Ancient Civilizations Numbering Systems and Foundation Calculation of Measurements
Numbers is one among the tons of discovery in human civilization, especially useful in the major called Mathematics, also that’s hard to predict the originality on when is the first idea of number or numbers does exist. The oldest information about numbers was recorded since the era of BC. It was written the first purpose of a number on numbering the populations; therefore, we could count it.
With more and more research permitted and developed in the last decades, then humanity can get more historical evidence, which has brought us to Egypt, especially on how the ancient Egyptians’ using this number on counting their captured enemies and their treasures. Within those forms of numbers, some of us also still can be seen as some form of symbols in Geometry form inside the Pyramid in Egypt, or if you can go to the Museum Ashmolean in Oxford England. With those shreds of evidence, it does prove, an additional the use of numbers as identification for a human to understand easier the amount of something, or to identify some form of geometrical object in nature has been existed since centuries ago, and perhaps before Ancient Egyptians using it.
Similarly, with prehistoric man, called caveman, which with the latest research about them, this caveman have known about number and numbers in order of doing, what current knowledge called as some simple math calculations in order of their survival at that time. This statement based on the current human research evidence on their founded cave, and their tools in such a form of geometrical tools, that has been used with some geometrical model, like most of the pentagon model are using as a spear for hunting, or a scraper or an ax, etc. Also, during their hunting, they will share the captured animal with their team, they will do a measurement without an idea of measurement based on the portion to be shared, or they will do a counting if they have some pet animal to be feed, etc. And as we know in the very early or beginning, it’s always been enough by general random measurements like a little, or a medium, or a lot. However, by the time of existences in humans and the development of their civilizations with more populations, then the needs of standards measurement and calculation, that can be used to counting and measured on how many is it in more detail, eventually got invented. This invention which also discovered in the use of number and numbers into a form of measurement and calculation has brought humanity on learning about a variant of traditional counting and measurement, and this is the new beginning of Mathematics as a way of thinking in systematics order that from what nature has told us.
Throughout, a logic and a way of thinking in humanity to survive, we could say, in the new beginning of the use of mathematics perhaps by a person that counting about the total numbers of their pet animals, by using a stone and/or by using a rope that been knot for one set of animals of one kind. In terms, each stone that has been collected or each knot that has been made in one rope can be identified as one set of animals of one kind by general. Then, along with that history of humankind, someone, somebody in non-recorded has been using a finger as well on doing the counting. With that, we could say, the idea of exchange has been existed without a realization, by terms like one knot indicate as ten medium stones, one medium stone as ten small stones, and one small stone indicate as ten fingers, then continue till the end. In a bit more detail, like the amount of the stones in one group of the stone pyramid has been looks like equal to ten fingers, then this can be used as a measurement of tens, or in the current method, this called as the initiate for two digits. The first set of stones has been made like a pyramid with a set of 10 small stones that can be saved and convert by one medium size of the stone, which one medium size of a stone indicate as 10 small stone. By now, we can see clearly the mean as ten fingers are one small stone, and ten small stones are one medium stone, so, if we have ten medium stones, means we have 100 small stones, in terms, we have 1,000 animals. Even though, based on historian prediction, the first measurement on one unit was using a tree branch as a stick, to indicate the measurement of one unit finger. With that ancient general standard measurement and calculation get discovered with purpose on easier to counting and measuring, and once we have conversion tools, then we know it now if we have three medium stones with seven small stones and eight sticks, they will be means three hundred and seventy-eight of things, in this case, we can say it is three hundred and seventy-eight of animals. With this idea that has been discovered, the current mathematics civilizations realized the form of what we called a digit, which has been formed unconsciously by that ancient time.
With that scenario, ten hand fingers or ten sticks will indicate the end of our fundamental counting and measurement with conversions as well. But not all cavemen were using the count of ten hand fingers or a summation from both hands as fundamental counting. Some cavemen at that time may be using a palm instead of a basic finger in their measurement. Even though, for most cavemen at that time, perhaps fingers at hand may easier, also according to our current 21st logic of thinking. Therefore, this group of palm user caveman just mixed it as one palm as five fingers. Moreover, some other caveman at that time also using all fingers they have, which from their hand and foot, which bring to total of twenty, and this become the third fundamental counting numbers of the third group of cavemen. Later on, till this study, then this variant method of counting and measurement will be called fundamental counting.
