Algebra Lesson: Properties of Real Numbers

Learning Objectives

• Recognize and apply the properties of real numbers.

Vocabulary of Properties

1. Reflexive Property

• States that two things are exactly equal.
  • Example: 5 = 5 or 3 + 4 = 3 + 4.

2. Symmetric Property

• Allows switching the left and right sides of the equal sign.
  • Example: If 3 + 4 = 7, then 7 = 3 + 4.

3. Transitive Property

• If two things are equal to the same thing, they are equal to each other.
  • Example: If 3 + 4 = 7 and 7 = 2 + 5, then 3 + 4 = 2 + 5.
  • Can extend with more equalities.

4. Commutative Property

• Allows changing the order of numbers in addition and multiplication.
  • Addition: 3 + 4 = 4 + 3
  • Multiplication: 3 × 4 = 4 × 3
  • Note: Does not apply to subtraction or division.

5. Associative Property

• Allows changing the grouping of numbers in addition and multiplication.
  • Example: (2 + (3 + 4)) = ((2 + 3) + 4)
  • Used in both addition and multiplication.

Specific Properties for Addition and Multiplication

6. Additive Identity

• Any number plus 0 equals the number itself.
  • Example: x + 0 = x

7. Additive Inverse

• A number plus its opposite equals zero.
  • Example: x + (-x) = 0

8. Multiplicative Identity

• Any number times 1 equals the number itself.
  • Example: x × 1 = x

9. Multiplicative Inverse

• A number times its reciprocal equals one.
  • Example: 4 × 1/4 = 1 or 4 ÷ 4 = 1

10. Zero Property of Multiplication

• Any number times 0 equals zero.
  • Example: x × 0 = 0

Summary Chart of Properties Applied

• Example 1: 2 + 8 = 8 + ?

  • Missing value: 2
  • Property used: Commutative Property.

• Example 2: If 3 + 8 = 11 and 4 + x = 11, then 3 + 8 = 4 + x.

  • Missing value: x = 7
  • Property used: Transitive Property.

Practice Problems

1. Determine value and property:

   • Value: 7
   • Property: Reflexive Property.

2. Determine value and property:

   • Value: 2
   • Property: Associative Property.

Evaluating Expressions Using Properties

• Expression Step by Step:
  • Step 1: 4 - 3 = 1 (Simplifying)
  • Step 2: 7 × 1 = 7 (Multiplicative Identity)
  • Step 3: 5 × 1/5 = 1 (Multiplicative Inverse)
  • Step 4: 7 + 1 = 8 (Simplifying)

Final Property Applications

• Final Steps:
  • First: 8 - 8 = 0 (Additive Inverse)
  • Second: 1 × 3 = 3 (Multiplicative Identity)

Conclusion

• Questions and clarifications are welcome regarding the properties discussed.