Integers: Concepts & Order of Operations

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Skill: integers

Integers: Concepts & Order of Operations

Negative numbers are not “broken” positives — they are directions. Learn to read them on a number line, combine them with sign rules, and handle the traps that negatives set inside order of operations.

1 Understand

The core idea in plain language.

The integers are …, −3, −2, −1, 0, 1, 2, 3, …. Every integer has an opposite the same distance from zero on the other side: the opposite of 5 is −5. The absolute value |−7| = 7 is just the distance from zero — always non-negative.

Adding on a number line means walking: start at the first number, face right for a positive second number and left for a negative one. −5 + 3 starts at −5 and walks 3 right, landing on −2. Different signs: subtract the smaller absolute value and keep the sign of the “bigger” number.

Subtracting is adding the opposite: a − b = a + (−b). That turns the nastiest case into an easy one: −5 − (−3) = −5 + 3 = −2. Minus a negative is plus a positive — always.

Multiplying and dividing: same signs → positive; different signs → negative. (−4) × (−3) = 12, 20 ÷ (−4) = −5.

Order of operations with negatives: the exponent binds tighter than a leading minus. −2² = −(2²) = −4, but (−2)² = 4. Parentheses decide what the exponent “sees”.

2 See it

Diagrams that make the idea visual.

Walking the number line: −5 + 3

Start at −5. Adding +3 means walk 3 steps right. You pass −4, −3, and land on −2.

Rule of thumb for different signs: subtract the absolute values (5 − 3 = 2) and keep the sign of the number with the bigger absolute value (negative wins).

−7−5−3−11+3 rightstart −5land −2
−5 + 3: start at −5, walk 3 right, land on −2.

−2² vs (−2)²: parentheses decide

In −2², the exponent applies to the 2 only: it means −(2 × 2) = −4.

In (−2)², the parentheses hand the exponent the whole −2: (−2) × (−2) = 4.

−2² = −4exponent sees only the 2(−2)² = 4exponent sees the (−2)
The exponent applies to whatever sits directly under it.

3 The key rule

Sign rules
same signs → positive · different signs → negative

(−)(−) = +, (+)(+) = +, (−)(+) = −, (+)(−) = − — for both multiplication and division. For addition: same sign → add and keep it; different signs → subtract and keep the bigger absolute value’s sign.

4 Worked examples

Follow each step. The pattern is always the same.

Example 1 Add with different signs
  1. Different signs: subtract the absolute values: 5 − 3 = 2.
  2. Keep the sign of the number with the bigger absolute value: −5 wins.
  3. Answer: −2.
−5 + 3 = −2

Check: Number line: start at −5, walk 3 right, land on −2. Correct.

Example 2 Subtract a negative
  1. Rewrite as adding the opposite: −5 − (−3) = −5 + 3.
  2. Different signs: 5 − 3 = 2, negative wins.
  3. Answer: −2.
−5 − (−3) = −2

Check: Minus a negative is plus a positive — the double negative became addition. Correct.

Example 3 Multiply with sign rules
  1. Signs first: negative × negative = positive.
  2. Magnitudes: 4 × 3 = 12.
(−4) × (−3) = 12

Check: Same signs always give a positive product. Correct.

Example 4 Order of operations with a negative
  1. PEMDAS: multiply before adding: 4 × (−2) = −8.
  2. Now −3 + (−8): same sign, add and keep: −11.
−3 + 4 × (−2) = −11

Check: Left-to-right would give (−3 + 4) × (−2) = −2 — wrong. Multiplication first. Correct.

5 Common mistakes

These errors show up on almost every quiz. Spot them now and they will never cost you points.

1. Adding different signs as if both were negative
Wrong
−5 + 3 = −8 — “two negatives make… wait.”
Right
Different signs: subtract, keep the bigger one’s sign: −2.
Only add the magnitudes when the signs match. Different signs always subtract first.
2. Treating minus-a-negative as subtraction
Wrong
−5 − (−3) = −8.
Right
Minus a negative is plus a positive: −5 + 3 = −2.
Two minuses in a row become a plus. Rewrite a − (−b) as a + b immediately.
3. Giving −2² the wrong sign
Wrong
−2² = 4 — “negative squared is positive.”
Right
The exponent sees only the 2: −(2²) = −4. Use (−2)² for +4.
A leading minus is not inside the power unless parentheses put it there.

6 Key vocabulary

Say these words like you mean them.

Integer
A whole number, positive, negative, or zero: …, −2, −1, 0, 1, 2, ….
Opposite
The number the same distance from zero on the other side. Opposite of 5 is −5.
Absolute value
Distance from zero, written |x|. |−7| = 7, |5| = 5. Never negative.
Sign rules
Same signs → positive; different signs → negative (for × and ÷).
Additive inverse
Another name for opposite: a + (−a) = 0.
Order of operations
Parentheses, exponents, ×/÷ left to right, +/− left to right — even with negatives.

7 Quick check

Try each one on paper first, then reveal the answer.

1. Compute: −8 + 5.
Answer
Different signs: 8 − 5 = 3, negative wins: −3.
2. Compute: 6 − (−9).
Answer
6 − (−9) = 6 + 9 = 15.
3. Compute: (−2)³.
Answer
(−2) × (−2) × (−2) = −8 (odd exponent keeps the negative).
4. Compute: −(3²).
Answer
−(3²) = −9. The minus is outside the power: −9.

Key points to remember

  • |x| is distance from zero — never negative.
  • Different signs when adding: subtract and keep the bigger absolute value’s sign.
  • a − (−b) = a + b — minus a negative is plus a positive.
  • Same signs → positive product/quotient; different signs → negative.
  • −a² = −(a²) but (−a)² = a² — parentheses decide.
  • Order of operations never takes a day off: exponents and ×/÷ still come before +/− with negatives.
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