NUMBER SYSTEM CONCEPTS,FORMULAS,DEFINITIONS,EXAMPLES,TRICKS PART-1

number system is very important chapter for all competitive exam . everything is included in it like all numbers,some special number,divisibility concept,unit digit,divisibility remainder concept,decimal and number of factor,key to remainder concept,lcm and hcf etc.

Number System:-

1) Natural Number:- All counting numbers are called natural numbers and it is denoted by N or number used for counting.

 Example: {1, 2, 3, 4..infinity)

 2) Whole Number:- The set of all natural number together with zero are called whole numbers and it is denoted by W or from zero to infinity.

 Example:- (0, 1, 2, 3, 4.....)

3) Integer:- The set of all natural numbers, zero and negative of all natural numbers are called integers and it is denoted by I and Z or all positive to negative number.

Example: ......-6,-5, -4,-3, -2, -1, 0, 1, 2, 3, 4,5,6....

* Negative integer is denoted by ={.....-5,-4,-3, -2, -1}
*Positive integer is denoted by I= {1, 2, 3, 4,5,6......)

4) Rational Number:- A number of the form p/q where p and q are whole numbers and q is not equal to zero.

Example:- 3/5, 2/3, -7/2, 0, 2, 7
*it can be negative, positive and also zero because we can write 0 as 0/1.
* it is denoted by Q.

5) Irrational Number:- Numbers which can neither be written as a recurring decimal nor in repeating decimal or (A number that can't be written in the form of p/q) or irrational numbers are not perfect square.

 Example:- 3.01001000100001, 1.23223222322223,√3,  etc.

6) Real Number:- The set of all rational numbers and irrational numbers are called Real Numbers or the number that gives positive result after squaring.

 Example:- 1,2/3, 0.6, 3.01001000100001 √з, π,  etc. 

7) Complex Number:- A number of the form (a + ib) is called a complex number where a and b are any real number and i is called iota (√-1).

 Example: √-3= 3 i, 7+ 3√-5= 7 +i 5, etc.

Some special numbers:-

8) Prime Number:- A number is called prime number that has only two factor (1 or itself) or number having only two factor.

Example:- 2, 3, 5, 7, 11, 13.. etc

* Note:  we can say the smallest(lowest) prime number is not 1.

the smallest prime number is only 2.

9) Composite Number:- Those numbers are called composite numbers which has more than two factors or except 1 and prime.

Example:- 4 because the factor of 4 are 1, 2 and 4.

10) Co Prime Number:- A pair of any two integers is called a Co Prime Number, if their H.C.F is 1.  

Example:- (7, 11), (5, 11) etc. 

Note:- (7, 11) is not a twin prime number.

DIVISIBILITY CONCEPT 

concept by the help of which we can check the divisibility or factors of a number without actual division:

CONCEPT :-  Divisibility by 2: Numbers whose last digit is either even or zero are divisible by 2.

Example:- 4,6, 14,42, 78 etc.

CONCEPT :-  Divisibility by 3: If the sum of digits of a number is divisible by 3, then the number is also divisible by 3.

 Example:- 12, 546, 99 etc.

CONCEPT :-  Divisibility by 4: If the last two digits of a Number is divisible by 4 or the number having two or more zeros at the end are divisible by 4.

Example:- 416, 780, 800 etc.

CONCEPT :-  Divisibility by 5: If the last digit of a number is divisible by 5 or 0, then the number is divisible by 5.

Example:-55, 10 etc.

CONCEPT :-  Divisibility by 6: If a number is divisible by 2 and 3, then number is also divisible by 6. 

Example:- 726, 732 etc.

CONCEPT:- Divisibility by 8: If the last three digits of a number are divisible by 8 or the last three digits of a number are zeros, then the number is divisible by 8

 Example:- 4432, 2000 etc.

CONCEPT :- Divisibility by 9: If the sum of all the digits of a number is divisible by 9, then the number is also divisible by 9.

 Example:- 9846, 855 etc.

CONCEPT :-  Divisibility by 11: If the difference between the Sum of digits at even place and Sum of digits at odd place is divisible by 11, then the whole number is divisible by 11.

 Example:- 54659, 968, 923587643 etc.

CONCEPT :-  Divisibility by 12: The number which is divisible by both 3 and 4 is also divisible by 12. 

Example:- 12, 588 etc.