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.