Signed Numbers (Integers & Absolute Value)

Why This Matters for Nursing: Temperature changes (dropped 3 degrees), fluid balance (negative 500 mL), and lab value changes all use positive and negative numbers. Understanding how they work prevents critical calculation errors.

What You Need to Know

Integers are whole numbers that can be positive, negative, or zero:

• Positive: +1, +2, +3... (or just 1, 2, 3)
• Negative: -1, -2, -3...
• Zero: 0 (neither positive nor negative)

Absolute value |x| is the distance from zero—always positive:

• |5| = 5
• |-5| = 5
• |0| = 0

🧠 Memory Trick

For adding signed numbers:

• Same signs → ADD and keep the sign
• Different signs → SUBTRACT and keep the sign of the larger absolute value

For subtracting: "Add the opposite" a - b = a + (-b)

"Two negatives make a positive" (when multiplying or dividing)

Rules for Operations

Adding Signed Numbers

Subtracting Signed Numbers

Add the opposite!

• 5 - 8 = 5 + (-8) = -3
• -3 - 4 = -3 + (-4) = -7
• -6 - (-2) = -6 + 2 = -4

Multiplying and Dividing

Same signs → Positive result Different signs → Negative result

✏️ Worked Examples

Example 1: Adding Same Signs

Problem: -15 + (-7)

Step 1 — Check the signs. Both numbers are negative. Same signs → add the numbers and keep the negative sign.

Step 2 — Add the absolute values (ignore the signs for now). 15 + 7 = 22.

Step 3 — Apply the sign. Both were negative, so the answer is negative: -22.

Answer: -22

💡 Think of it like debt. If you owe $15 AND you owe another $7, you owe $22 total. Same idea.

Example 2: Adding Different Signs

Problem: -12 + 8

Step 1 — Check the signs. One negative (-12), one positive (+8). Different signs → subtract and keep the sign of whichever number is farther from zero (bigger absolute value).

Step 2 — Find the absolute values (distance from zero, ignoring sign): |-12| = 12, |8| = 8.

Step 3 — Subtract the smaller from the larger. 12 - 8 = 4.

Step 4 — Apply the sign of the bigger absolute value. 12 > 8, and 12 was negative, so the answer is negative: -4.

Answer: -4

Example 3: Subtracting — Add the Opposite

Problem: 6 - (-4)

Step 1 — Convert subtraction to adding the opposite. Subtracting a negative is the same as adding a positive. Think: "minus a negative = plus a positive." 6 - (-4) becomes 6 + 4.

Step 2 — Add. 6 + 4 = 10.

Answer: 10

💡 "Two negatives make a positive" in subtraction: the minus sign and the negative sign cancel each other out, just like in language ("I'm not unhappy" = I'm happy).

Example 4: Multiplying Signed Numbers

Problem: (-8) × (-3)

Step 1 — Multiply the numbers, ignoring signs for now. 8 × 3 = 24.

Step 2 — Determine the sign. Same signs (both negative) → positive result.

Answer: 24 (positive!)

💡 Sign rule for multiplication: negative × negative = positive. Think: flipping direction twice brings you back to where you started.

Example 5: Nursing Context

Problem: A patient's temperature was 99.8°F. It dropped 2.4 degrees, then rose 1.1 degrees. What's the final temperature?

What we're solving: Start at 99.8, apply two changes.

Step 1 — Apply the drop. A drop means subtract (or add a negative): 99.8 + (-2.4) = 97.4°F.

Step 2 — Apply the rise. A rise means add: 97.4 + 1.1 = 98.5°F.

Answer: 98.5°F

Example 6: Absolute Value

Problem: |−8| + |3| − |−2|

Step 1 — Evaluate each absolute value. Absolute value just means "how far from zero?" — it's always positive.

• |−8| = 8 (the negative sign is stripped away)
• |3| = 3 (already positive)
• |−2| = 2

Step 2 — Now do the arithmetic. 8 + 3 - 2 = 9.

Answer: 9

💡 Pro Tips

• Think of a number line: Right is positive, left is negative
• Subtracting a negative = adding: 5 - (-3) = 5 + 3 = 8 (the two negatives "cancel")
• Two negatives multiplied = positive: Think "a double negative makes a positive" like in language
• Absolute value strips the sign: Just take the number without any negative

⚠️ Common Mistakes to Avoid

• Forgetting the sign: -3 + (-5) = -8, not 8
• Confusing subtraction and negative signs: -3 - 4 is NOT -3 + 4
• Wrong sign when multiplying/dividing: (-2) × 5 = -10, not 10
• Absolute value confusion: |-5| = 5, not -5

Quick Reference

Subtract = Add the Opposite

• a - b = a + (-b)
• a - (-b) = a + b

Absolute Value

• Always positive or zero
• |x| = x if x ≥ 0
• |x| = -x if x < 0

Signed numbers conquered! 💪 Next up: Algebraic Expressions & Simplifying