Algebra

Wikipedia is quite accurate to give definition of this Subject by including all its subbranches.

But what it actually is. Algebra is the true essence of core math where we start dealing with arbitrary symbols  to represent an object . Where in Arithmetic we are dealing with specific symbols like numbers (0-9 or more ) .   Arithmetic is Dealing with specific names and operations on Numbers . But Algebra is something beyond    that.

Here as I had already told we start to specify objects by giving them name and try to build operations on them . All objects are classified into an Algebraic Structure . That structure has basically set(s) of some elements as operands (On which the operation will be held ) and operators (Like + ,- ,* etc which will map from one operand set to another or that operand set  ).

What the thing it is actually ???

Ok ! let's see the following :-

Natural Numbers :- Set of Natural Numbers = N ={1,2,3,4.....}

Operators :- + ,*

Now we  are building  a algebraic structure like :- (N,+,*)

Now an Algebraic Structure may follow some property w.r.t some operation like :-

Closure :- x,y ∈ N => x+y ∈ N

Existence Identity :∃x  ∈ N s.t ∀y ∈ N x.y=y.x=y 

etc.

Can Human Do Continuous Mathematics

Well we collect information by our sensory Organs from Nature . All information are be taken as Discrete Data !
Yes ! We think we are viewing a continuous image when we are seeing our surrounding . But actually due to Perception of vision we see a sampled version of image :) Even we can't differ all frequencies but only can view a Bandwidth represent a frequency .

Similarly We have Perception of hearing , test etc. So we can detect only Sampled and Quantized Data A finite number of Symbols at a time . This is called Discrete data .

So it doesn't matter how our mind process them but actually our sensation , thought all are discrete but they occur so quickly that we can't understand the difference [ Just as when you are viewing this text your digital monitor pixels are turning off and on millions of times in a second ].
  

Solving Mathematical Problem

We , means Human and Computer both can Solve only finite problems [ Problems which can be solved by finite number of Computation(s) ] .

There are some problems which can't be solved by Any Human / Computer in Countable time [ They are called Undecidable Problem ] .

So we will solve problems Using following Steps : ->

1. Determine whether Problem is Decidable .

2. If Decidable then see whether it is finite or not.

3. If finite then solve it in finite steps else follow :-

              a) Induction       or 

              b) Contradiction  which one is Useful to reduce problem in finite Domain 

Induction:-

       

       Chose a property say P s.t. if it is valid for the whole set of problem then we can obviously say that the      

       problem is solved .  

      Find a Bijection f from problem set  to set of Natural Number N s.t Problem Set  = f(n)  

      where n ∈ N

      

       Basis: Prove it for a Small Cluster Of Problem Set .

       

       

       Induction Step : Prove that if for Problem set  f(1),f(2)...f(n) P holds then for f(n+1) also it holds.

       Now we can say that P holds for the Whole Problem Set . 

       

      That means Problem Solved.

Example :- Prove that f(n)=1+2+...n =n(n+1)/2 

                  Basis: n=1 it is true 

                  Induction Step : f(n)=n(n+1)/2 

                                          f(n+1)=f(n)+n+1= (n+1)(n+2)/2 i.e true

                                        

                 Proved.

Contradiction: Let the opposite is true . Then do the calculation with other Properties and  eventually you will  

                      find that a contradiction will arrive .

Example:- x=2 y=3 z=x+y then show z=5 

                let z!=5 

                x+y=2+3=5

                z=x+y

                5!=5 contradiction .

                 

                  

               

What is a Mathematical Problem

First of all we have to understand what is a mathematical problem.

It is typically a string which seek for more processing.

Like
      If  x+y=3 and x-y=1 then x=? y=?

Solving this problem means process this string/pattern till all questions resolved [ i.e Remove ? in this case ]

So basically  mathematics  is a mapping from set of Problem statement  A[say] to a set Of Solution B [say] under the Rule(s) S.

Thus,
Mathematical problem  M(say) | A---->B
                                                      S
Or         M | (A⨯S) ---- > B

Branches Of Mathematics

There are enormous branches of mathematics.

If we see from a schoolboy's vision we will find out :-

1. Arithmetic (Numbers and basic operations on Numbers)
2.Geometry
3.Algebra

Now in Higher Study Algebra and Geometry & Graphs took the main importance .

Algebra increase its span to :- Discrete , Continuous domain .

Set, Calculus  , Logic and Abstract algebra been introduced.

Geometry quickly mix up with Algebra and form  Graph ,Co-Ordinate Geometry ,Topology etc .

These are the main branches of Math but there are enormous subbranch and branching with different property are there .