Another reason, because the stones, the sticks, the leaves, and some other physical objects are not easier to be used in counting and measurement, then some people in a bit of more advanced era get started by designing symbols to describing their practical counting measurement to what is called as a symbolic geometry form, which will be called as numbers in a form of writing. For more than about 5,000 years ago, the current discovery has brought us to a civilization of Ancient Babylonian, which this Ancient Babylonian has developed this method of early writing, the model of writing that in a form of images, some form that has been using to indicating symbols of an object as standardizations at that era, which they wrote it on a wall or soft ground by using some sharp stone-like scraper, that has been sharpened at one edge. Also, with that writing method on geometrical form development in Ancient Babylonian, we discovered there are some kinds of special form in writing to identify some numbers on their civilizations to measurement and counting, like the geometric symbolic form of 1, 10, and 100. Slight example by some digital writing, like `Gamma` means 1, `-<` means 10, `Gamma>-` means 100. These symbols got invented to be used as an indication of numbers then Ancient Babylonians are using those symbols as needed, to help you more with the form of Ancient Babylonian, please check it here or this one, and more in our references below. Please look at the image provided in this article from outside resource, about the study we want to show about how those symbols have been forming for the first group on a number to numbering, especially with the amount of 100 and 1,000 get formed. Also, the Ancient Babylonian symbolic system for the number of three hundred seventy-eight. This method of writing numbers, later on, is called an addition because we need to be adding a symbol to get a final number. Besides, to demonstrate a higher value of numbers, then Babylonian using multiplication, like using the symbols that get a form to show about ten times hundred, to perform a thousand. If those symbols from the smaller groups of numbers show earlier than symbols of the bigger groups of number, then this identified about those groups need to get multiplied. And if a smaller group of number show earlier than another smaller groups of number, then those symbols need to be adding it.
Another discovery from an ancient civilization in the study of counting and measurement, called Ancient Egyptians, which have another method on counting and measurement, it called as hieroglyph method on writing and a hieratic system on their measurement and calculation. This system of hieroglyph shown very non-practical according to the current 21st-century model. An example like in the cover image on writing the symbolic form of 3,244 and 21,237. In most discoveries, the Ancient Egyptian using this numbering system to decorate their wall, especially like their monuments that you can found in current Egypt. Lastly, the Ancient Egyptians also using the system hieratic, which very efficient than hieroglyph, this system later on used by the Ancient Egyptian in their daily lives to live on counting and measuring, also further development needs to be found, accordingly about been said this Ancient Egyptians has discovered the idea of the connection between 3, 4, and 5 units earlier than Pythagoras. After all, this Ancient Egyptian also made their writing better on a Papyrus paper than the previous ancient civilizations.
Now, from those Ancient Babylonian and Ancient Egyptians, we move to Ancient Greeks, which based on discovery evidence in history, this Ancient Greek also developing several models on identifying numbers. Like modern Greeks, they developed all kinds of alphabets with some other addition in their symbolics. Each of those symbols identifies as some values. The first of nine symbols identifies as a group of numbers numbering from one to nine; then the next nine groups of symbols of a number on numbering show about the tens to the ninety; the last nine groups of a number on numbering show about hundreds to the nine hundred. Also, they have no symbols for zero. In Ancient Greek method for those civilizations to identify a thousand, Ancient Greeks method is adding a vertical line on the left side of the first nine symbols. So, `I \gamma` means 3000; `IZ` means 7000. A group of 4627 mean `I\DeltaXKZ` , the underline identifies about this alphabet create a number. Unlike the other two Ancient civilizations called Babylonian and Egyptians, Ancient Greek with also another ancient civilization called Ancient Jews, using nearly similar on their alphabetical into writing their numbers on numbering.
The other ancient civilization that discovers in the east, called Ancient Chinese also one of the civilizations at that time who’s using sticks systems symbols for counting a number. Unlike the other ancient civilizations, the Ancient Chinese are using a system placement on their symbols, in terms, some symbols will have a different value according to the position of their sticks. In a part of the cover image, we can see how ancient Chinese were writing their numbering system at that time, by category 1 to 9 and 10 to 90. You will also see in that system of numbering from one to nine, each stick is vertical as identification of each unit, and each combination of the vertical-horizontal line indicates five units. In the category of tens which is from ten to ninety, each horizontal stick indicates tens of unit. And each horizontal-vertical line indicates fifty units. Also, hundreds will be written similarly to the unit written symbolic system. So, the symbols of two vertical lines indicate as two units or two hundred units will depend on the position in that numbering system. Thousands have been written similarly with a tens of unit system, tens of thousands also similar with the unit and go on. Example like the value of 7,684 will be written like one vertical line at the top with two horizontal lines at the bottom in the first symbol, which indicate the value of 7, then one horizontal line at the top with one vertical line at the bottom, which indicate the value of 6, followed by one vertical line at the top with three horizontal lines at the bottom in the third symbol, which indicate the value of 8, and lastly four vertical lines which indicates as the value of 4. There is no symbol for the value of zero and just a space will be used to indicate that value of zero. Like 704, will be written like one vertical line at the top followed by two horizontal lines at the bottom, which indicates the value of 7, then space which is zero, followed by four vertical lines, which indicates the value of 4. If and only if the empty space between didn’t measure, then the reading will become 74 instead of 704.
