Algebra and General Mathematics: Basic Rules of Algebra

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(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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