate using a place value system. Instead of counting indefinitely using tally marks
or creating a unique symbol for every number
we might need, we use place values and a much smaller set of symbols to denote a large series of
numbers. They operate as such: Once a certain value is
reached, we "roll over" into the next place value and begin counting again, adding one more tick to the
next place value until we reach the last symbol, at which
point we "roll over" into the place value after that.
For the Base 10 Number System we use today, we can visualize
how numbers are represented using the image to the right.
We see that a number as large as 1,247 can be represented with as few as four symbols due to the place value
system. We see that a 1 in the thousands place represents a
thousand individual blocks or tallys, a 2 in the hundreds place represents having two groups of 100 blocks, and so on.
Seen below is how the value of each place value in a Base 10 Number System is determined. As we can see, each place value
is simply 10x, with the value of x increasing
as we add more numbers to the left. A 5 in the hundreds place would give us 5 * 102, which
would be notated as 500.
Likewise, the same applies going to the right, but in reverse. Numbers to the right
represent the value of x in 102 decreasing into the negative numbers. This
chart shows us the place values to the right. A 5 in the hundredths place would mean we have 5 *
10-2
which would be represented as 0.05. With this place value system, every number can be represented as a string
of the digits 0-9.
Expanding Understanding to Other Base Number Systems
Most of the information in the previous paragraph is taught to almost everyone in our modern society at a
fairly young age.
The ability to do any sort of math is based upon the ability to recognize and manipulate values in a given
number system.
However, number systems other than Base 10 are not used nearly as frequently, and thus are not thought about as
often. The basics
of every Base Number System remain the exact same as that of the Base 10 Number System. Seen below is the
layout of the Base 2 Number
System, one of the most commonly used Base Number Systems outside of Base 10. Here we can see that just as
Base 10's place values
are made through 10x, Base 2's place values are made through 2x. A 1 in the 64s place
would mean we would have
a number equal to 64, but would be expressed as 1000000. If we desired to represent a Base 10 number in Base
2, we would break that number
down into the highest powers of 2 and put 1s in those fields. For example, 34 in Base 2 can be expressed as
32 + 2, both powers of 2. (25 + 21).
As such, we would place a 1 in each respective location, leaving us with 100010.
A strange phenomenon occurs in all Base Number Systems less than 10 many digits simply are not used. In
Base 2 for instance, the only digits used are 0 and 1.
Every succeeding number either changes a 0 to a 1 or vice versa. Thus, the digits 2-9 go unused. The same is
true for each of the other Base Number Systems less
than 10 - the digits x-9 (where x is the Base Number System) go unused. We find a similar phenomenon taking
place in the reverse direction. All Base Number Systems greater
than 10 don't have enough digits between 0-9 to cover all of the numbers that we would be using. We wouldn't be
able to "roll over" into the next place value until we hit x1, with
x being whatever number system we happen to be working with.
This Schoolhouse Rock video goes over the basics of how a Base 12 Number System would handle not being able
to use "10" and "11" as symbols to represent the Base 10 values 10 and 11.
As shown in the video, in a Base 12 system, we would have to find a new way of expressing "10" and "11."
Sometimes mathematicians will use Greek letters, but more often than not, alphabetical letters
will be used (e.g., 10 and 11 would be represented as A and B respectively in a Base 12 system). This table shows both 10 and 11 being represented in Base 12 as well as what
the multiplication table would look like in such a system.
Here is a link to an applet that allows the user to input
numbers to see how that number would be represented
in a different Base Number Systems.
Significance and Application
Metric System
In most of the world, the standart system of measurement is the metric system, which uses the Base 10
Number System. For every unit of measurement, whether it
be distance (meters), volume (liters), or weight (grams), the prefixes to the right represent which power of
10 that object belongs to. For instance, 2 kilometers represents
2 * 103, or 2000 meters. This unit of measurement is far easier to teach, memorize, and understand
given our number system for most mathematical processes is in Base 10, and there have been several major pushes
in the last decades to switch the United States from its current Imperial System (which doesn't have
any basis in any Base Number System) to the metric system.
Binary Operating Systems
The basic coding instructions for computers are written in binary code, which uses a Base 2 number system to
send signals all throughout a computer. As mentioned above, a Base 2 number system consists
of only two digits 0 and 1. Binary code takes advantage of this and sends signals using the base 2 system:
"The 0s and 1s in binary represent OFF or ON respectively. In a
transistor, an "0" represents no flow of electricity, and "1" represents electricity being allowed to flow.
In this way, numbers are represented physically inside the computing device, permitting calculation"
(computerhope.com).
A byte can store 8 bits of information, meaning that a byte can store around 28 or 256 values. A
kilobyte is around 1,024 (210) bytes, a megabyte has 1,024 kilobytes, a gigabyte has 1,024
megabytes, and a terabyte has 1,024 gigabytes.
Hex Code
In computer coding, we can use a Base 16 number system to represent colors as well using the Hexadecimal
System. Computers
represent colors in a combination of red, green, and blue, and a computer can represent 256 shades of each
of these colors. The hexcode for colors is in
the form #000000, with each section of two numbers representing the shades of red, green, and blue
respectively being displayed in the output. However, if we
used a Base 10 system, the code would be 9 digits long. For example, the code for white would be #255255255.