Furthermore, Ancient Rome perhaps having its system of numbering from the Etruscan Civilization, the ancient Italian civilization. A geometric symbol that been use like an alphabetical system, such as “I” for one, “V” for five, “X” for ten, “L” for fifty, “C” for hundreds, “D” for five hundred, and “M” for a thousand. Then, two of “I” like “II” means two units, three of “I” like “III” means three units. And the combination of “I” before “V” indicates four units, and this is the system that indicates as a subtraction system, if there’s a smaller number come earlier than a bigger number, then a bigger number needs to be subtracted by the smaller number. Follow by “VI” means six, “VII” means seven, “VIII” means eight, a symbol of “IX” indicate nine based on the principle of the subtraction system. And the rest will do like, “X” for a ten, “XX” for a twenty, “L” for a fifty, “C” for a hundred, “D” for a five hundred, “M” for a thousand, “CMC” for a ten of thousand. Those are the system of Ancient Rome numbering system. If there is a smaller symbol from the next symbol, then the next symbol gets subtracted by the previous one, also if the symbol indicates bigger or similar than the next symbol, then those two values will be added to each other. Like “LX” means sixty, and ‘XL” means forty. This is the counting and measuring of Ancient Rome's discovery.
Also, in the Ancient Rome numbering system, they have still been using by the English language and some other current Western Civilization with some purposes or occasion. This numbering system in Ancient Rome has also been used to indicate a section of a chapter in a book. Also, this numbering system can be used in the clocking system, some monuments, clays, etc. Some people may indicate this Rome numbering system is difficult. For example, if we want to show the value of a thousand eight hundred and eighty-eight based on the Ancient Rome system, will become like “MDCCCLXXXVIII”, in the meantime in 21st modern math, we just write it as 1,888.
Another numbering system comes from the mixed of Ancient Hinduism-Arabian, because this numbering system came from India, long before the Christian era, then use by ancient Arabians, in which the Arabian defeat the Persian in the “VIII” century, and at that time this Hinduism-Arabian numbering system get introduce in every conquered territory. For this reason, also, we often see it at the Spain manuscript era “X”, in which these symbols become a standardization of the current numbering system after some evolution. By the coming of an era, many civilizations at that time, start to adopt this numbering system. Then in the era of “XV”, the symbols of this numbering system become a symbolic model that many of us, the humans, using it at this moment, as a set of number: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.
Based on the fundamental measuring system of us is tens. Therefore, sometimes the system also gets called a decimal system, as per “Decem” means “tens” based on the Latin word. The system is based on a positioning system, which indicates a different value in each position of that symbol. By placing one number each in front of or at the end of, then those groups of a number will increase ten times, as well as been seen below:
In a group of numbers like 4,962, the meaning of 4 means thousands, in terms four thousand, a number of 9 means hundreds, in terms nine hundred, a number of 6 means tens, in term of sixty, and a number of 2 means just unit of a number, in terms of two units. In a practical format of those 4,962 meaning as 4000 with 900 and 60 and 2. Another example like 704 in the previous example, the meaning of 7 in that group of number is hundreds, in terms seven hundred, then the 0 indicate of tens, but since the value is zero, then just zero tens, and 4 as a unit of a number, in terms of four-unit. In a practical format of those 704 meaning as 700 with 00 and 4. Because now we have a special symbol for zero, then this helps most of civilization to avoid the misunderstanding on reading a group of number like in the past by using just a space between a number like happened in most of the ancient civilizations.
With the latest numbering system that has around ten symbols, now most of the civilization at that time till now can write whatever form of group numbers in our convenient in whatever value of that numbers. Like in the example above, if we have just a number of 1 then once we add other symbols to the right side of this number of 1, like a number of 0, then the number of 1 will become a new group of numbers in the category of tens, in which symbolic as 10. With more and more addition of adding an extra number to the right like before, we make this value of 1 in the number of 1 to become a much bigger value as much as ten times than the originals. We also can write the much smaller values as much as ten times than the originals by adding a decimal point. The first number after the decimal point is identified as over tenth, the next will be over hundredth, then over thousandth, and go on. Like, 0.01 is one over hundredth, then 0.001 is one of a thousandth, etc. We will make this value as smaller as possible by adding one more symbolic number after the decimal point and a unit number, like 0.1 and not 01.0, or 0.01 and not 00.1, because the 0.01 and 00.1 has a difference in their value and meaning.
Of course, the biggerness or the smallness of this value in the numbering system indicates something, and we will try to expose it more the use of it in a practical way in our other article.
