NUMBER SYSTEM

NUMBER SYSTEMS 

INTRODUCTION:

                   NUMBERS
-------------------------------------------------------------
    |                                              |
REAL NUMBERS   IMAGINARY NUMBERS
   |                                        |
IRRATIONAL NOS.    RATIONAL NOS.
 |               |                       |                 |
SURDS  IRRATIONAL  Integers  Fractions 
(√2,3√5)  NOS. OTHER                that are
                 THAN SURDS                  not
     (e=2.718, π=3.14159.. etc)    integers

                        Integers
  ---------------------------------------------------------
|                                |                         |
Positive                zero            Negative 
Integers                                   Integers

Real Numbers 
 The set of all rational and irrational numbers forms the set of real numbers.
Ex:- -3, -5, √2, √3, 0 etc.

OR
All numbers that can be represented on the number lines are called real numbers. Every real number can be approximately replaced with a terminating decimal. 
    The following operations of addition, subtraction, multiplication and division are valid for both whole numbers and real numbers: [For any real or whole numbers a, b and c].
a) Commutative property of addition:  a + b = b + a .
b) Associative property of addition: (a + b) + c = a + (b + c).
c) Commutative property of multiplication: a.b = b.a
d) Associative property of multiplication: (a.b). c = a.(b.c)
e) Distributive property a multiplication with respect to addition: (a + b)c= ac + bc.
f) Subtraction and division are defined as the inverse operation to addition and multiplication respectively.
     Thus id a + b= c, then c- b= a and q=a/b then b.q = a(where b≠0).
     Division by zero is not possible since there is no number for which b.q equals to a non zero number a.


 Rational Numbers
A number that can be expressed in the form of x/y where x and y are dissimilar integers and y ≠ 0, is it rational number.
       Rational numbers have terminating decimal representation or non- terminating but recurring decimal expressions.
Ex:- 4/10= 0.4, 1=0 (Terminating)
1/3= 0.3333 (Recurring)

OR
A rational number is defined as number of the form a/b where a and b are integers and the b≠0.
   The set of rational numbers and encloses the set of integers and fractions. The rules given above for addition, subtraction, multiplication and division also apply on rational numbers.
       Rational numbers that are not integeral will have decimal values. These values can be of two types:
A)  Terminating (or finite)decimal fraction:
 For example, 17/4= 4.25, 21/5= 4.2 and so forth.
B) Non terminating and decimal fraction: 
Among non-terminating decimal fractions there are two types of decimal values:
 i) Non terminating periodic fractions:
These are non terminating decimal fractions of the type x. a₁a₂a₃a₄ .... aₙ a₁ a₂ a₃a₄..... aₙa₁a₂a₃a₄... aₙ. For example 16/3= 5.33333, 15.232323, 14.28762876 ..... and so on.
 ii) Non-terminating non periodic functions:
 These are of the form for x. b₁b₂b₃b₄ .... bₙc₁c₂c₃..... cₙ. For example 5.2731687143725186.
  Of the above categories, terminating decimal and non-terminating periodic  decimal fraction belong to the set of rational numbers.

Irrational Numbers 
Numbers that cannot be exposed in the form x/y where x and y are Integers and y≠0, are known as irrational numbers.
Ex:- √5, √7, √2, π, e
       These numbers when expressed in decimal form result in an answer where digits after the decimal point and non ending and non recurring.

OR

Fractions, that are non terminating, non periodic fractions, are irrational numbers.
   Some examples of irrational numbers are √2, √3 etc. In other words, all square and cube roots of natural numbers that are not squares and cubes of natural numbers are irrational. Other irrational numbers include π, e and so on.
    Every positive irrational number has a negative irrational number corresponding to it.
     All operations of addition, subtraction, multiplication and division applicable to rational numbers are also applicable to irrational numbers.
   Whenever an expression contains a rational and an irrational number together, the two have to be carried together till the end. In other words, an irrational numbers once it appears in the solution of a question will continue to appear till the end of the question. This concept is particularly useful in Geometry For example: If you are asked to find the ratio of the area of a circle to that of an equilateral triangle, you can expect to see a β/√3 in the answer. This is because the area of a circle will always have a p component in it, while that of an equilateral triangle will always have √3.
  You should realise that once an irrational number appears in the solution of a question, it can only disappear if it is multiplied or divided by the same irrational number.

 Integers 
[-∞,.....,-3, -2, -1, 0, 1, 2, 3,..... +∞] These numbers form the set of Integers. They are rational numbers which when reduced to the lowest form, do not contain a fractional part.

 Non-integer 
Non-integers are rational numbers which when reduced to their lowest forms, contain a certain fractional part.
Ex:- 2/3, 5/3, 7/9, ....16/21 etc.

 Natural numbers
The set of all positive integers from the set of Natural numbers.
Ex:- (1,2,3,........... ∞)

OR
These are the numbers(1,2,3 etc) that are used for counting. In other words, all positive integers are natural numbers.
        There are infinite natural numbers and the number 1 is the least natural number.
Example of natural numbers: 1,2,4,8,32, 23, 4321 so on.
      The following numbers are examples of numbers that are not natural: - 2,  -3, -31, 2.38, 0 and so on.

