# Algebra and General Mathematics: Basic Rules of Algebra

(Last Updated On: February 5, 2020) #### PROPERTIES OF REAL NUMBERS

1. Closure Property

Multiplication          :      a · b

2. Commutative Property

Addition          :      a + b = b + a
:      4m + n2 = n2 + 4m
Multiplication          :      ab = ba
:      (m – 3)n2 = n2(m – 3)

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

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

6. Multiplicative Identity Property

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

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

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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