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Identity and Equality Properties
Addition Properties
Additive Identity – For any number a, the sum of a and 0 is a.
EX: a + 0 = a
6+0=6
Multiplication Properties
Multiplicative Identity – For any number a, the product of a and 1 is a.
EX: a * 1 = a
3*1=3
Multiplicative Property of Zero – For any number a, the product of a and 0 is
0.
EX: a * 0 = 0
5*0=0
Multiplicative Inverse – For every number a / b, where a, b ≠ 0, there is
exactly one number b / a such that the product of a / b and b / a is 1.
EX: a / b * b / a = 1
3 /2 * 2 / 3 = 1
Properties of Equality
Reflexive – Any quantity is equal to itself.
EX: a = a
7=7
Symmetric – If one quantity equals a second quantity, then the second
quantity equals the first.
EX: If a = b, then b = a.
If 9 = 6 + 3, then 6 + 3 = 9.
Transitive – If one quantity equals a second quantity and the second quantity
equals a third quantity, then the first quantity equals the third quantity.
EX: If a = b, and b = c, then a = c.
If 5 + 7 = 8 + 4, and 8 + 4 = 12, then 5 + 7 = 12.
Substitution – A quantity may be substituted for its equal in any expression.
EX: If a = b, then a may be replaced by b in any expression.
If n = 15, then 3n = 3 * 15.
Distributive Property
Distributive Property- For any numbers a, b, and c, a( b + c) = ab + ac
and (b + c)a = ba + ca and a( b – c) = ab – ac
and (b – c)a = ba – ca.
EX: 3( 2 + 5) = 3(2) + 3(5)
3(7) = 6 + 15
21 = 21
Commutative and Associative
Properties
Commutative Property- the order in which you add or multiply numbers
does not change their sum or product.
EX: a + b = b + a and a(b) = b(a).
5 + 6 = 6 + 5, 3(2) = 2(3)
Associative Property- the way you group three or more numbers when
adding or multiplying does not change their sum or product.
EX: (a + b) + c = a + (b + c) and (ab)c = a(bc).
(2 + 4) + 6 = 2 + (4 + 6), (3 x 5) x 4 = 3 x(5 x 4)