What Is Mathematics

There are several  opinion about 'What is Mathematics ?' . I want to start from Some people's view :-

1. Wikipedia :- Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out   
    patterns  and formulate new conjectures.


2. Google Dictionary:- The abstract science of number, quantity, and space. Mathematics may be studied in its 
    own right (pure mathematics), or as it is applied to other disciplines such as physics and engineering (applied 
    mathematics).


3. Aristotle:- The science of quantity.


4.Comte:- The science of indirect measurement.


Now see what grammatically it mean :-


1. Mathematics (Eng) <mathematique (Old French )<  mathematica (Latin (ars)) means ‘mathematical (art)’ < 
    mathēmatikē (tekhnē) (Greek) from base manthanein means learn.
    
    So it means a way to Knowledge ( Learning ) [Or Knowledge itself ] .


2. Ganit [ Indian Sanskrit translation ] means Counting


To me mathematics is actually the way of generating symbols/patterns in consecutive time [ The generation of Symbol in consecutive time is  called  Computation ] .


What does it mean ? See the following:-


1+2=


Now you have 4 Symbols '1' , '+' ,'2', '=' which form a String [Sequence of Symbol] as depicted above . Now you know that this pattern asking you to put another Symbol on right of '=' .Also you know that 1 followed by a + and that followed by a 2 means 3 . So you will put 3 at right of '=' and extend that string to 1+2=3.


This is the basic how Our Brain / Thought Computes.


Similarly ,suppose I am going to tell you :- A box had 3 apples . I took 2 from it and put 8 in it . How many apples will remain in the Box?  ------ (1)


A student of mathematics will quickly MAP this into an mathematical problem 3-2+8 = ? And solve it by giving ans 9. 
    
Now look that previous String / Sequence of Symbol  which had been given name (1) is a Pattern . 
Your Obvious Question will be is it a mathematical expression ?


Obviously ans will be 'Yes' .


Does it contradict what you had learned till now ? It isn't a equation like x+y=z not a geometrical figure like a Circle so how a String can be a mathematical expression . 


Well don't fear because it doesn't contradict at all . Just recall your History of Understanding Mathematics .


You first learned Arithmetic with 10 Symbols {0,1,2,..,9} the Operand and initially 2 Extra symbol '+' and '-'  named as Operator and '=' '>' '<' . So then mathematics was limited in those symbols . 


Later you found out * and / and searched Natural Numbers , Integers , Rational numbers .


Later you increased your span by knowing Algebra [With arbitrary extra symbols ] and ease your Computation.
Then you found out root , power , log and So on... 


So day by day the set of mathematics increased to you by introducing new Symbol with new properties .


Thus a straight forward intuition is to insert more and more symbol in mathematical Domain . 


So ultimately you will reach what had I told earlier [the Definition of Mathematics]. 




Every Language actually is  a mathematical Expression [ Though meaning of Some Symbol may be Unknown or not well defined but that's a limitation of our knowledge / capability not the fault of Mathematics ] .


So Whenever we right / construct something in some language we actually form a mathematical expression .


You may think that this is just a mere generalization but actually has no Application . But truly this is the most applied mathematical stuff . Computer Science is a study of computation , it use computer as a machine which can do finite no of computation . In Computer Science Automata and Language , Algorithm and Complexity are totally deals with the above mentioned definition and these are  consider as  mathematics disciplines .


Then is all that we can know reside under the main fold of mathematics . Ans is no . Why ? Because all that we know is obviously a Sequence of pattern but the transformation  of the pattern in consecutive time is mathematics / Computation but not the patterns . 


Simply Suppose we saw that whatever body we saw  when we release them in Space it fall on Earth . Then mathematics will tell that :-
 ∀x s.t x is body and we had released them in Space <=> They fall on the Ground  ----(1)
  Number of body we released in space is Large Say N -------- (2)


 (1) and (2) => There is a High probability that  ∀x s.t x is body when they will be released in Space[By 
                       anybody not only we that is we who did/Saw that experiment ] they will fall on the Ground .

Now if N-> Inf we can Obviously say that for all body that been released in space will fall on the ground.


Similar way if we conduct any Scientific experiment we will have a Statistical data which will direct us to Build Some theorem . 


So there is two part -
1. Information Collected from Nature , 
2. Mathematical Stuff to Build a theorem . 


The Part 1. is the key factor which differs Science from Mathematics . 


What is the difference between  Math and Computation ?


Actually Computation means the Ongoing traversal between patterns and Math Says the way of the traversal . Thus Math is the way of Computation [ This part is called Algorithm ] and consist of Some General theorem About Some Abstract Structure We Build to Do ease this overhead of Computation [Mathematical Theorems ] . 


And Computation is a Algorithm in Execution .