Numbers are how we express quantity, and they are an integral part of the concept of ‘counting.’ Counting is a very basic concept in mathematics, and it stems from our realization that there are different ‘things’ of the same kind in the world. Because it is a very fundamental concept, it is exceedingly difficult to simplify the concept of counting! But we can describe the process of counting.
When we see a tree and characterize it as such, then we see another tree beside it, we want to express the fact that there is more than one such thing as a tree. We realize that the two trees are not the same thing. When we see a forest, we realize that there are more trees than just two. We realize that some forests are larger than others and have more trees than them. The need to express this concept of quantity is the basis of counting. In practical terms, is it better to have just one apple to eat or two apples? Or more? The existence of more than a single entity of anything at all is what made counting necessary.
Counting is one of the fundamental concepts in mathematics.
that reflect the use of tally marks.
Ancient people used very primitive ways of counting that we still use until today. Tally marks were the oldest ways of expressing quantity. The concept is very simple: for every ‘thing’ they wanted to count, they made a mark. This ‘one-to-one relationship’ between the marks and the things that needed to be counted made it easier for them to have a ‘feel’ of the quantity. It is easier to estimate the number of marks on a piece of bone than to estimate the number of trees in the woods or the forest. It is easier to have a ‘record’ of the days that passed since the previous full moon than to just remember them. Also, tally marks enabled ancient people to ‘communicate’ quantities. Imagine that one of those people wanted to tell another that they had 4 goats. What would they do? They have to either ‘show’ them the goats, or something that matched the number of the goats. Perhaps they used their fingers to indicate the quantity. Well, this is essentially the same idea as tally marks; a one-to-one relationship of marks and things to be counted. When numbers were big, they had to use something to hold these marks, like a piece of bone, wood or stone.
What if they wanted to say the number of goats to someone else? How would they do it? One way of doing it is to say “I have a goat, a goat, a goat, and a goat.’ Again, there is a one-to-one relationship between the word ‘goat’ and the actual goats. But what if they had forty goats?! If they used this method, it would take them much longer to say the number of goats out loud, and the listener would be more likely to get bored and lose track of the count as the word ‘goat’ is being repeated forty times. Imagine what would happen if they had four hundred goats! Eventually, they would have to invent a ‘word’ for certain quantities to make communications easier. In time, they would need to dedicate parts of the language to expressing quantities in a systemic way that can be used to express any quantity using a limited number of words.
First, people used not to consider numbers things on their own. They didn't have a word for ‘four,’ but rather different words for ‘four sheep,’ ‘four trees,’ ‘four cows,’ ‘four coins,’ etc... It seems that they later realized that they do not need to do so; that numbers are not tied up to any physical object. In a more recent stage in history, people started to use words to express numbers as objects. The concept of a number as the ‘count’ of objects of the same kind came to existence.
It took people a while to start considering numbers as abstract entities of their own.
So, what do we mean by ‘things of the same kind?’ From the perspective of an ancient person, would ‘a piece of meat’ be enough to describe objects to be counted? Imagine that you are a hunter in ancient times, and you went out on a long, exhausting hunt with your village peers. Now that you've hunted down some animal, it is time to divide the spoils and enjoy the meat. Would it be ‘fair’ in your point of view to be given the ribs when someone else is given the whole hind quarter of the animal?! I guess not! But why is it so? Because we realize that things need to be of the same ‘size,’ too, in order for them to be counted as equal. I highly doubt that anyone, even in the most ancient of times, would trade five cows for five rabbits! That's unless they have some plan in their minds that makes them believe that one rabbit is worth one cow. But even in this case, the ‘worth’ of one cow, in their mind, is the same as the ‘worth’ of one rabbit. An important and essential criterion in counting is that things are equal in whatever aspect they are being evaluated.
