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Mathematical Patterns and Nature. Number Systems Report

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Updated: Oct 17th, 2021

Introduction

A number system is the set of symbols used to express quantities as the basis for counting, comparing amounts, determining order, performing calculations, representing value and performing calculations. It is the set of characters and mathematical rules used to represent a number. Examples include the Arabic, Babylonian, Chinese, and Egyptian, Greek, Mayan, and Roman number systems. The ISBN and Dewey Decimal System are examples of number systems used in libraries. Even Social Security has a number system. (Cajori, 1993)

Roman Numerical System

Originally the Roman numbers were independent and used original symbols but currently they use Roman alphabet letters, a good example is Etruscans that used I Λ X ⋔ 8 ⊕ for I V X L C M. The letters I and X are letters in their alphabet. From these symbols, folk etymology say that the V usually represents a hand and the X is a combination of V’s One inverted. The Etrusco Roman numericals were derived from notches on tally sticks. For example I was the letter I scored across a stick. The fifth notch was double cut (V) and the tenth was cross cut (X). This then if one wants to get seven the symbol was IIIIVII and it was abbreviated as VII. The seventeen letter was then written as XVII. Zero didn’t have any symbol but they used the word nulla.

The tenth V or X along the stick received an extra stroke. 50 were written as N, И, K, Ψ, ⋔. But at the time of Augustus it had a ⊥ sign and later it was transformed to L. For 100 there were also various shapes like Ж, ⋉, ⋈, H, but the Ж took more wait. It then later changed to >I< or ƆIC but C finally took more weight because it was a letter and also it stood for centum which meant “hundred” in Latin.

500 was written as Ɔ superposed on a ⋌ or ⊢ — and by the time of Augustus it became D and finally it was written as D. 1000 was encircled or boxed X: Ⓧ, ⊗, ⊕, and by the time of Augustus it was a phi. This was a reason for 500 been half D to phi.

Romans used decimal system for whole numbers, and a duodecimal for fractions, for example, the divisibility of twelve was easier to handle than those based on tens. They used notations to indicate the twelfths and halves. For example, a dot indicated a twelfth. Dots then were used for fractions up the fifth. S was used to abbreviate a six twelfths and dots were then added for values of seven to eleven twelfth just like tallies in I to V. (Menninger, 1992)

Whole numbers in Roman Numerical System.

Symbol Value Name
I 1 (One) unus
V 5 (five) quinque
X 10 (ten) decem
L 50 (fifty) quinquaginta
C 100 (hundred) centum
D 500 (five hundred) quingenti
M 1000 (one thousand) mille

Fraction in Roman Numerical System.

Symbol Value Name
1/12 uncia “ounce”
•• 2/12 sextans “sixth”
••• 3/12 quadrans “quarter”
•••• 4/12 triens “third”
•••• 5/12 quincuncis “five-ounce”
S 6/12 semis “half”
S• 7/12 septunx “seven-ounce”
S•• 8/12 bes “twice”
S••• 9/12 dodrans “nineth-ounce”
S•••• 10/12 dextans “ten-ounce”
S••••• 11/12 deunx “less an ounce”
I 12/12 as “unit”

The number system is not additive or subtractive in its form but it was ordinal.

Greek Number System

This is the representation of Greek numerals using Greek alphabet. It is also called Milesian numericals, Alexandrian numericals or alphabetic numericals. In present Greece they are still used.

Before the use of Greek alphabet, Linear A and Linear B used different symbols with symbols ranging from 1, 10, 100, 1000 and 10000 and they used the following formulae: | = 1, – = 10, ◦ = 100, ¤ = 1000, ☼ = 10000. But, the easiest numerical system which was associated with alphabet was a set of acrophonic Attic numericals, they operated like the Roman numerals and used they following design Ι = 1, Π = 5, Δ = 10, ΠΔ = 50, Η = 100, ΠΗ = 500, Χ = 1000, ΠΧ = 5000, Μ = 10000 and ΠΜ = 50000.

