NATURAL NUMBER and PROPERTIES

 The  number  which starts from   1   is called the natural number 

  vane diagram of the natural number  below

                    Number line  of  natural number

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1    2   3   4  5   6    7   8    9    10    11   12   13    14     15      16      17        and so on

Properties of natural number 1 closure property 2 commutative property 3 Associative property we explain these properties under ( Addition, subtraction, multiplication, and division ) with a binary operation

Addition

1;  closure   property

x   and   y   are  any  two number   adding  the  result  is the natural number  called closure properties

Example

 x   +   y    =    z

 5   +    6  =     11             11    is   natural number

2:   Commutative property 

x   and   y   any two  natural   are   commutative   under   addition    then the property    is called  commutative  property   closed  under addition

Example

x  +  y  =  y  +   x

2  +7   =   7  +2

 9  =  9

3: Associative property (+) x1, y1, and z1 any three natural numbers are associative then the property is called associative under closed addition EXAMPLE x1 +( y1 + z1 ) = ( x1 + y1) + z1 9 + (10 +11 ) = ( 9 + 10 ) + 11 9 + 21 = 19 + 11 30 = 30

SUBTRACTION:

1: CLOSURE PROPERTY(-) x and y any two natural number is not closed under subtraction in natural number ( so closure property is not held under subtraction ) EXAMPLE X - Y = Z ( z is number) 4 - 14 = -10 - 10 ∉ ℕ

2: commutative property (-)

two natural number x and y does not hold commutative under subtraction

EXAMPLE if x =6 , y = 8 x - y ≠ y - x 6 - 8 ≠ 8 -6 -2 ≠ 2

3: Associative property(-) x1, y1, and z1 are three natural numbers is not holding associative property under subtraction EXAMPLE: x1 - ( y1 - z1) ≠ ( x1 - y1) - z1 any natural number x =9 , y = 10 , z = 11 9 - ( 10 - 11 ) ≠ (9 - 10 ) - 11 9 - ( -1 ) ≠ -1 -11 9 +1 ≠ -12 10 ≠ -12 not hold N ( proof)

MULTIPLICATION:

1; closure property(*) any two natural numbers x and y multiplied with each other is called closed under multiplication EXAMPLE X x Y = Z 3 x 8 = 24

2: commutative property (*) x and y are multiplied of natural number in commutative is called commutative called closed under multiplication EXAMPLE ; X x Y = Y x X 3 x 8 = 8 x 3 24 = 24

3: associative property(*) any three natural number x, y, and z are multiplied by associative then property is called associatively closed under multiplication EXAMPLE : X x ( Y x Z ) = ( X x Y ) x Y 7 x ( 8 x 9 ) = ( 7 x 8 ) x 9 7 x 72 = 56 x 9 504 = 504

DIVISION:

1; closure property( ÷)

any two natural numbers x and y divided the resultant value do not lie in a natural number called not closed under division

X ÷ Y = Z clearly Z ∉ N 2 ÷ 5 = 2.5 ( someone results not hold property not satisfy)

2: commutative property(÷)

x1 and y1 any two number their division result is not same we say commutative is not closed under division in the natural number EXAMPLE X1 ÷ Y1 ≠ Y1 ÷ X1 5÷ 10 ≠ 10 ÷ 5 .5 ≠ 2 (not hold under division)

3 Associative property(÷) X, Y, and z are any three natural number is not associative under division then the property is called not associative under division EXAMPLE : ( any no. x = 2 y = 4 z = 8 ) X ÷ ( Y ÷ Z ) ≠ ( X ÷ Y ) ÷ Z 2 ÷ ( 4 ÷ 8 ) ≠ ( 2 ÷ 4 ) ÷ 8 2 ÷ .5 ≠ .5 ÷ 8 4 ≠ 1/16 Hence equality does not hold under division associative property of a natural number

 Which  property  can apply for   👉👉  REAL NUMBER   and the final number is 

  👉👉  " complex  number "

 👉👉 PROPERTY OF COMPLEX NUMBER