1. Numbers (numerals: Duodecimal System)

1-4-5. Duodecimal is another system of numerals in which the base is twelve. Compared to 10, 12 has more factors so it can be divided more easily, besides it can be counted by fingers: excluding thumb, we have four fingers, each is divided into 3 parts or consists of 3 finger bones; so we can count up to 12. We can use the thumb moving over other fingers to count the amount.

We need to define two symbols for ten and eleven. Either A and B, or T and E is used to represent ten and eleven accordingly. Furthermore, a rotated 2 (ᘔ) and a reversed 3 (Ɛ) are proposed by Sir Isaac Pitman as symbols for ten and eleven respectively.

Duodecimal is also called dozenal system. That’s why we have a pack of dozen or half dozen products especially drinks. Now we can see why 13 is considered as an unlucky number. The 13^th product, let’s say can of drink, can ruin the divisibility of 12, moreover 13 is a prime number (it doesn’t have any factors but 1 and itself) which will be discussed later. So imagine you wanted to share 12 cans of drink between six friends, and now you had 13 cans, what would you do? Then the mythologies have amplified the unlucky power of 13.

That’s also why the words eleven and twelve are different from teen numbers. They stemmed from the Deutsche words elf and zwölf respectively.

In order to compare different numeral systems, we need to know the fractions and index form (that’s why I try to introduce them briefly), therefore, all different bases will be discussed elaborately after the history of numerals will have been discussed up to Hindu-Arabic numerals that we use these days. We will try to develop all arithmetic operations based on different bases, then we can compare their efficiency and drawbacks.

1. Numbers (numerals: Roman numerals)

1-4-4. I've deliberately avoided to define the base-system since index form is the pre-requisite and I had planned to explore mathematics step by step and to build the new concepts upon the pre-requisites. I’d planned that after developing the index form, I could get back to numbers and discuss base-system with a new perspective. I, however, found it almost impossible to go through Roman numeral and compare in to Egyptian numerals without a precise definition of the base. So I have to rely on the knowledge of my readers about exponent and index form which I’m certain of, then I can move forward. Chronologically speaking, Egyptians and Romans were aware of the base, so we just study it using Hindu-Arabic numerals which we presently use.

In a nutshell, since it’s going to be discussed thoroughly, the base-system is grouping numbers according to the progressive powers of the base. For instance Egyptian numerals follow the base system because they group objects based on 10, then 10 groups of 10s which is a hundred; 10 groups of 100s which is a thousand and so on. Tally mark numerals, on the other side, doesn't have such property because the grouping is limited to 5. We could say that it’d follow the base system if they grouped 5 bundles of 5; then 5 bundles of 25 and so on; but the base doesn't develop progressively.

Romans developed their numerals based on Tally marks. However, they used five and ten to group numbers. The other difference was using alphabetic letters to represent numbers and they developed it to 1000 in the first place. Here are the letters Romans used to write their numbers.

Symbol	Value
I	1
V	5
X	10
L	50
C	100
D	500
M	1000

In order to simplify the symbols, Romans found out that they can define new numbers by changing the position of digits which can be considered one of the first attempts to come up with the place-value or positional notation. VI represents 6 while IV represents 4. LX is 60 but XL is 40.

So 1974 in Roman numeral is written MCMLXXIV.

The last thing needs to be mentioned about Roman numeral is writing 5000 and larger quantities. They drew a bar on top of a letter to represent 1000 units of the value. So x̄ means 10000.

In conclusion, though working with Roman numerals, to some extent, is easier than Egyptian numerals or Tally marks, calculation with it is still a hardship which normally arises through multiplication and division. There’s still a need to ameliorate the numerals.

1. Numbers (numerals: Egyptian Numerals)

1-4-3. Egyptians were among the first people who came up with their hieroglyphic numerals around 5000 years ago. Unlike Tally marks system in which the base is 5 and it hadn't been developed to higher levels of grouping (such as 25, 125, 625 and etc.), the Egyptian numerals had been a base-10 system and it had been developed up to 1,000,000 which had been a huge improvement (since Egypt was a big civilization and they had a million-plus population, they needed to work with big quantities). Here you can see the symbols they’d used to represent the numbers.

https://www.easycalculation.com/funny/numerals/egyptian.php

You can see that like in Tally marks a stroke represents 1; and a heel bone, coil of rope, lotus, finger, frog and sitting man represent 10, 100, 1000, 10000, 100000 and 1000000 respectively.

