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).