Adding and Subtracting Negative Numbers, With a Number Line

To add a negative number, move left on the number line; to subtract a negative number, add its opposite instead. The Common Core State Standards (7.NS.A.1) state both ideas: p + q is the number located a distance |q| from p, in the positive or negative direction depending on the sign of q, and subtracting is the same as adding the additive inverse, p − q = p + (−q).

1. Start with the picture

Picture a horizontal number line with 0 in the middle, positives to the right and negatives to the left. The standards (6.NS.C.6.a) note that opposite signs sit on opposite sides of 0, and that the opposite of the opposite of a number is the number itself: −(−3) = 3.

Adding moves you in a direction:

• Adding a positive number moves right.
• Adding a negative number moves left.

2. Adding numbers with the same sign

When both numbers are negative, add their sizes and keep the negative sign.

Example 1. −3 + (−4). Start at −3 and move 4 left. You land at −7.

Example 2. −9 + (−4) = −13. Sizes: 9 + 4 = 13. Sign: negative.

3. Adding numbers with different signs

Subtract the smaller size from the larger size, and use the sign of the number with the larger size.

Example 3. −3 + 5. Start at −3 and move 5 right. You land at 2. Check by the rule: sizes 3 and 5, so 5 − 3 = 2, and 5 is bigger and positive, so the answer is +2.

Example 4. 4 + (−6). Sizes 4 and 6, so 6 − 4 = 2. The 6 is bigger and negative, so the answer is −2.

Example 5. −9 + 4 = −5. And 9 + (−4) = 5.

Opposites cancel. −8 + 8 = 0. The standards (7.NS.A.1.a, b) call this the additive inverse: a number and its opposite add to 0.

4. Subtracting: add the opposite

The standards say that p − q = p + (−q). So change the subtraction to addition and flip the sign of the second number, then use the addition rules above.

Example 6. 5 − 8 = 5 + (−8) = −3.

Example 7. −2 − 5 = −2 + (−5) = −7.

Example 8. 3 − (−4) = 3 + 4 = 7.

Example 9. −6 − (−2) = −6 + 2 = −4.

5. Distance between two numbers

The standards also say the distance between two numbers on the number line is the absolute value of their difference. For −7 and 5: |5 − (−7)| = |12| = 12. If you subtract in the other order you get −12, and the absolute value is 12 again.

6. Word problems

Temperature. The standards give −3 °C as warmer than −7 °C, which matches −3 > −7. How much warmer is 4 °C than −3 °C? 4 − (−3) = 4 + 3 = 7 degrees.

Account balance. The standards use −30 dollars as a debt of 30 dollars, with |−30| = 30. If you owe $30 and deposit $50: −30 + 50 = 20. You now have $20. If you instead spend another $15: −30 − 15 = −45, a debt of $45.

7. A longer expression

Example 10. −5 + 8 − (−3) − 10. Work left to right.

1. −5 + 8 = 3
2. 3 − (−3) = 3 + 3 = 6
3. 6 − 10 = 6 + (−10) = −4

Answer: −4.

Example 11, check by grouping. Rewrite as all additions: −5 + 8 + 3 + (−10). Positives: 8 + 3 = 11. Negatives: −5 + (−10) = −15. Total: 11 + (−15) = −4. It matches Example 10.

8. Common mistakes

1. "Two negatives make a positive" applied to addition. That idea only applies when a minus sign sits in front of a negative number (subtracting a negative), or when multiplying. −3 + (−4) is −7, not 7.
2. Dropping the sign. In −2 − 5, the 5 is subtracted. Treat it as −2 + (−5).
3. Reading 3 − (−4) as 3 − 4. The second sign changes it: the answer is 7, not −1.
4. Wrong sign on the answer. For different signs, the sign belongs to the number with the larger size, not the first number.
5. Treating −7 as larger than −3. On a number line, −3 is to the right of −7, so −3 > −7.

9. Key takeaways

• Adding a negative moves left on the number line; adding a positive moves right.
• Same signs: add sizes, keep the sign. Different signs: subtract sizes, use the sign of the larger.
• Subtracting means adding the opposite: 3 − (−4) = 7.
• Opposites add to 0, and distance equals |difference|.
• Check a long expression by regrouping positives and negatives.