Not all placement systems need tens as the foundation of calculation. Several kinds of basic calculations have been developed and introduced. To show the differences between the systems that have different foundation calculation, let us describe it as below scenario:
“A kid entered a first-grade class at the age of 10. After study about 22 years, the kid enters university at the age of 32 years old. The age of the kid is 41 years old once the kid graduated from the university, and the kid gets marriage at the age of 100, then the kiddies at the age of 240”.
Now, if we implied the placement system with the foundation of calculation “tens” to that scenario, the system will not be applied that much, of course, the kid has been peacefully life to live. If we used the system based on the fifth foundation, in defining those numbers in that scenario. In this system, we will only be using the symbols of 1, 2, 3, 4, and 0. By placing one space to the left we will increase the value about five times. The last number in a system of group number identified as a unit, the next numbers in the front of it identified as the fifth of it as 5 * 1, the second number after the last is a number of twenty-fifth as 5 * 5, the third number from the last number is a hundredth-twenty-fifth as 5*25, etc.
Let’s applied this system to the scenario case of numbers. Like 10 is one of fifth and one empty, and that’s five; 22 is two of fifth and two of units, and that’s twelve; 32 is three of fifth and two units, and that’s seventeen; 41 is four of fifth and one of the unit, and that’s twenty-one; 100 is one of twenty-fifth, fifth of zero’s, and one of zero, and that’s twenty-five; 240 is two of twenty-fifth, four of fifth, and one zero units, and that’s seventy. With interpretation like this system, then that scenario becomes mathematically logical makes sense.
Placement systems called “binary” systems have a foundation of numbering based on two. This system only using two symbols, that is 1 and 0. In this system by adding and/or moving one number to the left means doubling it like we will show it to you in the below table:
2 units => 21
4 units => 2 x 2
8 units => 2 x 4
16 units => 2 x 8
32 units => 2 x 16
Let’s try this numbering system, as like 101101 into sketch-like
32 units => 1
16 units => 0
8 units => 1
4 units => 1
2 units => 0
1 unit => 1
As we see by the table above 101101 in the binary system show us:
This binary system introduced by a philosopher and mathematician from Germany in the era of “XVII” named Gottfried Wilhelm von Leibniz because of the simplicity and because in his opinion, this system showed the creation of creativity. By identifying about 1 is the existences and 0 is the emptiness in order of them to completing each other to form such a model.
This binary system is also quite useful in the Physics Development theory. This system has been helpful in the development of calculating machines that have been called an electronic computers. This machine also works by electric energy positive and negative. If the flow of the electrical energy gets cut then the value of symbolic 0 in the system will be shown in the monitor of the machine, and if there is a flow of connection to the electrical energy in the system then the value of symbolic 1 will be shown in the monitor of the machine. This electric computer is able to calculate a complex calculation by using a symbolic number of “0” and “1” from the binary system. A model of modern computer technology has been called ILLIAC IV, an electric computer system that capable of processing a billion instructions in each second. The development of this electronic computer is evidence of the development of a system, a binary system that has been changing most of the modern aspects to some meaningful calculation.
Mayan civilization, which has succeeded develop an incredible civilization in the Mid of America at the first era, they are using systems foundation calculation of twentieth. There is evidence of this method has been used in multi-linguistic include in the French Language. The systems of numbering in French, that’s up to sixty-nine, similar to our current system. But the seventy that is in French is soixante-Dix (means sixty-ten); seventy-one is soixante et onze (sixty-eleven) etc till up to eighty. Then, eighty is quatrevingts (means four of twenty), followed by other like quatre-vingt-un which is four of twenty and one, quatre-vingt-Deux which is four of twenty and two, ninety which is quatre-vingt-dix (means four of twenty and ten), ninety-one which is quatre-vingt-onze (four of twenty and eleven), etc till up to hundreds. The English language also has evidence of measurement of system twentieth. Like been shown in the book of King James, if we read about the average time of a frame is three of twentieth and ten. Abraham Lincoln started his speech in Gettysburg on Nov 19, 1863, with the word “four of twenty and seven years ago”. The community of duodecimal in the US has suggested the use of duodecimal or a system foundation calculation of duodecimal, in which in this case there is twelve symbolic number to be used, like 1, 2, 3, 4, 5, 6, 7, 8, 9, t, e, and 0. With the meaning of t as ten and e as eleven. By placing a number one position to the left, in the system of duodecimal, then we multiplied the value of those numbers by twelve times, like
20.736 => 12 * 1.728
1.728 => 12 * 144
144 => 12 * 12
12 => 12 * 1
The numbers 6et4 in the system of duodecimal means (6*1.728) +(11*144) +(10*12) +(4*1). If we do the multiplication and the addition, then we will know about 6e+4 is equaled to 12.076.
In the end, there is a very minimum possibility that our current daily decimal measurement will be replaced by some other form of the foundation of calculation.
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