It is, therefore, intuitive that we do not count together things that are different in size or value. We do not count apples and oranges together; hence the famous saying ‘comparing apples to oranges.’ We do not count apples with half-apples together. Things that we count together have to match in both kind and size or value. Does this mean that we cannot count apples and oranges together at all?! Of course not! But it means that if we do so, we strictly consider them all to be of the same kind and value. So, in that case, we would be counting ‘fruits,’ not ‘apples’ or ‘oranges.’ Furthermore, we would consider that an apple is equal in value to an orange; so if we are dividing them on people, it doesn't matter if they get an apple or an orange.
Things counted together must be of the same kind and value.
Perhaps this was one of the reasons numbers were not considered entities of their own from the very beginning. People realized that ‘three stones’ are not equal in kind or value to ‘three sheep.’ They needed to make sure they are counting the same kind of thing. Perhaps later they started to have this consensus that counting must be of things of the same kind and value, and that's when they started considering numbers as abstract entities of their own. The essential condition of ‘same kind and value’ became implicit in the concept of counting, so much so that some of us have lost sight of it! I came across someone who was wondering if two drops of water merging into one drop made 2=1! They have simply lost sight of (or were trying to draw the attention to) the fact that counting must be of things of the same kind and size or value. One drop merging with another drop and making a larger drop does not make 2=1, because there are two small drops and one large drop. They do not have the same size; hence they cannot be counted together.
more easily when they are clustered.
We can consider numbers abstract entities of their own if, and only if, we pay attention to the meaning of numbers. The abstract nature of numbers allows them to be used for all material purposes, but they have to be used with the same meaning in the same context. On their own, numbers only mean ‘the quantity of things of the same kind and value,’ and nothing more. Once you start attaching them to physical objects, you have to make sure you do not deviate from this meaning of numbers, or you will be making mistakes.
Of course, people could have used any mark in one-to-one ratio to whatever they wanted to count in order to represent numbers. Choosing a line, however, seemed the simplest and most efficient solution. Dots may be missed because they can be small. Other than dots, lines are the simplest shapes that can be drawn. Also, they are easy to carve on harder material. Any piece of rock that is sharp enough can make a linear mark on bone or wood. For efficiency and clarity, lines seem to be the most suitable to use for marking numbers.
But again, writing numbers in the form of lines becomes quite useless when the numbers are big. Consider writing a relatively small number like 8 using linear tally marks. It becomes very difficult to easily recognize the number of marks lined up next to each other if there is more than 3 or 4 of them. That's why people from early on decided to group tally marks into ‘clusters’ in order to recognize their number more easily. The intuitive cluster that comes to mind is one of size 5, since we have 5 fingers in each hand and 5 toes in each foot. Clustering tally marks makes a huge difference in understanding and recognizing the numbers they represent. However, things become messy when we want to express even moderately large numbers. Consider writing the number 476 in tally marks. You'd have to write 476 lines. Furthermore, you wouldn't be able to evaluate the actual number with a simple look. You'd have to count the clusters! While this is certainly possible, it puts more limitations on how large of a number we can practically represent using this method. Consider writing 4,305,698 using tally marks. If you could write 5 marks per second (and that's a big IF,) you'd need a little less than 10 days of continuous writing—without a break to drink, eat, go to the washroom, rest, or sleep—to finish writing just this number! You can imagine how much material you'd need to write this number on using tally marks! That's why people needed to invent a more practical way of writing numbers.
The more efficient way of writing numbers is by using ‘numerals,’ which are symbols that represent numbers. The simplicity of tally marks lies in that no one needs to memorize anything at all to use them. If you can write a line and see a line, then you can use tally marks. But when people started thinking of symbols to represent numbers, they had to use different symbols for different numbers. Of course, the line, being very simple to write and read, was not lost altogether. In many civilizations around the world, the simple, straight line remained as a representative of the number 1. From one civilization to another, the symbols (i.e. numerals) that represented numbers and the methods used to write numbers different significantly. The story of how numerals evolved is beyond the scope of this article, but it is worth mentioning something about two important aspects of number systems.