Based on divisibility, there could be two types of natural numbers: Prime and  Composite.

Whole Numbers
The set of all natural numbers and zero(0), form the set of whole numbers.
(0,1,2,3,4,..........∞)

Common symbols for  Representation 

Number type                Set Notation 
* Whole number                W
* Natural number              N
* Integers                          I or Z
Real numbers                    R

x ∈ W implies x is a Whole Number.
x ∈ implies x is a Real Number and so on.

 Concept of i
 The square root of (-1) is denoted by i
i.e. i=-√1
      i²= -1
       i⁴= i² × i² = (-1)×(-1)= 1

Imaginary Numbers
The square root of negative numbers are not real, such numbers are called imaginary numbers.
Ex:- √-64 , = √-1 × √-64= √-1 × 8 = 8i

Complex Numbers
Numbers which an imaginary part are known as complex numbers. i.e. the imaginary part is non zero.
e.g: 2+ 3i, 4 - 5i
    Real number is a special case of complex number having the imaginary part zero(0).
A purely imaginary number has the real part zero(0).

Prime Numbers 
Number that are only divisible by themselves and one (1) are called Prime Numbers
e.g.,  2, 3, 5, 7, 11, 13, 17.......
 However it should be noted that 1 is not a Prime Number.
N. B: There are 25 Prime Numbers from 1 to 100 
Any prime number greater than 3 is of the form 6N+1 or 6N -1 
Ex:- 13= 6×2+1

OR
A natural number larger than unity is a prime number if it does not have other devisirs except for itself and Unity.

Note: Unity (1) is not a prime number.

 Some properties a Prime Numbers 
• The lowest prime number is 2.
• 2 is also the only even prime number.
•  The lowest odd prime number is 3.
• The remainder when a prime number P≥ 5 is divided by 6 is 1 or 5. However, if a number on a being divided by 6 gives a remainder of 1or 5 the number need not be prime.
• The remainder of the division of the square of a prime number P≥5 divided by 24 is 1.
• For prime numbers P> 3, p² - 1 is divisible by 24.
• Prime Numbers between 1 to 100 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97
• Prime Numbers between 100 to 200 are: 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199.
• If a and b are any two odd primes then a²- b² is composite. Also, a²+ b² is composite.
• The remainder of the division of the square of a prime number P≥5 divided by 12 is 1.

Short Cut Process
To Check Whether a Number is Prime or Not
 
To check whether a number N is prime, adopt the following process.
A) Take the square root of the number.
B) Round of the square root to the immediately lower integer. Call this number z. For example if you have to check for 181, its square root will be 13.. , Hence, the value of z, is this case will be 13.
C)  Check for divisibility of the number N by all prime numbers below z. If there is no prime number below the value of z which divides N then the number N will be prime.
Ex:
 The value of value √239 lies between 15 to 16. Hence, take the value of z as 16.
     Prime numbers less than 16 are 2, 3, 5, 7, 11 and 13, 239 is not divisible by any of these. Hence you can conclude that 239 is a Prime Number.

Composite Numbers:
A number which is divisible by numbers other than itself and 1 are known as composite numbers. Composite numbers can be broken into a set of Prime Factors.
Ex:- 15 = 3 × 5 ; 36 =2 ×2×3 ×3. etc.

OR
 It is a natural number that has at least one divisor different from unity and itself.
    Every composite number n can be factored into its prime factors. (This is sometimes called the canonical form of a number).
  In mathematical terms: n= pᵐ₁ . pⁿ₂ ...pˢₖ, where p₁, p₂ , ...pₖ are prime numbers called factors and m, n, ...k are natural numbers.
Thus,    24= 2³.3
              84= 7.3.2² etc
 This representation of a composite number is known as the standard form of a composite number. It is an extremely useful form.

Co-Prime Numbers
Two numbers are said to be co-prime it they are mutually prime to each other, i.e., they don't have a common factor between them except for one. The co-prime numbers may be either prime numbers or composite numbers.
e.g., 15, 16 though not Prime themselves, are mutually Prime to each other. Hence they are co-prime numbers.
      Two prime numbers are always co-prime to each other.

Test of Prime Numbers:
To check whether a number is Prime, we need to check divisibility by Prime Numbers till the Prime Number square exceeds the given number.
e.g.,  To check whether 359 is a Prime Number.
 step 1: Find a perfect square immediately greater than 359. The number is 361.
Step 2: Take the square root of 361. It is 19.
Step 3: Check whether any prime number within 1 - (19-1) is a factor of 359.
Since, no prime number in the range 1 to 18 is a factor of 359, hence, 359 is a prime number.

 VBODAMAS 
In resolving the value of a given expression the various operations must be performed in the given order.
1.  Vinculum or B                      V
2. Removal of a bracket in the order ( ), { } , [ ]                                             B
3. Of                                           O
4.  Division(÷)                           D
5.  Multiplication(×)                 M
6. Addition(+)                            A
7. Subtraction(-)                        S

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