From the 4th century BC the acrophonic system was replaced by an alphabetic system which was also called the ionic numeral system. The design was such that the units 1, 2, 3… 9 were assigned a separate letter, tens 10, 20… 90, a separate letter and the hundreds a separate letter. In total they required a total of 27 letters from the 24 letter Greek alphabet. This lead to the extension by three letters to obsolete letters: digamma ϝ, (stigma ϛ / Modern Greek στ) for 6, qoppa ϟ for 90, and sampi ϡ for 900. To differentiate numerals to letters they used a symbol which was similar to an acute sign called “keraia”.

The alphabetic system operates on additive rule in which the numeric values if they are added together, that is, 241 is represented by σμαʹ for (200 + 40 + 1). To show the values that are from1,000 to 999,999 the same letters are used again as thousands, tens of thousands and hundreds of thousands. A keraia which is usually on the left is used to distinguish them from the standard use. For example, 2008 is represented as ͵βηʹ (2000 + 8).

For the values 10,000 the Greeks used a myriad (M’) and for one hundred million they used myriad myriad (MM’). Archimedes the philosopher proposed a way to name high or big numbers such as sand grains on the beach and all other beaches on the world. This Greek system was used in sexagesimal position numbering system by the Hellenistic astronomers. This is through limiting position to a maximum value of 50+9 and included a symbol for zero. Its position was limited to the fraction part of a number which was sometimes refereed to as minutes, seconds and were not used for integral part of a number.

The zero symbols changed in time in its placement. During the second century a symbol that was used was a papyri and was a small single with an overbar with many diameters but later shortened to one diameter. This was similar to the modern o macron (ō). But, later the overbar was completely omitted. (Ifraim, 2000)

Symbols for Greek Lower Numbers.

Letter Value Letter Value Letter Value
αʹ 1 ιʹ 10 ρʹ 100
βʹ 2 κʹ 20 σʹ 200
γʹ 3 λʹ 30 τʹ 300
δʹ 4 μʹ 40 υʹ 400
εʹ 5 νʹ 50 φʹ 500
ϝʹ or ϛʹ or στʹ 6 ξʹ 60 χʹ 600
ζʹ 7 οʹ 70 ψʹ 700
ηʹ 8 πʹ 80 ωʹ 800
θʹ 9 ϟʹ 90 ϡʹ 900

Greek numbers of higher numbers.

Decimal Symbol Greek Numerical
1 I ἴος (ios)
5 Π πέντε (pente)
10 Δ δέκα (deka)
100 Η hεκατόν (hekaton)
1000 Χ χίλιοι (khilioi)
10000 Μ μύριοι (myrioi)

Roman, Greek and Hindu Arabic Number System

The majority of ancient peoples, however, including the Chinese, the Greeks, the Romans, and the Hebrews, used the decimal system.

The earliest numerical notation that was used by the Greeks was the Attic numerical system. It used a stroke for 1, and had symbols for 5, 10, 100, 1,000, and 10,000. And in about 500 B.C. the Greeks borrowed the Egyptian numeral system and used it to derive an alphabetic decimal system. This Ionic system was somehow sophisticated than the Egyptian system. Like the Egyptians, there wasn’t a provision for place value or a symbol for zero. (Cajori, 1993)

At the same time, the Romans also developed an alphabetic numeral system. The Romans used letters of the alphabet to represent numbers, and this system is still used in things such as page numbers, clock faces, and dates of movies. In general, the letters are placed in a decreasing order of value, for example, CXV11 = 117. Letters can be repeated one or two times to increase value, but the letters cannot be repeated three times, so XXXX is not used for 40 instead XL is used. Like in the Greek system, there wasn’t a provision for place value or a symbol for zero.