Base of a number system
People needed to invent new symbols to represent numbers, but how many symbols should they invent? Is it enough to invent one more symbol other than the straight line? Do we need 5 more symbols? Perhaps 20 symbols? Or 60 symbols? How many is enough? The system that we use nowadays is the decimal system, derived from the greek word that means ‘ten.’ And how many different numerals (i.e. digits) do we use to write numbers? We use 10 different numerals to represent all possible numbers; namely 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. So, the number of numerals that we need to represent any number in any number system is equal to the ‘base’ of that system. Using 10 different numerals is relatively more difficult than using simple lines to represent numbers, because we would have to memorize the meaning of each of these symbols. However, it gives us the opportunity to represent 8 with a single symbol rather than 8 separate lines.
Historically speaking, there have been systems with base 10 (most common), 20, and 60. Remnants of the 20-based and 60-based systems still exist in modern usage, but the details are beyond the current scope. What survived and flourished was our current decimal system. Was it because it had less symbols than other systems? Was it because it was easier to write than other systems? Was it because civilizations using the decimal system happened to be advanced in mathematics at a certain time in history? We will probably never know for sure, but it works fine for us, and we are likely to continue using it for all the foreseeable future.
But how can we use a limited set of numerals to represent all numbers? Do we add them together? Suppose that the symbol © represents the number 12. If we want to write 48, do we write ©©©©? What if we want to write 480? We would have to write 10 times as many symbols! This seems like a better idea than using simple linear tally marks, but it is certainly not the best idea! Some of the older civilizations used this method to represent numbers, but the best method that was invented was the ‘place value’ method, attributed to the Babylonians.
Place value
Without this concept, it would not have been practically possible to use only 10 different numerals to represent as many numbers as we could comprehend. The place value principle is quite simple. We will use the chosen symbols to represent the first 10 numbers. When we run out of symbols, we will start reusing them again, but with a different value. The new value depends on their place in the written number. When we write just ‘7’, this symbol means ‘seven.’ When we write ‘74,’ because the symbol ‘7’ lies in the second place from the right, it does not mean ‘seven,’ but rather ‘seven tens,’ which is seventy. When we write ‘751,’ the symbol ‘7’ comes in the third place from the right, so it means ‘seven tens of tens,’ which is seven hundreds. This means that for every place to the left of the first one, the value of the symbol is multiplied by 10 (the base of the number system) as many times as the shift to the left. For the second place—a shift of 1 place to the left—the value is multiplied by 10 one time. For the third place—a shift of 2 places to the left—it is multiplied by 10 two times, which is equivalent to multiplying it by 100. We eventually add the values of those symbols (numerals) to each other to come up with the number we want to represent. Simple and brilliant!
By giving the numerals a different value based on its place in the number, it is possible to represent any number with a limited set of numerals. Can we use the same principle to represent numbers using something other than the decimal system? Of course, we can! But again, this is beyond the scope of this article. Besides, why bother when we have a perfectly working decimal system?! But just to give you a taste of what can be done, in computer science, it is common to use the binary system (2-based), the octal system (8-based) and the hexadecimal system (16-based.) In calculations of time and angles, we encounter the sexagesimal system (60-based.) And in theory, any base can be used for any specific number system, given that the numerals in that system will be as many as the base. Using this with the place value, any number can be written concisely!
Zero
The number zero is the youngest of all numerals. Because numbers developed out of the need to count, and because we count things that exist, people did not really think of nothing as something that existed! They realized the concept of nothingness and some civilizations even had symbols to represent this nothingness, but they didn't consider it to be a number. For them, numbers represented only things that existed. Relatively recently, somewhere between the 5th and the 7th centuries in India, the number zero was born. In ancient number systems, there is no zero as a numeral; only as a concept. Nowadays, we could not do without the zero!
The paradox that numbers represent existing things, and that zero is non-existing is what delayed the introduction of the number zero. But if you think a bit about it, if the concept of nothingness did not exist, how come you can think of it?! If you think of a certain concept, the concept must exist, albeit only in your mind. So, zero does not represent something that exists, but the concept that nothing exists. Just as all numbers are considered abstract entities that do not necessarily correspond to real, existing things in the material world, the number zero exists as a concept with the rest of the numbers in the realm of mathematics.
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