First Tool of Mathematics - Algebra

We used addition all over our life. But we know that it is not sufficient. When we need to explain this process to our next generation we must explain much better than our previous understanding. We must tell it to them in a way by which they can understand it completely. And so we are trying to explain this process much simpler and simpler.

          In that way of making any process simple, we tried to make a common procedure by which it doesn’t have any specified values. We are trying to make the simple representation of the complete theory.  That simple mathematics analogue is called mathematical equation. Mathematical equation is the simple and short one line form that describes all mathematical theory precisely and accurately.                               But how will you represent this equation?

          We don’t anything about it.

          We don’t know how the equation will look like or what may be the things it have or what its property or for which values it is defined or what are the things need for describing that equation.                           Simply the whole thing we are going to study is unknown. Simply it is the form of unknown. But we know there are certain properties for this unknown that is common and universal and not dependent on anything. So step by step we solved the problem of unknown by defining the property of that unknown and by giving some shape for that unknown. This giving sign or symbol for unknown is one of the great methods to the deep understanding of any topic. It made a great renaissance in Mathematics at ancient times.  

          Thus our common and deep analyzing of mathematics was started with these giving symbols for unknown. This process of giving symbols for the unknown is greatly known as Arithmetic. Arithmetic is the tool which explains clearly about the mathematics. So let’s try to know how to use this tool.

In arithmetic we are not doing any special thing of calculation. We are just putting empty space instead of giving some specified values in addition process.

It is simple to derive that common case. For clearance take some examples. We will derive the common thing that is mathematical equation from these data because most of the equations and discoveries are found only by the deep analyze of the given data.  

e.g.    5             +              5      =          10

          29           + 2378434       +          1000    =          2379463         

          355         +   123323       +          80        =          123758

          4234       +     72344       =          76578  

          78764     +       6223       =          84987

          939843   +        790        =          940633

          8093098 +         88         + 10     +    100   =  8093296

Thus a common thing in this addition is that, ‘the amount which has to be added must have some finite amount of values’. And so we can explain that finite value easily by numbers. So, all additions were done with numbers.

When a man write any number like 1 or 2 or 3 at that time his mind already know the inner meaning of that number. Then why he writes because he just wants to give a shape for the inner meaning in his mind. That is how he gave the shape 1 for the inner meaning of some amount.

          In the same way when I need to make a common term for this addition, I don’t know which value will come to the process of addition process. But only thing I know is that there will be some amounts has to be added and that addition will be done by adding some two amounts at single time. But I don’t know what those amounts are.

Simply there will be nothing but we know that there will come something.

It is like taking a seat in the bus by putting an object. You can see some people in buses and other crowded places will place any object of their own in that seat to represent that the seat is theirs. Though the seat is empty, nobody will sit there because they know that somebody is going to come and sit there in that seat.  Although they don’t who is going to sit there, other people will not sit in that empty place. Here the object placed on the seat is the representative of that somebody else.

In the same way on making the common mathematical equation we will put some symbol instead of the object. And that symbol will represent the quantity which is going to come in that empty place.   

And mostly we will use the symbols x, y, z… to represent the empty place in mathematical operation.

Briefly symbols are the representative of the empty places which will be filled later by their belonged amounts in any Mathematical operations. This algebra was used almost in all the topics of mathematics.

Don’t waste your time at any moment on thinking about these symbols. You can give any shape or symbol for the representation. The only thing you need is that the reader who reads your mathematical operation must understand what you mean by that symbol. That is all. It is only used for the understanding of the writer and the reader. In other way they are nothing, just symbols.

          You must practice your mind in a way by which it can identify the inner meaning easily and accurately.

          That is why you can give any shapes or symbols with your convenience. You can give x or y or z or α or β or γ or δ or ε or ζ or any of these symbols - η ϊ ϋ Ϫ ϧ Ϧ Ϣ ϰ ϱ to represent that empty place.

These symbols are just the various scribbles of various people.

Anybody can use any scribble for their personal understanding. 

When you need to promote your scribbling to others, then you just need to specify what you mean by that scribble.  

Thus how algebra was made. So addition of any two unknown amounts can be written as

          X + Y = Z

X is some amount and Y is some other amount and Z is also some other amount.

And X, Y, Z all the three were represented by numbers.

