Operation and The Character of Integers

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1. Addition

Example : 3 + 4 = -1

The character of addition :

          a)    Commutative

a + b = b + a

          b)   Associative

a + ( b + c ) = ( a + b ) + c

          c)    Closed

Integers + integers = integers

          d)   Identity

a + 0 = a

2. Subtraction

a – b = a + (-b)

Example : 4 – 2 = 4 + (-2) = 2

The character of subtraction

          a)    Inverse / opposite by addition

          Opposite by a is –a

          b)   Closed

          Integers – integers = integers

          c)    Not commutative

          d)   Not associative

3. Multiplication

(+) x (+) = +

(+) x (-) = -

(-) x (+) = -

(-) x (-) = +

The character of multiplication

a)  Commutative

a x b = b x a

b) Associative

a x (b x c) = (a x b) x c

c)  Closed

Integers x integers = integers

d)  Identity

a x 1 = a

e)  Multiplication by zero = 0

f)  Distributive

·      For addition

a x (b+c) = (axb)+(axc)

·      For subtraction

a x (b-c) = (axb)-(axc)

4. Division

(+) : (+) = +

(+) : (-) = -

(-) : (+) = -

(-) : (-) = +

The character of division

a.   Inverse / opposite by multiplication

a : b = c

a x 1/b = c

a = b x c

b.  Identity

a : 1 = a

c.   Division by zero

0 : a = 0

a : o = ~~

5. Power

aⁿ = a x a x a…x a

until n.factors

a = basic number

n = exponent

aⁿ = rank number