(Last Updated On: February 5, 2020)

#### PROPERTIES OF REAL NUMBERS

**1. Closure Property**

**Addition : a + b**

**Multiplication : a · b**

**2. Commutative Property**

**Addition : a + b = b + a**

**: 4m + n2 = n2 + 4m**

**Multiplication : ab = ba**

**: (m – 3)n2 = n2(m – 3)**

**3. Associative Property of Addition**

**Addition : (a + b) + c = a + (b + c)**

**: (m + 12) + 3n2 = m + (12 + 3n2)**

**Multiplication : (ab)c = a(bc)**

**: (4m · 3n)6 = 4m(3n · 6)**

**4. Distributive Property**

**Right Distributive : a(b + c) = ab + ac**

**: n(m + 9) = mn + 9n**

**Left Distributive : (a + b)c = ac + bc**

**: (m + 9)n = mn + 9n**

**5. Additive Identity Property**

**a + 0 = a : 4m2 + 0 = 4m2**

**6. Multiplicative Identity Property**

**a · 1 = 1 · a = a : (5m2)(1) = (1)(5m2) = 5m2**

**7. Additive Inverse Property**

**a + (-a) = 0 : 7m2 + (-7m2) = 0**

**8. Multiplicative Inverse Property**

**a · 1/a = 1, a ≠ 0 : (3m2 + 4) [1 / (3m2 + 4)] = 1**

#### PROPERTIES OF EQUALITY

**1. Reflexive Property**

**a = a**

**2. Symmetric Property**

**If a = b, then b = a**

**3. Transitive Property**

**If a = b, and b = c, then a = c**

**4. Addition Property**

**If a = b, then a + c = b + c**

**5. Subtraction Property**

**If a = b, then a – c = = b – c**

**6. Substitution Property**

**If a = b, then**

__a__can be replaced by__b__in any expression involving__a__.**7. Multiplicative Property**

**If a = b, then ac = bc**

**8. Division Property**

**If a = b, then a/c = b/c, with c ≠ 0**

**9. Cancellation Property**

**If a + c = b + c, then a = b**

**If ac = bc, then a = b, provided c ≠ 0**

#### OPERATIONS WITH ZERO AND INFINITY

**1. a + 0 = a**

**2. a – 0 = a**

**3. a(0) = 0**

**4. 0/a = 0**

**5. a/∞ = 0**

**6. 0a = 0**

**7. a(∞) =**

**∞**

**8. a/0 = undefined**

**9. ∞/a = ∞**

**10. a0 = 1, a ≠ 0**

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