Calculus start with Number Theorem

Calculus means pebble a very small object . The meaning of calculus is to analysis a very large object by going to its smallest part . The target is to integrate differentiate and have some fantastic knowledge of numbers and measured objects .

Ok! first to proceed to calculus  we have a prerequisite of  basic Number theory .

People first started to count objects as 1 , 2 , 3 , 4 ... like this . To simplify counting they form numbers to be written with respect to a base or radix .

For example in decimal  / Metric system base is 10 . i.e if we write 15 it means 10 + 5  .                   1421 = 1000 + 400 + 20 + 1 etc .
In any number system with base r the number written as →
a_n a_(n-1)... a_0  means a_n * r^(n-1) + a_(n-1)*r^(n-2) + .... a_0 * r^0 .

Now Indian scientist first invented from the philosophical study that if we negate a from a  we got something and that is named as 0 ( A revolution ) .

The set N = {1,2,3...} is the set of natural number . Now we add 0 with it . But still we can't find        11 - 20 . So then the negative numbers came . So we had a larger set Z = {0 , ±1,±2...} formed and we called it the set of Integers .

Now to find numbers like 2/5 3/7 we further increase the set and introduce fraction . So the set with integers and fraction uniformly create R = {0 , ±1 ,±1/2  ,2 ,....} the set of Rational Numbers .

But then Also some number remained which can't be represent as ratio of two integers (Rational Numbers )  like √2 .They called irrationals  . Rational and irrationals uniformly create Real Numbers (Re say) .

Then Also some number remains like (√(-1)) . Scientist then introduce another number i = (√(-1)) and called number having i as Complex numbers (C say ) .

Now we have N < Z < R < Re < C [ < here represent  ⊂ ]

Properties : Closed means with respect to that operation on elements of set mention in the  row result will be member of that set .

Sets  Vs  Operation	+	-	*	/(except by 0)	^	√
N	Closed		Closed		Closed
Z	Closed	Closed	Closed
R	Closed	Closed	Closed	Closed	Closed
Re	Closed	Closed	Closed	Closed	Closed
C	Closed	Closed	Closed	Closed	Closed	Closed

Still some numbers are not known like 0^0 Something / 0 etc . IEEE tell them as NaN [ Not a Number ]

A set is called ordered / partial Order if we can Order the elements of the set somehow [Above are all ordered set so any to element a,b of Order set either  a < b or a = b or a > b ].

Cantor did so much work on the measure of the sets .
1. Two sets are equinumerous iff they have bijection .
2. A set is infinite iff it has bijection to some of its proper subset . [ All above sets are infinite ]
3. A set is countable iff it has a bijection with some subset of N. [ Set N,Z,R are countable ]
4. A set is dense if between any two element of that set there is another element of that set . [R,Re,C are dense]

For showing uncountability of one infinite set we use Cantor Diagonalization Principle .


Calculus is the analysis of real numbers . Calculus means algebra with Concept with Limit , Continuity and infinity .

As we already know what is a function we will start with Limit .