Understanding Properties of Real Numbers

Sep 3, 2024

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 ร— 4 = 4 ร— 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 ร— 1 = x

9. Multiplicative Inverse

  • A number times its reciprocal equals one.
    • Example: 4 ร— 1/4 = 1 or 4 รท 4 = 1

10. Zero Property of Multiplication

  • Any number times 0 equals zero.
    • Example: x ร— 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 ร— 1 = 7 (Multiplicative Identity)
    • Step 3: 5 ร— 1/5 = 1 (Multiplicative Inverse)
    • Step 4: 7 + 1 = 8 (Simplifying)

Final Property Applications

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

Conclusion

  • Questions and clarifications are welcome regarding the properties discussed.