Introduction to Geometry

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

Geometry is the name given to the mathematical method used to solve real life problems. It is the word related to the earth and its measurements. To know about the each and every beautiful structure and shape of this universe, geometry helps us a lot.  

It was used in the processes like building houses, making structures and many other things which involved with shapes. So they started to improve it from their ancient time. 
It was later improved due to the geometrical needs that faced by humans in their life. 
To overcome every problem he was supposed to use geometry. So he is able to solve many physical problems easily.

To learn geometry, we defined many definitions. To explain big things we started from the small things. 
We learned lower levels to the higher levels.

When you focused on the objects around us, you can notice that most of the objects are not in perfect shapes like rectilinear or circular. They will be in different and different shapes. They will not fit with each other. By using the curved shape stones we can’t make a good house without holes because if you used curved shapes it will not fit correctly. 
For that reason we don’t want to handle natural objects that has imperfect shapes. 
They know that they must alter the stones into some perfect shapes to use it in our real life. 
You can see it on bee house. Great example for closed and well altered shape is bee house. Perfect polygons are used by bees naturally.     

For all these reasons, we tried to alter the stones in perfect shapes. Also in perfect shapes, we are using the special shapes like square, rectangle, and polygon. Only then we can find the properties like areas, volumes easily.  

I will tell you another problem which involved in curved shapes and non-special perfect shapes. That is, if the shapes are not special means the process of making the parts that used in fitting of the building will become tough. If the first shape is not uniform and if we want to make the next shape then that second shape must be in the shape that can be perfectly fit with the first shape. So, if this process goes on continuously you can’t make the third plate until placing the first two plates. The process will take more time. So we must try to make the process simple. Simple means the plates must be produced easily like the technique used in the factories.   

We are trying to find a common shape which is easy to build much kind of houses and also the process of making the shapes must be fast. So which is the most common and easy to enclose completely?

You can see that most of the answers are squares, rectangles, rhombus, parallelogram, hexagon and many other polygons. Here the shape means it must also define the sizes ( length, breadth, area, volume).

 Lengths are the main thing used. When we tried to fit the stone shapes it must satisfy the condition that it must be in same length which is suitable for fitting. The shapes should satisfy the lengths for good fitting. You can’t make a big stone shape for a small hole. In the same way areas and volumes are also used. 

Take any solid figures.  To make it simple, blast it into smaller parts. Comparing to the curved 3 dimensional shape, straight line 3D shape will be better. And if it is in the 2 dimensional shape or plane surface shape there 2D lines will be better. Now if you go more deep it will split into smaller parts like small dust. Here we got a shape. We can assume that it is the basic shape for all shapes.

It is called point……… The definitions goes on.

To define it mathematically we must go one by one.

Various kinds of geometries are there.

To a little deep in geometry we will first go to the Euclidean Geometry. You can study the Euclid's elements books by clicking the link. The full book is given here.

Euclid's elements books (this site will take little time to show)

Number System

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

When humans started living together as groups, they happened to make some calculations for their needs either for food or for weapons or for other things.

And to avoid the expenditures in their works, they occurred to make accurate calculations based on the number of people and their needs. So it was often raised a same question that is “How much”.

And the answer for the question “How much” made us to count these things.

And that is how counting process was begun. It was developed further by the help of a newly created number system called positional number system.

Number system is essentially created to improve the process of counting. First it was used hands and other signals for counting. But there is a problem that it couldn’t be used hands and fingers to represent all the values of different amounts. There is always lies a certain limit for verbal communication.

That is the reason why humans tried to develop a new kind of communication.

And the new type of communication is Non-verbal communication that is used only by humans in our world.

They gave new kind of symbols and signs in scriptural writing form. By the means of writing communication, humans have been developed a lot. 

Writing is the most powerful way of communication in time. It is the only thing that lives in time and gives its information to the future generation. Writing made the difference of humans from animals.

That is why in later period it was improved the writing system and it was given scriptural form for most of the languages in the world. In the same way in mathematics, scriptural form was given to the number system.

Giving scriptural letters for numbers made the number system very easy as we are using it today in our daily life.   

Note: We should be very careful that the scripts of the number system were given only after defining the meaning of the values in the number system.

e.g. the scriptural form “1” is just the symbol given for the value one.  

When they started to give signs for the values of amounts, they couldn’t make separate signs for each and every value. For infinite values we can’t give infinite number of signs. We just need to represent all the values in number system using some certain limit of scripts.

And the problem of infinite signs was resolved for such a long period as number system was developed since many years.

Finally a beautiful and a simple solution were found out.

And it was called the positional number system.

First it should be tried to familiarize with this Positional number system as it is the basic fundamental of mathematics. 

