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A Brief History of Numbers and Number Systems
Gaby De Jesus
Franciscan University of Steubenville
EDU 331
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One of the quintessential things mathematicians are always seeking to find is where
things come from. When it comes to formulas and theorems, a derivation or a proof is always
sought; however, when it comes to numbers and number systems – the very basic thing
mathematics is composed of – it seems that all current day users of numbers never ponder how
these systems were developed. The modern numbers system has not always been, and it is
because of the invention of other number systems throughout different cultures and times that it
is what it is today.
The Egyptian culture is one of the first recorded cultures that developed and utilized its
own number system. Their system dates back to 3,000 BCE, and was written on papyrus along
with other Egyptian writings. All written communication at this time was expressed in
hieroglyphs, so it follows naturally that numbers were represented by hieroglyphs as well. The
Egyptian number system was a decimal system meaning that it had a base of ten, so each power
of ten was illustrated by a different symbol. The number 1 was represented by a stroke, 10 by an
arch, 100 by a coil of rope, 1,000 by a lotus plant, 10,000 by a finger, 100,000 by a tadpole or
frog, and 1,000,000 by a man with his arms raised above his head. Using these base symbols, all
other numbers could be composed by simply using addition. For example, the number 432 would
be represent by four coils of rope, three arches, and two strokes. However, the largest power of
ten would appear on the right because the Egyptians wrote from right to left. The strictly additive
nature of this number system is its biggest advantage because it left no room for
misinterpretation (Weibull, n.d., p. 5).
About 1,100 to 1,200 years after the Egyptians, the Babylonians arose with their own
number system between 1,900 BCE and 1,800 BCE. Similarly, their numbers were derived from
their writing system of cuneiform that consisted of wedge shapes. Writing was done using a
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stylus on clay tablets that were then left in the sun to dry. Unlike the Egyptians, the Babylonian
number system was sexagesimal with base 60 rather than 10. The choice of base 60 is reflective
of their way of life at the time. Babylonians broke each day into 24 hours, each hour into 60
minutes, and each minute into 60 seconds (Troutman, n.d.). Because of this, having a number
system with base 60 was most practical and useful to them. The number 1 was represented by a
downward pointing wedge (), and the number 10 was represented by a left pointing wedge
(). Larger numbers could be written by using a combination of these two wedge shapes.
However, the Babylonian system arranged these wedge shapes into groups and utilized place
value. The right most place designated ones (or anything less than 60), the second place
designated 60 to the first power, and the third place designated 60 to the third power, and so on
and so forth. The value of a larger number would be calculated by multiplying the numbers in
each place value by their respective power and adding the products together. For example, the
number 142 would look like and be interpreted as 22 + (2 601). This number
system lacked a symbol for zero, but it was clear that they used the idea by leaving spaces blank.
One of the biggest problems with this number system is that its use of symbols in different
groups made it extremely difficult to correctly interpret the actual value of the number being
written (Edkins, 2006).
At the turn into the Common Era, the Romans were developing their number system
which, just like the Babylonians, was based on their lettering system. Numbers were portrayed
by letters. 1 was represented by I, 5 by V, 10 by X, 50 by L, 100 by C, 500 by D, and 1,000 by
M. There are some connections that can be made between these symbols. For one, a V is the top
half of an X, and 5 is half of ten. Additionally, the C for 100 and M for 1,000 were taken from
the roman words centum and mille respectively. As made apparent by the base symbols, the
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Roman number system was also a decimal system, and larger numbers were composed through
addition. For example, the number 2,017 would be written as MMXVII and interpreted as 1,000
+ 1,000 + 10 + 5 + 1 + 1. The simplicity of this number system reflects Roman culture in the
way that not much emphasis was placed on mathematics, but rather, the focus was on creating
the world’s largest empire. Surprisingly, the practice well known today of using subtraction to
write numerals (i.e IX as 9) was not developed until after the fall of the Roman Empire (Weibull,
n.d., p. 10). Although quite basic, this number system is still used in multiple contexts today
which shows that its ease of use has made it more valuable.
More widely dominant than the Roman number system is the Hindu-Arabic number
system which is the one used today. This system originated in India and reached Europe through
the writings of two famous Eastern mathematicians – al-Khwarizmi and al-Kindi (The Editors of
Encyclopaedia Britannica, 2017). The Hindu-Arabic number system consists of ten digits
(0,1,2,3,4,5,6,7,8,9) that, when placed in different combinations, can represent all possible
numbers. Like the Egyptians and Romans, this is a decimal number system. The use of base ten
and of ten digits comes from the ten fingers as the term “digit” is a synonym for the word finger.
Just as the Babylonian number system utilized grouping and place value, so too, does the Hindu-
Arabic number system use grouping and place value. A group of ten 1’s is the same as one 10, a
group of ten 10’s is the same as one 100, a group of ten 100’s is the same as one 1,000, etc. Each
of these groups has its own place value. Ones are the right most place, then tens, hundreds,
thousands, etc. The value of larger numbers can be calculated by multiplying each place value by
its digit and adding all the products together (“Hindu-Arabic numeration system,” n.d.). For
example, 571 is (1005) + (107) + (11). One of the most beneficial characteristics of this
number system is that very small amounts of symbols are required to communicate large
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numbers. Additionally, the development of Hindu-Arabic numerals makes other fields of
mathematics, such as algebra and calculus, possible. It is perhaps because of these reasons that
this is the number system used today.
Such historical information on the development of numbers and number systems can be
very useful in the classroom for two reasons. Firstly, utilizing this information will provide
students with a real world application of mathematics by exposing them to numbers within the
context of societies throughout the world. By seeing that numbers were necessary for these
ancient cultures to function and thrive, students will hopefully be able to realize the practicality
behind mathematics. Secondly, the teaching of the history of number systems in the mathematics
classroom lends itself perfectly to taking advantage of cross-curricular instruction. Cross-
curricular instruction is characterized by bringing the content from other classes into a certain
classroom. If the history of the Egyptian number system in the math classroom was being taught
at the same time students were learning about the Egyptian Empire in more detail in the history
classroom, students would greatly benefit because the cross-curricular nature of the teaching
would allow them to make connections and bridges between the information from the two
subjects.
The Hindu-Arabic number system used today would not be as refined and practical if it
weren’t for the previous number systems developed by the Babylonians, Egyptians, and Romans
among others. Although unique and occurring over vastly different time periods, these number
systems were also similar in the fact that they utilize place value, addition and/or multiplication
for composition of larger numbers, and a base relevant to what was needed at the time. As taken-
for-granted and overlooked as numbers and number systems may be, the differentiation in
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education and the leaps and bounds that have been made in the field of mathematics and would
not have been possible without them.
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References
Edkins, J. (2006). Babylonian Numbers. Retrieved September 30, 2017, from
http://gwydir.demon.co.uk/jo/numbers/babylon/.
Hindu-Arabic numeration system. (n.d.). Retrieved September 30, 2017, from http://www.basic-
mathematics.com/hindu-arabic-numeration-system.html.
The Editors of Encyclopaedia Britannica. (2017, September 08). Hindu-Arabic numerals.
Retrieved September 30, 2017 from https://www.britannica.com/topic/Hindu-Arabic-
numerals.
Troutman, J. (n.d.) The Babylonian Number System. Retrieved September 30, 2017, from
http://www.math.wichita.edu/history/topics/num-sys.html#babylonian.
Weibull, N. (n.d.). An Historical Survey of Number Systems. Retrieved September 30, 2017,
from http://www.math.chalmers.se/Math/Grundutb/GU/MAN250/S04/Number_Systems.
pdf.
