MATHEMATICS IN EASY STEPS : NUMBER AND NUMBER SYSTEM . KNOW ABOUT DEVISIBILITY RULES

        

Mathematics is a easy subject but maximum find them so hard to learn and remember Mathematics functions and rules. That all because we not give time to understand basics of Mathematics rule. So learn here the basic rules step by step.

The first step to find understanding is learn about number and number system. Also divisibility rules so useful in solve questions. We find us uncomfortable if we not know as which number original number can be divide , that can be make easy if we know divisibility rules. Here all will describe in short notes so you can understand soon.

1. Types of Numbers :

Following are the basic types of numbers :-

1. Natural Numbers  (N) = (1,2,3,4,...)

2. Whole Numbers  (W) = (0,1,2,3,4,...)

3. Integers  (Z) = ( … , -3,-2,-1,0,1,2,3 , … )

4. Real Numbers  (R) = (23.057 , 0.057 , 0.5555 , 5/7 ,  √3 , -15 , 20 etc.)

5. Rational Numbers  (Q) = ( 1/2 , -3/5 , 2/9 , 0/2 , 99/100 , 1.56 etc.)

6. Irrational Numbers  (P) = (√2 , √3 , √5 , 0.343434 , Π , etc.)

7. Complex Numbers ⟨ ( a+b i )form ⟩ =( 4+5i , -3+4i , 2+√2i etc. )

other related numbers are :-

8. Imaginary Numbers

9. Discrete and Continuous Numbers

10. Prime Numbers

11. Composite Numbers

1. Natural Numbers  (N) :

    These are positive integers. Zero (0) not a natural number.

     (N) = (1,2,3,4,...)

2. Whole Numbers  (W) :

    These are the set of natural numbers with add "Zero" (0) .

    (W) = (0,1,2,3,4,...)

3. Integers  (Z) :

    These are the set of whole numbers with included negative natural numbers.

    (Z) = ( … , -3,-2,-1,0,1,2,3 , … )

4. Real Numbers  (R) :

    These are the numbers which can be use decimal also. Fractions also write in decimal form. It also includes all the irrational numbers such as  Π , √2 , √3 etc. 

All integers are real numbers but not all real numbers are integers.

Each real Numbers can be represented on number line.

Real numbers are combination of all numbers such as rational , irrational , decimal etc.

Real numbers includes all the positive and negative integers , whole numbers , fractions , repeating decimals , terminating decimals etc. complex numbers and imaginary numbers are not real numbers.

     (R) = (23.057 , 0.057 , 0.5555 , 5/7 ,  √3 , -15 , 20 etc.)

5. Rational Numbers  (Q) :

    These are the fractions where numerator or denominator numbers are integers that includes both positive and negative fractions but denominator never be zero (0) , but numerator can be zero.

Decimal numbers can be represented in fractional form.

Example : 0.256 , -256/1000 , -35/100 etc.

    (Q) = ( 1/2 , -3/5 , 2/9 , 0/2 , 99/100 , 1.56 etc.)

6. Irrational Numbers  (P) :

    These are the numbers which can't be possible to express in the form of P/Q (Fractional form).

    (P) = (√2 , √3 , √5 , 0.343434 , Π , etc.)

7. Complex Numbers ⟨ ( a+b i )form ⟩ :

    The numbers which in the form of ( a + b i ) is called complex numbers ; where "a" and "b" are real number and " i '' is an imaginary number.

Example : 4+5i , -3+4i , 2+√2i etc. are complex numbers.

8. Imaginary Numbers :

    Imaginary numbers are use in complex numbers . The square root of negetive number is represented by letter " i ".

Example : √(-2) , √-5 etc.

# SOLUTION OF IMAGINARY NUMBER :

    If X = √ (-81) then we can solve that as below -

        X = √ ( 81 x -1) = √81 x √(-1) = 9 x √(-1)

            = 9i

Hence X = 9i

9. Discrete and Continuous Numbers :

    The Natural numbers , Whole numbers , Integers , Rational numbers are referred to as discrete numbers. Discrete numbers can be countable.