Positional number system

If it was defined a symbol “a” for some value and “b” for two times the first value and “c” for three times the first value. Then the way it can be specified four times the first value from these a, b and c is 4 = a + c or b+ b.   They used  ‘+’ sign for representing the addition process. Here “a” and “b” and “c” are in the first place [the first place will be understood only after what is meant by the second place in positional number system].

It was understood that all the values in mathematics can be explained by the addition process between the values as it was mentioned above.

While representing the number system, we don’t want to use this addition sign between each and every numbers as it makes the representation of number system as a lengthy one.

To make it simpler and shorter, we represent the number system on the basis of changing the positions of the values. By changing the positions using a simple rule, we derived simple notations for all amounts of values in the Number system.

They put the same sign “a” in second place [that is near to the left side of the first symbol] to represent further values in the number system. With this method we can represent any amount of values using just three symbols of three different values.  

And to indicate the difference between placing the symbol “a”in the first place and placing it in the second place, we created a new symbol to represent nothing. Thus the nothing sign is created just to represent the difference between the position places of the number system.

And the symbol is written as “0” which we are using it now in all over mathematics.

Now we can denote the fourth value in number system using the already defined three values “a,b,c”

To represent the values that greater than the value of “c”, it was made a convention that these signs “a,b,c” should be placed left side of the first place. But it should be careful that the first place denoted using zero. Only then the second place easily can be identified.

In this convention of placing “a” in the second place represents the value four for one time.

It means one times the fourth value because “a” means one time.

In the same way if we place b in the second place it represents the number 8 as it is equal to two time the four. 

So the fourth value is represented as = a0.

[where “a0” is a number that is just as same as 10 representing two in binary number system and 10 representing ten in decimal number system.]

Be careful that the numbers coming in the first place denotes its value by unit values but the same numbers in the second place represents the different values with respect to the number of symbols used and the value of the base which the number system was defined.

Here it was represented the second place in left side order (I mean placing “a” left side of 0 ) . No matter happens if the second place is at right side order since it is just a convention we are following and not a rule.

Who knows? In some other planet aliens may be using the number system with right side order representations.

E.g. Instead of writing hundred as 100 they may be using it as 001

Both are correct if it was defined as same as the positional number system. Positions can be either left side or right side order.

            In the same way addition, multiplication, division and all other mathematical operations will be defined in a different way, if it was changed the order of the number system to right side.

But the physical meaning of the answer will not change by any means of the representation of number system. It remains same for all kind of number systems such as unary, binary, ternary. Quaternary,.. etc.  

If I add one and two the answer will be only three in all the number systems.

“a0” means “a value in second place + 0 value in first place”. Here 0 is in the first place and “a”is in the second place so it represents the value of four one time and 0 means nothing. [Note that“a” means value of one and the second place means value of four and so “a” in second place implies one time the value four]. Adding the value four one time and nothing gives the answer of value four.

Now you can define the value five as, 

We already know “a”=1 when “a” is in the first place but it is equal to “a”=4 when it is in the second place. Then the value of five can be rewritten as,

“a in second place + a in the first place” or simply as “aa”

and six is written as “a in second place + b in the first place” or simply “ab” for the reason that “b=value of two” when it is in the first place.

If b is in the second place it represents 8 (b means two times and second place means value of four, so two times the fourth value = value of eight)

So the value of eight represented as = b0 that is analogous to the representation of the value four using “a”.

And with this new kind of thinking, we gave scriptural letters for all the numbers in the number system. In later ages, the symbols were changed and developed in the number system.

Finally it came to a decision that it was agreed to introduce a number system that has ten symbols for global use and simplicity among all the places.

And it was called the decimal number system. And that is what we are using it now all over the world which we have studied it in our first chapter of elementary school mathematics.

It is 1,2,3,4,5,6,7,8,9,0.

This is like abacus or sand table method. When the first places are filled with the given signs the next place will be filled by same signs but with the newly assigned values.

If I place nine beads in the first place (if first place denotes the value one) it represents the number 9. If I place one bead in the second place, it represents number 10 in decimal number system. Instead of placing 10 balls in the first place we placed only one ball in the second place so the process has become simple.   

But we can use any number system with any number of scriptures greater than zero.

The name given to the number of beads or scriptures used in a number system is Base or Radix.

If I said base 2 means there are only two symbols which are allowed to use.

Base 3 means 3 symbols are allowed and so on.

Why we are using the number system with base 10 globally? Why did we used that convention?

I think the answer lies in our hands. When we are counting for the first time we will use our hands and it has 10 fingers.

Hence we choose the number system with base 10 as a conventional one.

But In computers we were using the number system with base 2. It means only two signs. Why base 2? 

Computers don’t have 10 fingers or other things. In most of the electronics, there are only two process that can be identified easily and it is the on and off process.

Only two signals are used in all these electronics for communications.