Working with the Egyptian numerals was also as difficult as working with Tally marks (drawing or carving the symbols was a tedious job), moreover the calculation with these numerals was a herculean task. You can try multiplying 89 by 597.  (it might take several hours to do this simple multiplication)

In conclusion, though compared to tally marks, the Egyptian numerals simplified writing numbers and especially doing arithmetic with big amounts, we cannot say that it was an efficient numeral system. However its base-10 grouping has become the pillar of the Hindu-Arabic numerals that we use.

1. numbers (numerals: basic arithmetic operations)

1-4-2-1. Speaking of evolution of mathematics chronologically, necessitates to bring up the first arithmetic operations because they were conceived while or ahead of the time that people invented and developed numerals: addition, subtraction and multiplication.

Addition is one of the undefined concepts (we know that a word can’t be defined with its synonyms). We’re simply adding a notch to a in order to write b (It shows that how intertwined adding and counting are), however it might have taken time that people came to awareness about what they were doing, therefore they called the process addition.

Subtraction on the other hand is taking an amount away from another (can we state that subtraction is an undefined concept too?). Both addition and subtraction were conceived when people found the change in quantity of things around. For example they wanted to know how many sacks of rice are left when they cooked or sold out two sacks.

To count eeeeee they needed to add five to itself six times. That’s why they came up with multiplication to simplify addition.

So we see that the concepts of basic arithmetic operations were conceived by people who used the tally marks, and they were one of the reasons for evolution of numerals, apart from writing or reading them.

P.S. I found out that the tally marks font can't be shown on some systems which don't support it, so in the second paragraph if, instead of one notch and two notches, you had a and b, they'd mean one and two respectively; and in the forth paragraph if you had six e's, it'd mean six fives. 

1. numbers (numerals: Tally marks)

1-4-2. Carving numbers on wood or stone made them more abstract, instead of using objects to represent numbers they used symbols; so the easiest symbol to be carved is a straight line such as a, b, c, or d. These numerals were the first symbols invented and they are called tally marks. They notched the quantities as they counted objects; however for large quantities they needed to recount the notches, which was not easy. Imagine that you’re asked to count 1000 banknotes. You’d count them one by one, but what would you do if you doubted the current number let’s say if it’s 500 or 501. Then you should start over counting which is tedious and tardy. Even if you used tally marks, you’d need to count them.

The easy solution for this problem is bundling the banknotes, normally we count up to ten, and then we fold the tenth banknote around the bundle. That’s exactly what they did with the fifth notch, they carved it across the four notches (d) to create e.

Why 5? Because we have 5 fingers attached to a hand.

1. numbers (numerals: from objects to symbols)

1-4-1. Imagine that you’d been a shepherd, living before invention of numerals. You probably would've wanted to know if you’d lost some sheep, maybe a wolf had preyed on them. So you'd need an equivalent amount which could be remained unchanged.

Your fingers were probably the first things you could have found. Making a one to one correspondence between your fingers and the number of the sheep enabled you to record the quantity. However, you know that it wouldn't be applicable if the quantity exceeded ten or twenty (if you used your foot fingers too); moreover, there are unexpected and unfortunate accidents in which somebody might lose one or some of their fingers.

So, one of the first things that you could find around abundantly was stone. You could gather pebbles to record the quantity. So it’s not unexpected that the word calculation stemmed the Greek word “calculus” (plural: calculi) which means “pebble”. However using pebbles has two main disadvantages:

1. they take up space: it’d be difficult to keep them, especially to carry them around if you wanted to trade your sheep

2.  one or some of them might be lost (pranksters might have been in that era too)

If you'd been one of the geniuses of that epoch, you would've found out that you could have carved a notch into a stone or wood. It doesn't take that much space and it’s rather permanent. Of course you could lose the stone or wood but the recorded quantity can be hardly changed.

