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➗ Math · Numbers and Algebra · Lesson 3 of 8 · 6 min

Order of operations, negative numbers and absolute value

🧮 Two people can work 8 − 2 × 3 and get 18 and 2. Only one of them gets the point on test day.

💡 Key idea

PEMDAS, with multiplication/division and addition/subtraction each worked left to right as pairs.

🧠 The idea

The ORDER OF OPERATIONS is PEMDAS: Parentheses (and other grouping) first, then Exponents, then Multiplication and Division from left to right, then Addition and Subtraction from left to right. Multiplication does not always beat division; whichever comes first reading left to right goes first: 12 ÷ 3 × 2 = 4 × 2 = 8. With NEGATIVE NUMBERS, adding a negative is subtracting (5 + (−8) = −3), and subtracting a negative is adding (5 − (−8) = 13). When you multiply or divide, two signs that match give a positive and two signs that differ give a negative: (−4)(−3) = 12, but (−4)(3) = −12. Watch exponents: (−3)² = 9, but −3² = −9, because the exponent applies only to the 3. ABSOLUTE VALUE, written |x|, is a number's distance from zero, so it is never negative: |−7| = 7. Treat the bars like parentheses and work inside them first.

✏️ Worked example

Evaluate 18 − 3 × (4 − 6)² ÷ 2.
Step 1: Parentheses: 4 − 6 = −2.
Step 2: Exponent: (−2)² = 4.
Step 3: Multiply and divide left to right: 3 × 4 = 12, then 12 ÷ 2 = 6.
Step 4: Subtract: 18 − 6 = 12.
Answer: 12.

📌 Takeaways

  • Parentheses, exponents, then × ÷ left to right, then + − left to right.
  • Subtracting a negative is adding; matching signs multiply to a positive.
  • Absolute value is distance from zero, never negative.

📖 Terms in this lesson

Order of operations: The rule for evaluating expressions: parentheses, exponents, multiplication and division left to right, then addition and subtraction left to right; for example, 2 + 3 × 4 = 14.

✅ Check yourself

Question 1 of 3

What is 20 − 4 × 3 + 2?

  1. 50
  2. 10
  3. 14
  4. 6
Show the answer and rationale

Answer: B

Multiply first: 4 × 3 = 12. Then work left to right: 20 − 12 = 8, and 8 + 2 = 10. The answer 50 subtracts first (16 × 3 + 2), and 6 adds 12 + 2 before subtracting, breaking the left-to-right rule.

Question 2 of 3

What is −6 − (−10)?

  1. 4
  2. 16
  3. −16
  4. −4
Show the answer and rationale

Answer: A

Subtracting a negative is adding: −6 + 10 = 4. The answer −16 treats the expression as −6 − 10, and −4 gets the size right but the sign wrong.

Question 3 of 3

What is |3 − 11| − |−2|?

  1. 6
  2. −6
  3. 10
  4. −10
Show the answer and rationale

Answer: A

Work inside the bars first: |3 − 11| = |−8| = 8 and |−2| = 2. Then 8 − 2 = 6. The answer 10 adds the absolute values, and −10 forgets that absolute value is never negative.

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