2.3 Rational Number Operations & Order of Operations

2.3 Rational Number Operations & Order of Operations

Quick Summary: Rational numbers encompass integers, fractions, terminating decimals, and repeating decimals. Evaluating expressions containing multiple rational numbers requires strict application of the Order of Operations (PEMDAS/GEMDAS). Remember that Multiplication and Division (and Addition and Subtraction) share equal priority and must be solved left-to-right.

In healthcare, clinical decisions rely on calculating multi-step mathematical expressions—such as computing net fluid balance (Intake minus Output), calculating child dosages based on Body Surface Area (BSA), or interpreting body temperature variations. Mastering PEMDAS and signed number arithmetic guarantees accuracy on complex TEAS 7 problems.

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1. What Are Rational Numbers?

A rational number is any number that can be written as a quotient or ratio of two integers in the form a/b, where a is an integer and b is a non-zero integer (b ≠ 0).

Sets of Numbers Breakdown

 ┌─────────────────────────────────────────────────────────┐
 │ REAL NUMBERS                                            │
 │  ┌───────────────────────────────────────────────────┐  │
 │  │ RATIONAL NUMBERS (a/b, b ≠ 0)                     │  │
 │  │  - Integers: ...-3, -2, -1, 0, 1, 2, 3...           │  │
 │  │  - Fractions: 1/2, -3/4, 22/7                      │  │
 │  │  - Terminating Decimals: 0.25, -1.875              │  │
 │  │  - Repeating Decimals: 0.333..., 0.1818...          │  │
 │  └───────────────────────────────────────────────────┘  │
 │  ┌───────────────────────────────────────────────────┐  │
 │  │ IRRATIONAL NUMBERS (Cannot be written as a/b)     │  │
 │  │  - π (3.14159...), √2, √3                           │  │
 │  └───────────────────────────────────────────────────┘  │
 └─────────────────────────────────────────────────────────┘

• Integers: Whole numbers and their negative opposites (..., -3, -2, -1, 0, 1, 2, 3, ...). Note that any integer n can be written as n/1.
• Terminating Decimals: Decimals that end after finite digits (e.g., 0.625 = 5/8).
• Repeating Decimals: Decimals that repeat a sequence of digits infinitely (e.g., 0.444... = 4/9).
• Irrational Numbers (Contrast): Numbers with non-terminating, non-repeating decimal expansions (e.g., π ≈ 3.14159..., √2 ≈ 1.41421...). These are NOT rational numbers.

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2. Arithmetic Rules for Signed Numbers

Clinical data frequently uses positive and negative signs (e.g., net intake/output balances, temperatures above or below 0°C, weight gain vs. weight loss).

Addition Rules

• Same Signs: Add the absolute values and keep the common sign.
  Examples: (+5) + (+3) = +8; (-6) + (-4) = -10.
• Different Signs: Subtract the smaller absolute value from the larger absolute value, and keep the sign of the number with the larger absolute value.
  Examples: (-9) + (+4) = -5; (+12) + (-7) = +5.

Subtraction Rule ("Add the Opposite")

To subtract a signed number, change the subtraction sign to addition and change the sign of the number being subtracted (the subtrahend).

$a - b = a + (-b)$ $a - (-b) = a + b$

Worked Example 1 (Clinical Subtraction): A patient's core temperature rises from -1.5°C relative to baseline to +2.3°C relative to baseline. What is the total temperature change?

1. Setup change expression: New - Original = (+2.3) - (-1.5).
2. Apply "Add the Opposite": +2.3 + (+1.5) = +3.8°C.

Result: The temperature changed by +3.8°C.

Multiplication and Division Rules

• Same Signs: The result is ALWAYS Positive (+ × + = +, - × - = +).
  Examples: (-4) × (-5) = +20; (-18) ÷ (-3) = +6.
• Different Signs: The result is ALWAYS Negative (+ × - = -, - ÷ + = -).
  Examples: (-6) × (+3) = -18; (+24) ÷ (-4) = -6.

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3. Order of Operations (PEMDAS / GEMDAS)

When an arithmetic expression contains multiple operations, you must evaluate them in a standardized, hierarchical sequence using the acronym PEMDAS (or GEMDAS, where G stands for Grouping symbols).