This is fairly difficult to manipulate, but in a
Base 16 system, 256 numbers can be represented using only 2 digits. The digits for Base 16 are 0-9 and A-F,
meaning FF is the highest double digit number in Base
16 which representing the number 255. We then have 256 numbers between 00-FF, meaning that the code for white ends up
being #FFFFFF. The code for the background of this
section is #A6A6A6. A6 in Base 10 is 166, meaning that this background is shade 166 of red, green, and blue.
With Base 16, representing information such as color is
much easier to manage.
References and Resources
Binary. (n.d.). Retrieved from https://techterms.com/definition/binary.
Darling, D. J. (2004). The universal book of mathematics: from Abracadabra to Zenos paradoxes. Edison, NJ: Book Sales Inc.
Denise Schmandt-Besserat. (n.d.). Retrieved from https://sites.utexas.edu/dsb/tokens/tokens/.
Egyptian numerals. (n.d.). Retrieved from http://mathshistory.st-andrews.ac.uk/HistTopics/Egyptian_numerals.html.
Lamb, E. (2014, August 31). Ancient Babylonian Number System Had No Zero. Retrieved from https://blogs.scientificamerican.com/roots-of-unity/ancient-babylonian-number-system-had-no-zero/.
McCallum. (2012, March 10). Math History Mysteries Part 1: Counting using Tally Sticks. Retrieved from http://annmccallumbooks.com/math-history-mysteries-part-1-counting-using-tally-sticks/.
What is Binary? (2019, October 7). Retrieved from https://www.computerhope.com/jargon/b/binary.htm.
Y, D. (2019, August 5). The Lebombo Bone: The Oldest Mathematical Artifact in the World. Retrieved from https://afrolegends.com/2019/05/17/the-lebombo-bone-the-oldest-mathematical-artifact-in-the-world/.
The image to the left is the Ishango Bone,
found deep
in Africa. The bone itself is dated as 20,000 years old and is one of the first examples of a tally system
coming into play.
Scientists presume that this bone was used to mark off days on a calendar, and it is one of the earliest
examples
we have
currently of a tally system being used. An older bone, known as the "Lebombo
Bone",
dates back even further to between 44,200 and 43,000 years old. According to "The Universal Book of
Mathematics," the 29 notches
on the Lebambo Bone may have been used as a lunar phase counter. In that case African women may have been
the first mathematicians
because keeping track of menstrual cycles requires a lunar calendar. This makes the Lebambo Bone the
world's oldest known
mathematical artifact. Through the discovery of ancient artifacts like this we can see, even thousands upon
thousands of years in the past, manking
needed a way to express
amounts, and the tally system was a vital first step along that path.
However, as one might expect, the tally system is not perfect when it
comes to accounting for all objects in a given space. Just as counting fingers quickly becomes overwhelming,
using
tally marks becomes unwieldy with larger and larger numbers. Accounting for larger groups of sheep with
tallies could be
just as difficult as counting the sheep. A solution to this problem existed in the Middle East, where small
clay tokens were
used to represent quantities. Different shapes represented different commodities, like oil or sheep, and
notches on the tokens
represented different values. The image to the
right denotes an amount of grain, with the circular notches representing a large measure of grain and the
wedges representing
a small measure of grain. Here, this token would represent four large measures and four small measures of
grain.
Ancient Egyptians had a writing system
(hieroglyphics)
dating from about 3000 BC, and from this writing system, we have found evidence of a number system. Their
number system was
a Base 10 system, similiar to the one we are familiar with. The symbols to the right were used to denote
powers of ten up to
10,000,000. Over time, and with the introduction of papyrus, the number system evolved to a more dynamic
writing system that allowed
for the expression of larger numbers using less symbols. Seen here, more
symbols
were added to denote more numbers,
such as the introduction of symbols for each individual multiple of 100. Numbers like 9,999, which would
originally have take 36 hieroglyphs to express, could now be expressed with 4 symbols.
Meanwhile, the Babylonians used a number system consisting of Base 60,
meaning each number less than 60
needed its own numerical symbol, as seen on the left. As can be seen, clusters of ten were combined together
to more easily facilitate expression. However, the Babylonians did not have a symbol for zero.
Interestingly, the Babylonians
did not have a symbol for zero. For example, to express the number 60, the symbol for one would be used. A
space would be included after the "1,"
leaving both numbers looking almost exactly similiar. Though Base 60 seems
much too difficult to use,
we can see how their number system affects us to this day, such as in the 60 seconds and minutes of angular
measurement, in the 180 degrees of a triangle, and in the 360 degrees of a circle.
For the Base 10 Number System we use today, we can visualize
how numbers are represented using the image to the right.
We see that a number as large as 1,247 can be represented with as few as four symbols due to the place value
system. We see that a 1 in the thousands place represents a
thousand individual blocks or tallys, a 2 in the hundreds place represents having two groups of 100 blocks, and so on.
Metric System