1. numbers (a need to move forward)

1-3. We learn numbers so early that we might not be able to recall it, so we take the ability to work with numbers for granted. We also live in a world which has been quantified and digitized so we might find it difficult to imagine a world without number. However, there are still people who haven’t developed the concept of number and they hardly use it. This video shows how a member of Walpiri tribe expressing 4.

This behavior has been observed in many primitive tribes whose members were able to count up to a certain level, let’s say 2, 3 or 6; and beyond that they use the word “many”. Dobrizhoffer observed that counting bores primitive people.

“Primitive man not only does not know arithmetic, he avoids it. His memory proves to be a major handicap; counting bores him, so he is unwilling to use it. “When members of a primitive tribe return from an expedition hunting from wild horses, nobody asks how many wild horses they have brought back. The question asked is, “How large an area would the herd you brought back occupy?”

How can inability to understand, work with, or express the large numbers, be justified? Since most of the 6 year old kids are able to count up to 100, can we conclude that there is a faculty in our brain, which enables us to understand concept of numbers, therefore primitive people lack it?

In order to answer the question, we should look through the process of creating the concept of number step by step. At the first step, we perceive and conceive the difference between two different amounts. To some extent animals can compare two quantities and select one of them. This video shows the ability of pigeon to distinguish the quantities and select the smaller value (in order to get rewarded).

Is pigeon able to compare every two values? We have to consider that comparing 1 object with 2 objects is easier than comparing 8 objects with 9 objects. Through a research Gatton found that if he took 1 of the 4 eggs out of a nest, “the bird didn’t seem to mind it, but if he took two eggs, then the bird destroyed its nest”. You might find it difficult to compare 10 flowers to 11 flowers, eventually you would need to count each group in order to compare their quantities.

Primitive people might not be able to recognize the difference between two close quantities, so they don’t use different words for them, or the difference between the quantities doesn't affect their living therefore, they haven’t felt the need to come up with proper names for each number.

Evolution of numbers was propelled by individual and social needs. Even in ancient Greece when geometry was flourishing, mathematicians almost disregarded arithmetic since they thought recording and working with numbers is a slavish job (their slaves were in charge of reckoning). Development of numbers can be directly related to the civilizations in which people needed to deal with the surplus of their grains, to share something evenly with a group of people, to deal with the inheritance, to measure the quantities such as time, length and mass, and many other needs. If you ate whatever you hunted or obtained, you wouldn't have any belonging to be counted; besides, if you dwelt in a barren land you wouldn't be able to see the periodic growth and decay of the plants, therefore you wouldn't come up with the idea of agriculture.

Jean Piaget, whose name was mentioned earlier, conducted several experiments on many kids to monitor the cognitive stages of development. These two videos demonstrate his experiments.

We can see that the concept of quantity and specifically the number (in the coin experiment) needs to be refined in the kids' brains, since they can't still separate it from the size or the change of shape. It shows that creating the concept of numbers and quantity needs a huge development in our brains (what we take for granted) that if it didn't take place we couldn't understand the numbers, therefore there couldn't be almost any developments in our societies. 

In conclusion, we can say that the ability to understand the numbers is not built in; even though our brains have been evolved through centuries so we can understand them faster than our primitive ancestors, it has been acquired when we were small kids and it can be reinforced and developed when we need the numbers for counting or calculations.

1. numbers (mathematical number vs. physical number)

1-2. Story of numbers started when our ancestors found out there was a difference between one apple and two apples. In order to communicate clearly, they needed to come up with two different names for them. We don’t know what they called them, but we are sure that soon after they faced another problem which was a new word for three apples. It took a while that they found out the similarity between them, so they decided to disintegrate the name into two parts: one part to indicate the difference and the other to reflect the similarity. Then they said one apple, two apples, and three apples.

However, there was another issue that they used different adjectives for different objects; for example the adjectives that they used for apples were different from swords or cats. Another big leap was needed that they conceived the similarity between two men and two apples.

