An operation is a rule , or set of rules, for processing one or more object.

An operator is the symbol used to show which operation is to be done.

examples : + - x ÷

A binary operation is one which combines two objects to make a third.

Addition, Subtraction, Multiplication and Division are all examples.

A commutative operation is one in which the order does not matter.

Addition is commutative since 3 + 2 = 2+ 3

Subtraction is not, since 3-2 ≠ 2-3

Multiplication is commutative since 3 x 2 = 2 x 3

Division is not, since 3÷2 ≠ 2÷3

A binary operation is associative if order and brackets make no difference.

Addition is commutative since 3 + 2 +5 = 3 + ( 2 + 5)

Subtraction is not, since 5-3-2 ≠ 5-( 3 - 2)

Multiplication is commutative since 3 x 2 x 5 = 2 x (3 x 5)

Division is not, since 12÷ 3÷2 ≠12÷(3÷2)

The identity for a binary operation on a set is an object in the set

which is combined with an operator on any second object to produce

a result equal to that second object.

e.g. a + 0 = a = 0 + a

so 0 is the identity for addition.

Addition

Commutative Law a + b = b + a

Associative Law (a + b) + c = a +( b + c)

Identity elements a + 0 = a = 0 + a

Multiplication

Commutative Law a x b = b x a

Associative Law (a x b) x c = a x( b x c)

Identity elements a x 1 = a = 1 x a

Subtraction

Commutative Law a - b ≠ b - a

Associative Law (a - b) - c ≠ a - ( b - c)

Identity elements a - 0 = a ≠ 0 - a

Division

Commutative Law a ÷ b ≠ b ÷ a

Associative Law (a ÷ b) ÷ c ≠ a ÷ ( b ÷ c)

Identity elements a ÷ 1 = a ≠ 1 ÷ a

Bases

	b^6	b^5	b^4	b^3	b^2	b	Units
Binary b = 2	64	32	16	8	4	2	1
Ternary b = 3	729	243	81	27	9	3	1
b = 4	4096	1024	256	64	16	4	1
b = 5	15625	3125	625	125	25	5	1
b = 6	46656	7776	1296	216	36	6	1
b = 7	117649	16807	2401	343	49	7	1
Octal b = 8	262144	32768	4096	512	64	8	1
b = 9	531441	59049	6561	729	81	9	1
Decimal b =10	1000000	100000	10000	1000	100	10	1
b =11	1771561	161051	14641	1331	121	11	1
Duodecimal b =12	2985984	248832	20736	1728	144	12	1
b =13	4826809	371293	28561	2197	169	13	1
b =14	7529536	537824	38416	2744	196	14	1
b =15	11390625	759375	50625	3375	225	15	1
Hexadecimal b =16	16777216	1048576	65536	4096	256	16	1
b =17	24137569	1419857	83521	4913	289	17	1
b =18	34012224	1889568	104976	5832	324	18	1
b =19	47045881	2476099	130321	6859	361	19	1
b =20	64000000	3200000	160000	8000	400	20	1

	b^3	b^2	b	Units	b^-1	b^-2	b^-3
Binary b = 2	8	4	2	1	1 2	1 4	1 8
Ternary b = 3	27	9	3	1	1 3	1 9	1 27
b = 4	64	16	4	1	1 4	1 16	1 64
b = 5	125	25	5	1	1 5	1 25	1 125
b = 6	216	36	6	1	1 6	1 36	1 216
b = 7	343	49	7	1	1 7	1 49	1 343
Octal b = 8	512	64	8	1	1 8	1 64	1 512
b = 9	729	81	9	1	1 9	1 81	1 729
Decimal b =10	1000	100	10	1	1 10	1 100	1 1000
b =11	1331	121	11	1	1 11	1 121	1 1331
Duodecimal b =12	1728	144	12	1	1 12	1 144