Why do some infinite series add up to a finite number?

This is a classic question that trips up a lot of people, but the explanation is beautiful once you see it. **The intuitive approach:** Think of multiplication as repeated addition. 3 × 4 means adding 3 four times: 3 + 3 + 3 + 3 = 12. Now what does (-3) × (-4) mean? Adding -3 negative-four times? That doesn't make intuitive sense, which is why this is confusing. The better way to think about it is using the **distributive property**. We know that (-3) × (4 + (-4)) = (-3) × 0 = 0. By distribution: (-3) × 4 + (-3) × (-4) = 0. We know (-3) × 4 = -12, so -12 + (-3) × (-4) = 0, which means (-3) × (-4) must equal 12. **The number line interpretation:** Think of negative as "reverse direction." A negative times a positive reverses the direction once. A negative times a negative reverses the direction twice, bringing you back to positive. It's like saying "turn around, then turn around again" — you're back facing forward. **Why this matters:** If negative × negative were negative, then the distributive property would break. Math would become inconsistent. The rule isn't arbitrary — it's forced by the requirement that arithmetic be logical and self-consistent.