Appendix D: Properties of Operations, Equality and Inequality
FLORI DA’ S B. E. S. T. STANDARDS : MATHEMATI CS
Properties of Operations
Properties of Equality
The table below illustrates the properties of operations. For each property, the variables 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐 stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number system, the real number system and the complex number system. Property of Operation Associative property of addition Commutative property of addition Additive identity property of zero Existence of additive inverses Associative property of multiplication Commutative property of multiplication Multiplicative identity property of one Existence of multiplicative inverses Distributive property of multiplication over addition
Property of Equality
Example
Reflexive property of equality
(𝑎𝑎 + 𝑏𝑏) + 𝑐𝑐 = 𝑎𝑎 + (𝑏𝑏 + 𝑐𝑐)
Symmetric property of equality
𝑎𝑎 + 𝑏𝑏 = 𝑏𝑏 + 𝑎𝑎
Transitive property of equality
𝑎𝑎 + 0 = 𝑎𝑎 0 + 𝑎𝑎 = 𝑎𝑎
For every 𝑎𝑎 there exists −𝑎𝑎 so that 𝑎𝑎 + (−𝑎𝑎) = 0 and (−𝑎𝑎) + 𝑎𝑎 = 0. (𝑎𝑎 × 𝑏𝑏) × 𝑐𝑐 = 𝑎𝑎 × (𝑏𝑏 × 𝑐𝑐)
𝑎𝑎 × " = 1 and
Subtraction property of equality
Division property of equality
For every 𝑎𝑎 ≠ 0 there exists !
Addition property of equality
Multiplication property of equality
𝑎𝑎 × 𝑏𝑏 = 𝑏𝑏 × 𝑎𝑎 𝑎𝑎 × 1 = 𝑎𝑎 1 × 𝑎𝑎 = 𝑎𝑎
The table below illustrates the properties of equality. For each property, the variables 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐 stand for arbitrary numbers in a given number system. The properties of equality apply to the rational number system, the real number system and the complex number system.
!
"
× 𝑎𝑎 = 1.
𝑎𝑎 × (𝑏𝑏 + 𝑐𝑐) = (𝑎𝑎 × 𝑏𝑏) + (𝑎𝑎 × 𝑐𝑐)
!
"
so that
Substitution property of equality
Example 𝑎𝑎 = 𝑎𝑎
If 𝑎𝑎 = 𝑏𝑏, then 𝑏𝑏 = 𝑎𝑎.
If 𝑎𝑎 = 𝑏𝑏 and 𝑏𝑏 = 𝑐𝑐, then 𝑎𝑎 = 𝑐𝑐. If 𝑎𝑎 = 𝑏𝑏, then 𝑎𝑎 + 𝑐𝑐 = 𝑏𝑏 + 𝑐𝑐. If 𝑎𝑎 = 𝑏𝑏, then 𝑎𝑎 − 𝑐𝑐 = 𝑏𝑏 − 𝑐𝑐. If 𝑎𝑎 = 𝑏𝑏, then 𝑎𝑎 × 𝑐𝑐 = 𝑏𝑏 × 𝑐𝑐.
If 𝑎𝑎 = 𝑏𝑏 and 𝑐𝑐 ≠ 0, then 𝑎𝑎 ÷ 𝑐𝑐 = 𝑏𝑏 ÷ 𝑐𝑐.
If 𝑎𝑎 = 𝑏𝑏, then 𝑏𝑏 may be substituted for 𝑎𝑎 in any expression containing 𝑎𝑎.
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