Now we will analyze deeply about this equation to get more information about addition.

What are the possible amounts or numbers that can be filled in these symbols?

At that time we have described number system from 1. And it don’t have any finishing value. So they wrote the whole number system as {1, 2, 3, 4….. }

Only these set of numbers were found at those times.

  

          And so they described the possibility of X, Y, Z as the set of numbers {1, 2, 3...}

The reason because X and Y are some amounts which can be written by the belonged number in that set of numbers and also sum of those amounts will give other amount and its belonged numbers which also lies in that set.

          Hence X, Y, Z all the three will have their belonged numbers of their amounts in that set. 

Accordingly a special name was given for that set of numbers {1, 2, 3…} namely Natural numbers [simply N – Natural numbers]. Maybe they had thought that in Nature there are only one set of numbers that is Natural numbers.  But after many various numbers were found.

Thus,

Various numbers were found on the basis of deep analysis of various mathematical operations. 

Beginning of Arithmetic

Mathematics is neither created nor discovered. It was created from the discovery of the universe

Arithmetic is the study of basic operations which used in everyday life. It is deeply connected with our real life needs. It is the first and foremost topic used by humans. It includes many operations like addition, subtraction, multiplication, division, etc.

First operation we are going to do is addition.

Addition

Addition is the method of telling many amounts together into one amount without caring the properties of that amount. The amount can be anything. We just stated the amount of how much in total. And we have already known that Numbers can be used to represent any amounts. In the same way we used numbers to represent the addition of these amounts also. We used numbers in addition by just putting the “+” sign between two amounts.

And so it was represented as,

            1^st amount + 2^nd amount

Later we converted these amounts into their belonged numbers by taking a reference amount

Since it became

             1^st number + 2^nd number

But always we can’t represent the addition or total of these amounts like this because it doesn’t tells any extra information.

We know that if we add two amounts the resultant will be also some amount. And so the resultant can be also represented by numbers. So the process was represented in a mannered way by using only simple numbers.

            Hence we stated the process of addition of two amounts by numbers and by putting two symbols between them. One of the symbol is, the known symbol which is addition symbol “+” and the other one is equal symbol “=” which is the great symbol used in almost all the topics of mathematics. It represents that any two things which lies either side of that symbol is equal in some specified property or quality. The specified property can be anything which we had considered in that problem.

            It is not compulsory that both things lies either side of equal symbol must be same kind of things. But it must satisfy at least one property exactly.

e.g.      No: of: apples = No: of: tables  

Here though apples and tables are not same kind of objects, we compared those things by taking only amounts. We equalized only one property namely the no: of: amount on both sides but not any other property of that object.

Therefore equal symbol means that the property taken on either side of that symbol is exactly same or similar to each other.

Thus addition was represented completely by numbers and “+”, “=” signs. Now we will do addition using the already explained number system, decimal value number system.

Decimal value number system with Base 10 is a conventional one used all over the world. And it is convenient for everyone in basic operation like addition, subtraction and multiplication. Simple additions were added using fingers and stones. But large additions can’t be done with fingers so they were added using abacus and in our age electronic instruments like calculators. All these things based on these place value system. In these system numbers were added from left to right as one decimal place to 100 and 1000 decimal place values. It is similar to abacus. In this number system there are totally ten symbols.

Here in the number system with base 10, one decimal place will be filled if the value of total 10 comes in that decimal place.  

When the first decimal place of value 1 was filled once then the number 1 will be added in the second decimal place of value 10. If the first decimal place was filled twice then the number 2 will be added in the second decimal place. Similarly other places will be added by following the same rule of decimal value.

e.g.      199 + 299 = ?

In decimal one place two nines are there so if we two nines then it will fill ten symbols at once and amount of 8 ones will remain. So add the number 1 on with the total addition second decimal and place remaining 8 ones on first decimal place. So 9 + 9 = 18, 8 on first decimal place and 1 is on second decimal place.

In second decimal place there are again two nines and so again 9 + 9 but now we must add one came from the fulfill of first decimal place once. So 9 + 9 + 1 = 19

And as same as previous calculation put 9 on second decimal place and add 1 on third decimal place. In third decimal place add 1 [which came from second decimal place] to 1 + 2. And it gave 3. Thus the answer of the addition is 199 + 299 = 498.    