But,

Real numbers can't be counted and are continuous numbers.

10. Prime Numbers :

    These are the natural numbers which have no factors other then 1 and itself. 1 is not a prime number. The smallest prime number is 2.

    Example : 2,3,5,7,... etc. are prime numbers.

11. Composite Numbers :

     Composite Numbers have some factors of a respective numbers such as 4, 15 are composite numbers because 4 is divisible by 2 ; similarly 15 is divisible by 3 and 5.

2. NUMBER SYSTEM :

A number system is defined as a system of writing to express numbers.

Number system is the mathematical representation of numbers of a given set of digits or other symbols in a systematic manner.

Following are the basic number systems :--

1. Decimal Number System

2. Binary Number System

3. Octal Number System

4. Hexadecimal Number System

In the number system , each number is represented by its base.

If the base is 2 , its called binary number system ;

If the base is 8 , its called octal number system ;

If the base is 10 , its called decimal number system ;

If the base is 16 , its called hexadecimal number system.

DECIMAL NUMBER SYSTEM :--

    In decimal number system we use the base 10 . It also called base 10 number system. Use numbers 0,1,2,3,4,5,6,7,8,9.

For representing decimal system we follow this rule ; for example :-

(342)₁₀ = ( 3x10² )+(4x10¹)+(2x10⁰)

             = 300+40+2

(348.258)₁₀ = ( 3x10² )+(4x10¹)+(8x10⁰)+( 2x10^-1 )+( 5x10^-2 )+( 8x10^-3 )

                     = 300+40+8+0.2+0.05+0.008

3. DIVISIBILITY RULES (DIVISION RULES) :

(1). Divisibility by 2 :

        If the ones digit of a number is either 0 or even number , then the given number will be divisible by 2.

Example : 2, 14, 246 , 298 etc.

(2). Divisibility by 3 :

        If the sum of all digits of number is divisible by 3 , then the given number will also be divisible by 3.

Example : 1608 ; Here 1+6+0+8 = 15 ; which is divisible by 3 so 1608 also divisible by 3.

(3). Divisibility by 4 :

       If the given number's tens and ones digit joining number is divisible by 4 , then given number will be divisible by 4.

Example : 954948 ; Here last 2 digits (48) is divisible by 4 so 954948 is divisible by 4.

(4). Divisibility by 5 :

       If the ones digit of a number is either 0 or 5 , then the given number will be divisible by 5.

Example : 485 , 590 are divisible by 5.

(5). Divisibility by 7 :

       Substract 2 times the last digit from remaining number . Repeat the step as necessary up to 2 digit number. If result is either 0 or divisible by 7 , then the given number will be divisible by 7.

Example : 321468 → 32146 - (2x8) = 32130→3213-(2x0)=3213→

                                  321-(2x3)=315→31-(2x5)=21; 

                  Which is divisible by 7 ; so 321468 also be divisible by 7

(6). Divisibility by 8 :

       If the given number's hundreds, tens and ones digit joining number is divisible by 8 , then given number will be divisible by 8.

Example : 433872 ; Here last 3 digits (872) is divisible by 8 so 433872 is divisible by 8.

(7). Divisibility by 9 :

        If the sum of all digits of number is divisible by 9 , then the given number will also be divisible by 9.

Example : 4085289 ; Here 4+0+8+5+2+8+9 = 36 ; which is divisible by 9 so 4085289 also divisible by 9.

(8). Divisibility by 11 :

       If the difference between the sum of even numbers and odd numbers are either 0 or divisible by 11; then the give number will be divisible by 11.

Example : 50525827 ; Here start from ones the sum of even numbers = 2+5+5+5 = 17 and sum of odd numbers= 7+8+2+0 = 17 

Here, 

Sum of even no. - Sum of odd no. = 17-17 = 0 

So, given number will be divisible by 11.