 ┌─────────────────────────────────────────────────────────────┐
 │ P / G  │ Parentheses & Grouping Symbols [ ], ( ), { }, │   │
 │        │ Absolute Value | |, Fraction Bars           │   │
 ├────────┼─────────────────────────────────────────────┼───┤
 │   E    │ Exponents & Roots (x², √x)                  │   │
 ├────────┼─────────────────────────────────────────────┼───┤
 │ M / D  │ Multiplication & Division                   │   │
 │        │ (EQUAL PRIORITY — Solve Left to Right!)     │ ◄─┼─ CRITICAL!
 ├────────┼─────────────────────────────────────────────┼───┤
 │ A / S  │ Addition & Subtraction                      │   │
 │        │ (EQUAL PRIORITY — Solve Left to Right!)     │ ◄─┼─ CRITICAL!
 └────────┴─────────────────────────────────────────────┴───┘

CRITICAL TEAS WARNING: Multiplication does NOT take precedence over Division, nor does Addition take precedence over Subtraction! You must evaluate Multiplication and Division strictly from left to right as they appear in the expression.

Worked Example 2 (Left-to-Right Priority Trap): Evaluate 24 ÷ 4 × 2.

• Incorrect (Doing Multiplication first): 24 ÷ (4 × 2) = 24 ÷ 8 = 3. (WRONG!)
• Correct (Left to Right): (24 ÷ 4) × 2 = 6 × 2 = 12. (CORRECT!)

Comprehensive Step-by-Step Expression Evaluation

Worked Example 3: Evaluate the expression: 18 - 2 × (4 + 3)² ÷ 7.

1. Parentheses (P): Solve inside parentheses (4 + 3) = 7. Expression becomes: 18 - 2 × 7² ÷ 7
2. Exponents (E): Evaluate 7² = 49. Expression becomes: 18 - 2 × 49 ÷ 7
3. Multiplication & Division (M/D left to right):
   • First operation left-to-right is multiplication: 2 × 49 = 98.
   • Next operation left-to-right is division: 98 ÷ 7 = 14. Expression becomes: 18 - 14
4. Addition & Subtraction (A/S left to right):
   • 18 - 14 = 4.

Final Answer: 4.

Worked Example 4 (Nested Brackets & Exponents): Evaluate [(120 - 40) ÷ 4] + 3 × (5 - 2)².

1. Innermost Groupings:
   • (120 - 40) = 80
   • (5 - 2) = 3 Expression: [80 ÷ 4] + 3 × 3²
2. Evaluate Bracket & Exponents:
   • [80 ÷ 4] = 20
   • 3² = 9 Expression: 20 + 3 × 9
3. Multiply: 3 × 9 = 27. Expression: 20 + 27
4. Add: 20 + 27 = 47.

Final Answer: 47.

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4. TEAS Calculator Strategy for PEMDAS

Because the TEAS 4-function calculator lacks parentheses keys ( ) and exponent keys x², you cannot type a long PEMDAS expression in all at once.

Calculator Protocol for Multi-Step Expressions:

1. Work out parentheses and exponents manually on scratch paper first.
2. Write down intermediate values on your provided scratch paper.
3. Execute the final multiplication, division, addition, and subtraction steps on the keypad in strict left-to-right order.

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5. Healthcare Application: Intake & Output (I&O) Net Fluid Balance

In nursing, calculating a patient's Net Fluid Balance over a shift requires adding all fluid intake (positive rational numbers) and subtracting all fluid output (negative rational numbers).

$\text{Net Fluid Balance} = \text{Total Intake} - \text{Total Output}$

Clinical Scenario: During a 12-hour shift, a patient's records show:

• Intake: Oral fluids 850 mL, IV Normal Saline 1,200 mL, IV Antibiotics 250 mL.
• Output: Urine output 1,450 mL, Jackson-Pratt drain 180 mL, Emesis 120 mL.

What is the patient's net fluid balance in mL?

Calculation:

1. Calculate Total Intake (Positive): Intake = 850 + 1200 + 250 = 2,300 mL
2. Calculate Total Output (Negative): Output = 1450 + 180 + 120 = 1,750 mL
3. Calculate Net Balance: Net Balance = (+2,300 mL) - (1,750 mL) = +550 mL

Interpretation: The patient has a positive fluid balance of +550 mL (retaining fluid).