So it was used base 2 number system for computers.

The same way it can be used many number systems with various bases. But Base 0 is meaningless as it cannot be represented anything with nothing.

Now it can be found a relation between these bases and the places were numbers are placed. Here the first place represents    (Base) to the power 0 and the second place represents (Base) to the power 1 and third place represent (Base) to the power 3 and continuously goes on.

For example in base 10 number system,

First place = 10 to the power 1,  

Second place = 10 to the power 2,

            Third place= 10 to the power 3… etc.

e.g. the number 6875 in base 10 number system.

5- it is in the first place so it is equal to 5*10 to the power 0 = 5*1 = 5

7- it is in the second place so it is equal to 7*10 to the power 1 = 7 * 10 = 70

8- it is in the third place so it is equal to 8*10 to the power 2 = 8*100 = 800

6- it is in the fourth place so it is equal to 6*10 to the power 3 = 6*1000 = 6000

And so, the total is 6000 + 800 + 70 + 5 = 6875

Until now it was just finished the beginning of the definition of the Number System.

After the creation of the decimal number system, it was further debated and developed by our scientists.   Using this number system it was defined addition, subtraction, multiplication, division, powers and roots etc.

And because of the newly created operations in mathematics, many new kind of numbers were found out and it changed the view of mathematics.

E.g. Rational numbers, Irrational numbers and Complex numbers, etc.  

Think in this topic:

How will you define a number system with base 1?

What is the meaning of base 1?

Tiny spots

Looking above the dark sky is the seed of our knowledge.  The feeling that comes when you see the night sky is truly amazing… 

When we are resting outside alone, we will happen to see the sky.

What will come to your mind?

Mind of elders will always think about their family, society and some other things. Mind of youngsters will always think about enjoying their lives with friends and others.

But the mind of the Scientists and the children will always think in a same way.

They will ask questions.. "the great questions of philosophy and science".

To feel the nature, please do the following. Go straight to your upstairs of your house and lie down, then watch the sky.

 Forget that you are in upstairs. Forget that you are in your house. Forget that you are in the earth.

But deeply focus on the night sky.

Now ask, where are you?

Your school mind will say

"I am in the earth and the earth was in the solar system"

 If you are little matured youngster learned science,

Your mind will say that you are in the universe.

But what is meant by “Universe”?

Does it have a shape?

            Does it have an end?

What was the limit of this universe? .

And moreover coming to the starting question,

Where is this universe?

These question goes on continuously without limits. 
You may ask these as many times you want.

 But don't lose the focus.

It is possible to make you very small. Very small like atoms!!!!!

You can ask me “How is it possible? Do you find a machine or something else like in the film “Honey I shrunk the kids" …..!!!!!!!!!!!!!!

I never find any machines. 
But I will just try to make you realize.

What I am going to do is like convex lens or magnifying glass.

When you see the letters by convex lens it looks large.

Though the letters are not really large, the lens makes you to think like that by creating an illusion.

In the same way I can make you small rather than an atom with the question “Where”.

I will magnify the surroundings using this magnifying glass.

As I said before if you are in the upstairs focused on the sky with the question “where”, slowly the process of magnifying starts in your brain.

When you asked the first “Where”, “where are you living?”

You will mention your house and your street.

When you compare your size with the size of the street you are like a small rat. 

If you ask another “Where” that is “where is your street?” 

You will mention your city.  Comparing with your city, you will be tiny like an ant.

Then again ask the next “Where”.

Now you will mention the full details of your position that is, your state, your country, your planet, your solar system, your galaxy and then finally the universe.

Now If you think that you are standing outside the earth, you will not see any creature in the world. May be you can see the Great Wall of China like a small line. Then you can see the big countries and the continents and the great oceans. But you will never able to see a human.

The size of the human becomes small as the size of the bacteria or amoeba comparing with the size of the earth.

I have magnified only about the small convex lens by using the size of Earth.

But I can magnify like a microscope by taking the solar system as the tool - The sun.

The size of the sun!!!!

1,300,000 Earths will fit inside the Sun.

 Also there are stars which larger than the sun.

Like the ones known as blue stars. We can put billions and billions of suns in that star.

And what about the earth?

How many earths can fulfill those blue stars?

But don't stop..  After all, there is galaxies. A galaxy has billions of blue stars and much more star systems.

And finally comes the Universe which has billions and billions of galaxies.

You can ask whether there is something else after this “Universe”.      

I don't know the answer. May be you can find.

Universe is similar to the infinity in mathematics.

We don't know what the final number in the number system. But to represent the largest number we made the symbol infinity.

Similarly we named the term Universe to represent the largest known space.

Now try to compare the size of a human and this whole universe...

What will come to your mind?

You have become incredibly small rather than anything else.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!