Number System

Number System
Numbers along with arithmetic operators.

Number 
Concept of counting or measuring. example 4 cows, 6 trees.

(a) Natural Numbers (N)
Numbers of the type 1, 2, 3, 4 ...

(b) Whole Numbers (W)
Numbers of the type 0, 1, 2, 3, 4 ... 

(c) Integers (Z)
Numbers of the type  ... - 3, - 2, - 1, 0, 1, 2, 3, 4 ... 

(d) Rational Numbers (Q)
A number of the type p/q where p and q are integers and q ≠ 0.

(e) Irrational Numbers (I)
A (Real) number ‘i’ is called irrational if it cannot be written in the form  p/q
where p and q are integers and q ≠ 0. 

(f) Real Numbers (R)
Combination of real rational and irrational numbers

(g) Complex Numbers (C)
Numbers of the form a + bi where a, b are real numbers and i = squareroot - 1
Lemma 
A proven statement used to prove other statements

Euclid's Division
Given two positive integers a and b,( a >= b) there exists unique integers q and r satisfying a = bq + r, 0 < = r < b.

Algorithm
An algorithm is a series of well defined steps which gives a procedure for solving a type of
problem.

Euclid's Division algorithm
This is used to obtain the HCF of two positive integers, say a and b, with a > b, follow the steps below:

#  Apply Euclid's division lemma, to a and b. So, we find whole numbers, q and r such that a = bq + r, 0 < = r < b.
#  If r = 0, b is the HCF of a and b. If r not equal to 0,  apply the division lemma to b and r.
#  Continue the process till the remainder is zero. the divisor at this stage will be the required HCF.

Prime Numbers
A number which has exactly two factors. example 2, 3, 5.

Composite Numbers
A number which has more than two factors. Example 4, 6, 8.


Is 1 a prime number?
No. 1 is neither prime nor composite because it has only 1 factor.


Is Every whole/integer no. rational ?
Yes, because every whole no. can be written with denominator 1 that is in the form of p/q where p and q are integers and q is not equal to zero.
examples 3 = 3/1, - 9 = - 9/1

Is Every rational no. a whole no./integer  ?
No. 7/5 is a rational number but not a whole no./ integer.


Fundamental Theorem of Arithmatic
Every composite number can be expressed as a product of primes, and this factorisation is
unique, apart from the order in which the prime factors occur. 

Correct proof of Fundamental Theorem of Arithmatic
Correct proof of Fundamental Theorem of Arithmatic was given by Carl Friedrich Gauss.


Relation of 2 numbers with their LCM and HCF
LCM × HCF = Product of numbers

Theorem on Prime numbers and factors
Let n be a prime number. If n divides a^2, then n divides a, where
a is a positive integer.