This addition process is same to all other number system with various bases.

For e.g. we will do base 2

In base 2 only the numbers 0 and 1 are used so add 110 + 111 + 111 =?                

First decimal place becomes 0+1+1.  Here adding one two times filled the decimal place once because base is 2 and the remaining is zero. And also because of filling first decimal place once add 1 on second decimal place. So second decimal place becomes            1+1+1+1 and it will be filled two times and no remaining so add two one at third decimal place so the third will have 5 ones. And the representation of 5 in base 2 system is 101. Representation of other numbers in base 2 system was described in number system topic. It is similar to the number system with base 10. Thus the answer is 10100.

You will get the same answer if you convert the question into base 10 and added.

So you must understand that number system with base 10 is nothing different from other bases.

We will see the symbol methods of converting numbers from one base to other bases after learning multiplication, division and other things.

Note: You can think why I am describing the basic things we had known. But I will describe because I have already said that all these things said here must be understandable to everyone. It must be understandable also to the people who don’t anything else about science and mathematics. 

  

That is how many large scale and small scale measurements were added with the use of these numbers. This addition can be used to any large number or small number but no matter about its size.                    

         And also here we consider about only two numbers but it can be done to any number of amounts. But at a time only two amounts will be added. And similarly other amounts will be added by adding any two and two amounts. Thus addition was done in ancient times. 

Deduction of the greatest Axioms

Detectives

Scientists are doing similar jobs like detective agents. Both have to find the mystery and both have to prove their verdicts in a correct manner by which others can easily understand it. And especially it must satisfy the condition that it must not be proven wrong. The proof found by the detectives in science is called axioms. Let’s see the detection of those axioms.

An axiom is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy.^[1] The word comes from the Greek ἀξίωμα 'that which is thought worthy or fit,' or 'that which commends itself as evident.'^[2][3] As used in modernlogic, an axiom is simply a premise or starting point for reasoning,^[4] and equivalent to what Aristotle calls a definition.^[5] Axioms define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true. Therefore, its truth is taken for granted within the particular domain of analysis, and serves as a starting point for deducing and inferring other (theory and domain dependent) truths. An axiom is defined as a mathematical statement that is accepted as being true without a mathematical proof.

I took this paragraph about axioms from Wikipedia. I don’t know whether I understood the term axiom accurately or not. But anyway I am going to use that word. 

The word axiom is self-evident truth with human knowledge. We believe that some idea is true in this universe and all other things are happening on the basis of that idea. There are so many basis ideas. We are using these ideas in science, mathematics and in almost all the topics. You can see many mathematical axioms in Euclid’s elements book.

We people don’t realize many axioms which we are using though we used those axioms in our daily life. I think one of the famous axioms is Newton’s gravitation law. You can’t ask why gravity acts but we are observing that two masses are attracting in Nature and also we are trying to explain that phenomenon in a mathematical way.

Many questions were unanswered. Like you can’t ask why human life is formed and why brain had given to us and also why and what for we are living here. We don’t know answers for these questions. For example another axiom lies behind the formation of human lives. Question is ‘how human life is formed?’ You can’t know predict the accurate history though you are predicting that we formed by the evolution of monkeys with the known scientific data.       

Simply Axioms are the foundation to our knowledge of this universe. Our entire knowledge of this universe is axioms only. All the things are axioms. You can’t say anything is true. Nothing is real. Science, our knowledge and all other things are depended only on these axioms. I will try to explain it by stating some great axioms.

The greatest axiom forever is the axiom behind the question ‘Why’. It is the fundamental axiom for our knowledge. The axiom is ‘There will be some reason behind anything or any natural phenomenon in this universe’.    

Simply

“There will be some reason behind anything and everything”.

This axiom was believed by humans from a long time. It is the axiom which made all the scientific inventions and discoveries. We believed that there will be some reason behind all these things. We took this thing as a self- evident truth.  

You can ask why there must be some reason behind anything. Though it looks as a foolish question it has meaning. But the answer will be philosophical and ethical.

That is why I said that word ‘Nothing is real’ because

All the things we are seeing also an axiom. The entire universe we had seen also an axioms because all these entire things are self-evident truths with our human knowledge.

But all these creations were formed by the only belief of this great axiom. 

So thanks for this beautiful axioms