Evaluating Expressions and Nonnegative Rational Numbers

Evaluating Expressions

• Definition: The process of calculating the value of an expression by substituting variables with numbers.

• Steps to Evaluate:

  1. Substitution: Replace each variable in the expression with the given value.
  2. Order of Operations: Follow PEMDAS/BODMAS rules:
     • Parentheses/Brackets
     • Exponents/Orders
     • Multiplication and Division (from left to right)
     • Addition and Subtraction (from left to right)
  3. Simplification: Perform the operations to simplify the expression to a single numerical value.

• Example: Evaluate (3x + 5) when (x = 2).

  • Substitute: (3(2) + 5)
  • Calculate: (6 + 5 = 11)

Nonnegative Rational Numbers

• Definition: Numbers that can be expressed as a fraction (\frac{a}{b}) where:

  • (a) and (b) are integers
  • (b \neq 0)
  • (a \geq 0) (ensures nonnegativity)

• Characteristics:

  • Includes whole numbers and positive fractions.
  • Examples: (0, \frac{1}{2}, 3, 4.75)

• Properties:

  • Closure: Nonnegative rational numbers are closed under addition and multiplication.
  • Ordering: Nonnegative rational numbers can be ordered on a number line where (0) is the smallest.
  • Density: Between any two nonnegative rational numbers, there exists another nonnegative rational number.

• Operations:

  • Addition: (\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd})
  • Multiplication: (\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd})

• Applications:

  • Used in various fields such as finance, engineering, and everyday calculations involving ratios and proportions.

Evaluating Expressions

• Expression Evaluation: Involves determining the value of an expression by substituting variables with specific numerical values.
• Evaluation Steps:
  • Substitution: Each variable in the expression is replaced by its respective value.
  • Order of Operations: Follow PEMDAS/BODMAS rules to maintain correct calculation order:
    • Parentheses/Brackets first
    • Then Exponents/Orders
    • Followed by Multiplication and Division (left to right)
    • Finally, Addition and Subtraction (left to right)
  • Simplification: Execute the operations from above to condense the expression to a final numerical value.
• Example Illustration: For the expression (3x + 5) with (x = 2):
  • Substitute to get (3(2) + 5)
  • Calculate to achieve (11)

Nonnegative Rational Numbers

• Definition: Nonnegative rational numbers are numbers that can be represented as the fraction (\frac{a}{b}), where:
  • (a) (numerator) and (b) (denominator) are integers
  • (b) cannot be (0)
  • (a) must be greater than or equal to (0) to ensure the number is nonnegative.
• Characteristics:
  • Consist of whole numbers and positive fractions.
  • Examples include (0, \frac{1}{2}, 3,) and (4.75).
• Properties:
  • Closure Property: Nonnegative rational numbers maintain closure under both addition and multiplication; combining these numbers keeps the result within this set.
  • Ordering: They can be arranged on a number line, with (0) being the least value.
  • Density: For any two nonnegative rational numbers, another nonnegative rational number can always be found in between them.
  • Operations:
    • Addition: When adding two nonnegative rational numbers, (\frac{a}{b} + \frac{c}{d}) results in (\frac{ad + bc}{bd}).
    • Multiplication: When multiplying two nonnegative rational numbers, (\frac{a}{b} \times \frac{c}{d}) results in (\frac{ac}{bd}).
• Applications: Nonnegative rational numbers are essential in finance, engineering, and daily calculations involving ratios and proportions.