Algebra and General Mathematics: Basic Rules of Algebra

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. 0^a = 0

7. a(∞) = ∞

8. a/0 = undefined

9. ∞/a = ∞

10. a⁰ = 1, a ≠ 0

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