Areas and Multiplication

Representation of areas was further developed by putting lengths instead of putting balls. So it gave accurate measurements of areas.  

Hence the area of a rectangle is stated as the space covered by l – length up to the perpendicular length b namely breadth.

Since Area of a rectangle is the multiplication of length and breadth it is equivalent to l * b or b * l.

It gave the unit of (metre)^2 for areas because  l, b  both are lengths. Unit of the length is metre. 
Hence the units is metre * metre = metre^2.       

Thus we measured areas using lengths.

Multiplication is the large continuous application of same addition process. It was made only to simplify the addition process. When we are concerned the additions of same amounts many times we tried to calculate it easily by remembering it.

Adding 6 for 5 times

6 + 6 + 6 + 6 + 6

This can be written simply as 5*6 or 5 X 6

And thus multiplication was stated by putting the symbol ‘ X ’  between the two numbers where multiplication has to be done.  The commutative property of multiplication is already explained in the area of a rectangle.

          e.g.         5*6= 5 rows of 6 columns balls = 30 balls

It is also equivalent to the multiplication of 6*5 because 6 rows of 5 columns balls = 30

Thus the area of any rectangle is l*b or b*l.

After mugging up easy multiplications we often used it in large multiplications.

e.g.  5 *36

We can do 36*5 or 5*36.

Being same multiplication also done as same as addition and so in multiplication the calculation starts from 1^st decimal place to the other decimal places.

First multiply the 1^st decimal of two numbers and so 5*6 = 5 times 6 or 6 times 5= we know the mugged up answer is 30. Hence we put 0 in the 1st decimal place of the answer and it left 3 in the 2^nd decimal place. Next the multiplication is done in the 2^nd decimal place. So, multiply 5 with 3 which is the addition of 3 for 5 times or 5 for 3 times gives 5*3=3*5 =15

And already there is number 3 which was left in the 2^nd decimal place by the multiplication of 1^st decimal place. So the total in 2^nd decimal place is 15 + 3 = 18.  

Here 8 will be filled in second decimal place and 1 was left in 3^rd decimal place. Hence there is no number in 3^rd place of the question we will put 1 in the 3^rd decimal. Therefore 5*36 is 180.

Thus we can do multiplication between any two numbers using the same rules mentioned above.

The common form of multiplication is x*y=z

The multiplication was used widely in all areas.

To make more simple, in multiplication we will put bracket signs around the numbers instead of putting multiplication sign. 
It is      (x)(y) = (y)(x) = z   instead of  x*y = y*x = z

Hence it is only the simple addition process and so

x, y, z ϵ N.

Being the same multiplication doesn't have anything extra information from addition.  

All the rules which undergoes for addition will also undergoes for multiplication. 

Some properties are…

Commutative property

                                       x*y = y*x

Associative property

                                           (x*y)*z = x*(y*z)

Large applications of addition in Areas

          From the previous posts we know that addition is the basic operation of our knowledge. 
           
         In nature, everything is made of group of small units and also those small units are the group of smaller units than their sizes. Small units of same kind or different kind combined together and made different objects. Henceforth we represent this combination using the operation addition because addition process itself derived from the nature.

          Representation of objects using addition is similar to the representation of measurements in units, where we will use some specified measurements (units) to represent the desired measurements.

 e.g.   10 meters [desired length]  = 10 * 1 meter [meter is the fixed reference length and so it is called the unit of length]

Here the desired length is the addition of fixed reference length 10 times.

          Another example:6.023*10 to the power 23 [6.023*10^23] particles makes the measurement of one mole.

Here the number 6.023*10 ^ 23 is the representation of the addition of 6.023*10^ 23 units.

          Almost all the thing in this universe follows addition. All these combinations made addition process very important in our life. That is why we are concerned about the simplification of addition.

          The process itself can’t be simplified or changed since processing method is natural. But we can make some remembering or mugging up for some specified problems of additions to make the calculations easily.

Specified additions are those additions that we used often in our daily life. So we are interested in the calculations that used essentially in our life.

One of that essential uses is Multiplication.

           Multiplication is first formed due to its great need in finding areas of simple shapes.

The uses of simple shapes are that, we can calculate its areas and volumes easily. Knowing precise areas and volumes we can make accurate arrangements and calculations in building construction, architecture design, etc. That is why perfect and simple shapes are preferred in most of the constructions.

Before any construction work, we should have studied about these measurements of areas and volumes.  

e.g. Size of the bricks using in construction works will have same amount of area and volume. This simplicity in areas and volumes made the arrangement of bricks easy and simple. Due to accurate measurements of those bricks they will fill the volume of the building shape properly without leaving any space.

          One of those single shapes is rectangle, a perfect closed shape. Each and every property about a rectangle is well defined. For its simplicity we used if often in our life from ancient times. And area of rectangle is defined step by step like the improvement of units.

          Area is the measure of space enclosed by any closed shape in 2 D plane with its limits.

The area can be represented in many ways. One of the way is we can represent area by saying the amount of total balls covered in that space by putting many balls in that area without leaving so much space. So we can denote that area as the space covered by x amount of balls in that plane.

         Here in the above example we denoted the area as the space covered by 5 rows of 6 balls = 30 balls. Though there is some space left in that area it is not a problem because I am just explaining how the representation of area was made.

But when we need more accurate measurements, instead of representing area by balls we represented it by lengths. Representation of areas were developed using accurate,  precise and well defined lengths.

That we will see in the next post.

Associative property

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

In our description of addition we took most of the examples as the addition of two amounts because anything can be related first between only two things and all other things can be related only after relating first two things in any operation. In the same way addition of any amounts can be added only after adding any two amounts in the given amounts. 

For e.g. to add 1 + 3 + 8 + 10 + 15 + 68 first we will take any two numbers in the given numbers. We can take any two from among these like 1+3 or 1+8 or 8+10 . After adding any two we will add other left numbers following the addition of the first two numbers. In the same way we can add all 5 numbers in 5 times by taking any two numbers taking in order.

This property of adding any two numbers by our own choice made the property called associative.

Hence associative property commonly and simply can be written in the equation form as,

If there are given three numbers a, b, c then from associative property

          a + (b + c) = (a + b) + c

And a, b, c are in whole numbers also can be written as a, b, c ϵ N

The detailed meaning of “ϵ” will be explained later in sets.

This associative property gave another property. In associative property we never mentioned the order of taking numbers. So if there are two numbers then from associative property you can take any two numbers but it is not mentioned in which order. 

So if we add a + b or b + a there is not going to happen any change. We just want pair of sets to add. And it needn’t be in some order in addition. This property is known as commutative property.

It was stated as 

            a + b  =  b + a             and      a, b ϵ N

 

                                                                     

                              

Spontaneously we got two properties which have to be checked in any operation given as associative and commutative property. 

After studying an operation in mathematics we will use that operation in various other mathematical topics. We will apply these operations in all mathematic topics to check whether it is applicable for all the topics or for only the specified topics. That is why the above two properties are so important because we will check it many operations. There are also many properties similar to the above properties.

Many properties like inverse property, associative property, commutative property and other properties will be discussed deeply in binary operations.

Addition also behaves like a property. That is why we are applying addition in all topics like functions, vectors, complex numbers, trigonometry, etc. Addition was used in all these topics.

E.g. Sin (A+B), A vector + B vector, a + ib, f(x) + f(y).

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.