BASIC MATHS

            
1)   Number system
2)   Fraction (p/q)
3)   Rational Numbers (Q)
4)   Irrational Numbers (Qᶜ)
5)   Real Numbers (R)
6)   Complex Numbers (C)
7)   Even Numbers
8)   Odd Numbers
9)   Prime Numbers
10) Composite Numbers
11) Co-Prime Numbers/Relatively
       Prime Numbers
12) Twin Prime Numbers
13) Numbers To Remember
14) Divisibility Rules
15) LCM And HCF
16) Factorisation
17) Cyclic Factors
18) Remainder Theorem
19) Factor Theorem
20) Ratio And Proportion
21) Intervals
22) Basic Concepts of Geometry
23) Basic Concepts of
       Mensuration
24) Indices and Surds.

-------+---++++++++---------------+++++--

1).        NUMBER SYSTEM

a) Natural Numbers (N) =
                               (1, 2, 3 ......∞).

b) Whole Numbers (W) =
                        (0, 1, 2, 3,.......∞).

c) Integers (I) =
       (-∞,.....-3, -2, -1,0,1,2,3.....∞).

d) Positive Integers (I⁺) =
                         (1, 2, 3 .....∞).

e) Negative Integers (I⁻) =
                       (-∞, ...-3, -2, -1).

f) Non-negative Integers =
                           (0, 1, 2, 3.......).

g) Non-positive Integers =
                     (-∞, .....-3, -2, -1, 0).

h) Even Integers =
               (...-6, -4, -2, 0, 2, 4,6...).

I) Odd Integers =
                      (-5,-3,-1,1,3,5.....).

****Note
*       Zero is neither positive nor
         negative.
**     Zero is even number.
***   Positive means > 0
**** Non-negative means ≥ 0
                 ------        --------

2) FRACTION (p/q) :::

a) Proper Fraction = 3/5 : Nʳ< Dʳ.

b) Improper Fraction =
                                5/3 : Nʳ > Dʳ.

c) Mixed Fraction : 2+3/5.

d) Compound Fraction: 2/3/5/6.

e) Complex Fraction : 7/3.

f) Continued Fraction: 2+     2.     
                                             2+ 2/+....
  This is usually written in the more
   compact form 2+1/2+. 1/2+ .....

                ---------       ---------

3. RATIONAL NUMBERS (Q)

    All the numbers that can be
    represented in the form of p/q,
    where p and q are integers and
    q ≠ 0, are called rational
    numbers. Integers, Fractions,
    Terminating decimal numbers,
    Non-terminating but repeating
    decimal numbers are all rational
    numbers.
    Q = (p/q: p , q ∈ I and q ≠ 0, ).

** NOTE **

* Integers are rational numbers, but
  converse need not to be true.
** A rational number always exists
   between two distinct rational
   numbers, hence infinite rational
   numbers exists between rational
   numbers.

4) IRRATIONAL NUMBERS (Qᶜ)

   There are real numbers which
   can not be expressed in p/q
   form.
   Non-Terminating non repeating
   decimal numbers are irrational
   number e.g. √2 , √5,√3, ³√10, e ,π.

    e⇔ 2.71 is called Napier′s
    constant and π⇔3.14.

* Note *

* Sum of a rational number and an
   irrational number is an irrational
   number e.g. 2+√3.
** If a ∈ Q and b ∉ Q, then
    ab = rational number, only a=0.
*** Sum, difference , product and
   quotient of two irrational
   numbers need not be an
   irrational number or we can say,
   result may be a rational number
   also.

5) REAL NUMBERS (R)

     The complete set of natural and
      irrational number is the set of
      real numbers, R = Q ∪ Qᶜ.
      The real numbers can be
      represented as a position of a
      point on the real number line.

6) Complex Numbers. (C)

    A number of the form a + ib,
    where a, b ∈ R and i=√-1 is called
    a complex number. Complex
    number is usually denoted by z
    and the set of all complex
    numbers is represented by :
    C=((x+iy) : x, y ∈ R, I =√-1).

      N ⊂ W  ⊂ I ⊂ Q ⊂ R ⊂ C

7) EVEN NUMBERS :

     Numbers divisible by 2, last digit
     0,2,4,6,8 & represented by 2n.

8) ODD NUMBERS :

    Not divisible by 2, last digit
    1,3,5,7,9 represented by (2n ± 1)
a)  even ± even = even.
b)  even ± odd = odd.
c)  odd ± odd = even.
d)  even x any number= even
                                            number.
e)  odd x odd = odd.

9) PRIME NUMBERS :

    Let ' p ' be a natural number, ' p '
    is said to be prime if it has
    exactly two distinct positive
    integral factors, namely 1 and
    itself. e.g 2 ,3, 5, 7, 11, 13, 17, 19,
    23,29,31 .....

10) COMPOSITE NUMBERS :

     A number that has more than
     two divisors .

**Note **

*    ' 1 ' is neither prime nor
      Composite
**  '2' is the only even prime
      number.
*** '4' is the smallest Composite
       number.
****Natural numbers which are not
        prime are Composite numbers
        (except 1).

11) CO-PRIME NUMBER/ RELATIVELY PRIME NUMBERS :

     Two natural numbers ( not
     necessarily prime) are Co-Prime,
     if their H. C. F. is one e.g. (1,2),
     (1,3),(3,4),(5,6) etc.

**Note**

*  Two distinct prime number (s)
     are always Co-Prime but
     converse need not to be true.
** Consecutive natural numbers
     are always Co-Prime numbers.

12) TWIN PRIME NUMBERS :

     If the difference between two
     prime numbers is two, then the
     numbers are twin prime
     numbers. e.g. {3,5 }{5,7},{11,13}
     etc.