Integers & the Number Line
Integers are whole numbers that can be positive, negative, or zero. Understanding them unlocks algebra, temperature, debt, and elevation problems.
What Are Integers?
The set of integers is: … −3, −2, −1, 0, 1, 2, 3 … Positive integers are to the right of zero on the number line; negative integers are to the left.
Key rules:
• Absolute value |n| = distance from zero (always ≥ 0)
• Opposite of n = −n
Adding & Subtracting Integers
Same signs → add and keep the sign.
Different signs → subtract the smaller absolute value and keep the sign of the larger.
Examples
−5 + (−3) = −8 (both negative, add)
−8 + 3 = −5 (subtract 3 from 8, keep negative)
6 − (−2) = 6 + 2 = 8 (subtracting a negative = adding)
Multiplying & Dividing Integers
Simple sign rules:
- Positive × Positive = Positive
- Negative × Negative = Positive
- Positive × Negative = Negative
Same rules apply to division.
(−4) × (−3) = 12 | (−12) ÷ 3 = −4
Real-World Uses
Integers appear everywhere: temperature below zero (−10°C), sea level elevation, bank balances, football yardage. Any situation with opposites needs integers.
Picturing It on the Number Line
If a sign rule ever slips your mind, the number line will rebuild it for you. Adding a positive moves right; adding a negative moves left.
Why −8 + 3 = −5
Start at −8. Adding 3 moves you three steps to the right, landing on −5. You moved toward zero but did not reach it, which is why the answer stayed negative.
Why 6 − (−2) = 8
Subtracting means moving left, but the number itself is negative, so the two reversals cancel and you move right instead. Taking away a debt makes you richer.
Comparing and Ordering Integers
On a number line, anything further right is larger. That single rule handles every comparison, and it is where negatives trip people up.
−2 > −7, because −2 sits to the right. A smaller-looking negative is actually the bigger number.- Every negative number is less than every positive number.
- Zero is greater than all negatives and less than all positives.
Order of Operations With Negatives
Evaluate −3 + 4 × (−2)
- Multiplication comes first:
4 × (−2) = −8 - Then add:
−3 + (−8) = −11
Doing the addition first would give −2, which is wrong. Order matters more once negatives appear.
Common Mistakes
- Assuming a negative answer is always smaller in size.
−20has a larger absolute value than 5, even though it is the lesser number. - Losing a minus sign when copying a problem down. Rewrite carefully — this causes more lost marks than any concept.
- Applying the "two negatives make a positive" rule to addition. It only holds for multiplication and division.
- Treating the absolute value bars as if they cancel a negative in front.
−|−4|is−4.
Summary
- Integers are whole numbers, positive, negative, or zero — no fractions or decimals.
- Same signs add and keep the sign; different signs subtract and take the sign of the larger absolute value.
- Subtracting a negative is the same as adding a positive.
- In multiplication and division, matching signs give a positive and mixed signs give a negative.
- Absolute value is distance from zero, so it is never negative.
FAQ
Is zero positive or negative? Neither. Zero is its own category.
What is the absolute value of −7? 7 — it is simply the distance from zero.
Are all whole numbers integers? Yes, and integers also include the negatives. Every whole number is an integer, but −4 is an integer that is not a whole number.
Is 3.5 an integer? No. Integers have no fractional or decimal part.
Why do two negatives multiply to a positive? Multiplying by a negative reverses direction on the number line. Reversing twice puts you back where you started, facing positive.
Can the absolute value of a number be negative? Never. It measures distance from zero, and distance cannot be negative.
Quick Quiz
Test what you just learned. Choose the best answer for each question.