The Arabic numeral system (also called the Hindu numeral system or Hindu-Arabic numeral system) is considered one of the most significant developments in mathematics. It was developed in the 4th and 3rd century B.C. Most historians agree that it was first conceived in India (the Arabs themselves call the numerals they use “Indian numerals”) and was then transmitted to the Islamic world and then via North Africa and Spain, to Europe. A place value decimal system, it used symbols for each number from one to nine. The Indians gradually developed a way of eliminating place names, and invented the symbol sunya [empty], which we call zero. During the 7th century A.D. the Arabs learned Indian arithmetic from scientific writings of the Indians and the Greeks. In the 10th century A.D. Arab mathematicians extended the decimal numeral system to include fractions. Leonardo Fibonacci, an Italian mathematician who had studied in Algeria, promoted the Arabic numeral system in his Liber Abaci (1202). Conversely, the system was not widely used in Europe until the invention of printing. (McSeveny, 2003)

Uniqueness and Difference of Hindu Arabic number System

The Arabic numbers usually consists of ten numbers and are also in sets of tens. This is completely different compared to other kinds of number systems. They also write their numbers from right to the left.

The commonly referred to as the West Arabic numerals are currently with the East Arabic numerals and similarly stem from the Hindu figures making them the forerunners of the Western figures.

The Hindu system can be described as a pure place value system, the reason as to why a zero is required in this system. (Cajori, 1993) Within the context of the Indo-European civilizations it is only the Hindus who have always applied a zero.

All the Roman and Greek numerical numbers had no placement for zero or its value. This then made it difficult for calculations. Even though calculations was not highly valued compared to the Arabs who were business oriented and were supposed to have ways to work out their monetary calculations.

The Roman and Greek numerical system values and figures are read from left to right while for Hindu Arabic system the values are read from right to the left. The Greeks and Romans fancied geometry and figurative numbers and were not concerned with computation and they perceived and tried to prove that there was no need for zero. An example was that there was nothing like a zero sheep if you count the flock of sheep. They usually used counting boards and abacus for computational purposes instead of using numerical. This then meant that they had to use many and repeated symbols to show a certain number.

Calculations using Greek symbols was cumbersome, for example, alpha + alpha = beta instead of the Arabic system (1+1) = 2. The Greek system also had many symbols with each denoting a different value this then made it hard during the memorizing of these digits. Hence this shows that they had to use a lot of symbols for each character and in total they had 27 symbols or names.

Hindu Arabic numerical system had a positional number kind of system which used a base ten. They also used a dot to signify a zero position and were also used as a place holder and for computational purposes. They had symbols for numbers from one to nine with no repetition. The use of zero as an example is when they use “1 sata, 5” which means 105. They had decimal place to show the tenths.

These three number systems are also associated with denoting the number of times that a thing happens. Even though there manipulations for calculations are diverse and others are not easy to calculate with, for example, multiply 378 by 378 in Roman form, if possible, takes a lot of time to work out or else impossible to solve.

Roman numerical system can be added and subtracted whereby letters are used to show certain base number, for example, X for ten and D for 500 and the other letters are denoted from combination of numbers. This then made it long to denote a simple value and work on it when compared to Arabic system.

In the Arabic number system there are fewer numbers that are required because the presence of zero makes digits to start again e.g. 10, 20… 100 etc, hence more suitable for calculations. While the Roman numerical system were long, for example, writing XCIX for 99 and there was also frequent repetition such as the X in the above example (Hayashi, 1995).

All these three numerical systems are still used today in various applications. In the Arabic numerical system it’s normally used in calculators and telephones. The Roman numerical system is still used in the number system of years and also in the wall clocks. The Greek numerical system is usually used in physics and mathematics like words such as alpha, omega and beta. Their day to day applicable situation differentiates them between various fields of study and research. Use of one type of symbol from these numbers shows the direction that one takes to solve a certain issue.

References

Cajori, F. (1993). A History of Mathematical Notations. New York: Dover Publications.

Hayashi, T. (1995). The Bakhshali Manuscript, An ancient Indian Mathematical treatise. Groningen: Egbert Forsten.

Ifram, G. (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. New York: John Wiley & Sons.

McSeveny, A. (2003). New Signposts Mathematics 7. Sydney: Pearson Longman.

Menninger, K. (1992). Number Words and Number Symbols: A Cultural History of Numbers. New York: Dover Publications.

Victor, J. (2007). The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton, NJ: Princeton University Press.

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