An integeris just a whole number that can be positive, negative, or zero — −3, 0, 7, and so on. The arithmetic students already know from elementary school still works, but now there’s a direction to keep track of. The good news: every rule below comes from the same picture, a number line, so you never have to memorize them as disconnected facts.
Picture the number line first
Draw a line with 0 in the middle, positive numbers to the right, negative numbers to the left. Adding a positive number moves you right. Adding a negative number — or equivalently, subtractinga positive one — moves you left. That single picture explains addition and subtraction of signed numbers completely. You’re not learning new math; you’re just tracking which way you’re walking.
Adding and subtracting
Take −5 + 3. Start at −5 on the line and move 3 steps to the right: you land on −2. Now take −5 − 3. Subtracting 3 means moving 3 steps to the left: you land on −8. The trickiest one is subtracting a negative, like 4 − (−6). Subtracting a negative is the same as adding its positive — think of it as “taking away a debt,” which leaves you better off. So 4 − (−6) = 4 + 6 = 10.
Multiplying and dividing — the sign rule
Multiplication and division follow a short, reliable rule: same signs give a positive answer, different signs give a negative answer.So (−4) × (−3) = 12, but (−4) × 3 = −12. The same rule applies to division: (−12) ÷ (−3) = 4, and (−12) ÷ 3 = −4.
Here’s the intuition behind it, not just the rule: multiplying by a negative number flips direction. Multiply a positive by a negative once, and you flip once — the result is negative. Multiply two negatives, and you flip twice — two flips bring you back to positive. That’s why “a negative times a negative is a positive” isn’t an arbitrary convention; it’s what happens when you flip direction twice.
The mistake I see most
Students almost always handle −5 + 3 fine but stumble on −5 − (−3)— the double negative. The fix is always the same: rewrite “subtracting a negative” as “adding a positive” before doing anything else. −5 − (−3) becomes −5 + 3, which is −2. Doing that rewrite as a separate first step, before any arithmetic, eliminates almost all sign errors.
Why this matters more than it looks like it should
Negative numbers show up constantly once algebra starts — solving equations, distributing a negative across parentheses, graphing below the x-axis. A shaky grip on signs here doesn’t just cost points in pre-algebra; it quietly costs points in every course afterward, because the sign mistakes blend in with “algebra mistakes” and get harder to isolate. Fixing it now is some of the highest-leverage tutoring time there is.
Want to work through it together?
If sign mistakes keep creeping into your work, that’s an extremely common and a completely fixable gap. We’ll find exactly where the number line picture breaks down for you and rebuild it, so the rules stop feeling like a list to memorize.