(9). Divisibility by 13 :

       Add 4 times the last digit from remaining number . Repeat the step as necessary up to 2 digit number. If result is divisible by 13 , then the given number will be divisible by 13.

Example : 5161 → 516 + (4x1) = 520→52+(4x0)=52→The result is divisible by 13 , so the original number will be divisible by 13.

(10). Divisibility by 17 :

       Substract 5 times the last digit from remaining number . Repeat the step as necessary up to 2 digit number. If result is either 0 or divisible by 17,then the given number will be divisible by 17.

Example : 94843 → 9484 - (5x3) = 9469→946-(5x9)=901→90-(5x1)                                 =85 ; The result is divisible by 17 , so the original number will be divisible by 17.

(11). Divisibility by 19 :

       Add 2 times the last digit from remaining number . Repeat the step as necessary up to 2 digit number. If result is divisible by 19,then the given number will be divisible by 19.

Example : 1132871637 → 113287163+(2x7) =          113287177→11328717+(2x7)=11328731→1132873+(2x1)                    =1132875→113287+ (2x5)=113297→11329+(2x7)= 11343→1134+(2x3)=1140→114+(2x0)=114→11+(2x4)=19  ; The result is divisible by 19 , so the original number will be divisible by 19.

(12). Divisibility by 23 :

       Add 7 times the last digit from remaining number . Repeat the step as necessary up to 2 digit number. If result is divisible by 23 ,then the given number will be divisible by 23.

Example : 1207339 → 120733+(7x9) =120796→12079+(7x6)=12121→1212+(7x1)=1219→121+ (7x9)=184→18+(7x4)= 46 ; The result is divisible by 23 , so the original number will be divisible by 23.

(13). Divisibility by 29 :

       Add 3 times the last digit from remaining number . Repeat the step as necessary up to 2 digit number. If result is divisible by 29,then the given number will be divisible by 29.

Example : 45327 → 4532+(3x7) =4553→455+(3x3)=464→

46+(3x4)=58 ; The result is divisible by 29 , so the original number will be divisible by 29.

(14). Divisibility by 31 :

       Substract 3 times the last digit from remaining number . Repeat the step as necessary up to 2 digit number. If result is either 0 or divisible by 31,then the given number will be divisible by 31.

Example : 473153 → 47315-(3x3) =47306→4730-(3x6)=4712→

471-(3x2)=465→46-(3x5)=31 ; The result is divisible by 31 , so the original number will be divisible by 31.

(15). Divisibility by 37 :

       Substract 11 times the last digit from remaining number . Repeat the step as necessary up to 2 digit number. If result is either 0 or divisible by 37,then the given number will be divisible by 37.

Example : 97791 → 9779-(11x1) =9768→976-(11x8)=888→

88-(11x8)=0 ; The result is 0 , so the original number will be divisible by 37.

(16). Divisibility by 41 :

       Substract 4 times the last digit from remaining number . Repeat the step as necessary up to 2 digit number. If result is either 0 or divisible by 41,then the given number will be divisible by 41.

Example : 21730 → 2173-(4x0) =2173→217-(4x3)=205→

20-(4x5)=0 ; The result is 0 , so the original number will be divisible by 41.

(17). Divisibility by 43 :

       Add 13 times the last digit from remaining number . Repeat the step as necessary up to 2 digit number. If result is divisible by 43,then the given number will be divisible by 43.

Example : 10750 → 1075+(13x0) =1075→107+(13x5)=172→

17+(13x2)=43 ; The result is divisible by 43 , so the original number will be divisible by 43.

(18). Divisibility by 47 :

       Substract 14 times the last digit from remaining number . Repeat the step as necessary up to 2 digit number. If result is either 0 or divisible by 47,then the given number will be divisible by 47.

Example : 9635 → 963-(14x5) =893→89-(14x3)=47; The result is divisible by 47 , so the original number will be divisible by 47.

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