To understand the importance of this abstraction, we should know that our brain is removing all components of apples and men to find one similarity between them which is their quantity, apart from that they are completely different. We can’t even imagine it because we’re removing their shape, size and every feature; we only understand it. So they invented one set of adjectives for all different quantities which was a huge development.

Here one is not a number. It’s merely an adjective which is serving a noun. When one as a number was born that our ancestors were able to separate it from the nouns which they had counted. Physical one is different from mathematical one, physical one is meaningless without a noun or unit; if you told somebody that I'd been waiting for you for one, how could they know if you meant one second, minute, hour or even year, however in math nobody needs to know what one is referring to. That’s why they invented a different symbol for it: 1. 

This segregation enabled humankind to develop a different system for numbers which is the foundation of mathematics. This was the moment that the plane took off.

1. numbers (pre-requisite: set)

1-1. We discussed how a concept is created by our mind. It abstracts the similarity from several things in order to group them. In mathematics we call these concepts SET. Dog is a concept for the animals with particular similarities, in mathematics dog is defined as a set of animals which have those common properties.

Set is one of the basic concepts which cannot be defined; however almost everybody can understand it. Set of mathematicians, set of bicycles, set of psychological disorders, and set of smiles.

Members or elements are things that belong to a set. Gauss is a member of mathematicians, but Lionel Messi is not one of its elements. Since we use this format frequently that a particular member is an element of a particular set, we replace “is an Element of” with this symbol ∈ (stretched E). So translated into math language, we write Gauss ∈ mathematicians. We could even simplify it more, if we used Gauss and mathematicians frequently. Therefore, we could write that G ∈ m (In the context G and m must be defined). “Messi is not an element of mathematicians” can be simplified as M ∉ m. You can see how a sentence can be written in mathematical language more simply and easily. 

Like the words in English that we memorize for communication,  we should keep in mind the symbols in order to read them correctly, otherwise they look like some weird, mysterious or even terrifying codes.

  

If we change the arrangement such that we say "mathematicians" consists the element named Gauss, we write it m ∋ G, or m ∌ M (for mathematicians which doesn't have an element called Messi). Most of the times we use the first arrangement " the member is (or is not) an element of the set".

0. Introduction (what we talk when we talk about math)

0 – 5: Water is a chemical compound with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms that are connected by covalent bonds. Water is a liquid at standard ambient temperature and pressure, but it often co-exists on Earth with its solid state, ice, and gaseous state, steam (water vapor).

That is a definition for water. Does it motivate anybody to learn swimming? Definitely no. The best way to define swimming is jumping in water. You can find many definitions for math throughout Internet, but they are not as attractive and mesmerizing as solving a math problem. So instead of defining math, I’ll try to portray it.

Mathematics is a language in which we speak of quantifiable problems. It has its own words, sentences, grammar and punctuation. The big difference between English and math is if you made mistake in an essay, it would probably result in a weak essay, not a wrong one; however in math, your solution would be certainly wrong. We can analogize communicating in English and math to driving a car and an airplane respectively; if airplane pilot made a minor mistake, it would end up in a deadly accident. That’s why math is the most challenging subject for majority of the students.

The second aspect of this portrait is the elegance and efficiency of math in order to solve problems. If solving a problem can be analogized to a journey between two places, math is the aerial travel. In this analogy, land is the reality or the physical approach, and sky is the realm of mind.

Imagine that you’re asked to count the number of planted trees in a land. You could count them one by one and if there were 50000 trees, it would take a long time (if you could count one tree per second, it would take 15 hours approximately). However, if you could find a pattern (such as grid plantation), you would need to count the rows and columns (say 500 by 100) so instead of tallying 50000 tress you’d need to count 600 trees, then your brain could process the numbers (in the sky!) to find the answer (it would roughly take 10 minutes)

Without science and math as its language, who would trust to get on an airplane? If an engineer announced that this plane had been designed and manufactured after 1000 experiments (without applying mathematical approaches), would you trust them?

So, wherever we talk about a quantity, we can see math, coming along.