Explaining ?3 × ?3 = +9: The “Removing a Debt” Story

1. Negative as “Owing” In many real-world contexts, a “negative” amount of money can be thought of as owing money (a debt).
2. What Does ?3 Represent? ?3 dollars could represent “owing 3 dollars.” In an example, imagine you owe someone $3—that’s ?3.
3. What Does ?3 Times ?3 Mean? Multiplying by ?3-3 again can be seen as “taking away (or removing) that owing situation 3 times.” Another way to phrase it: A negative sign can be read as “the opposite of.” So ?3 times ?3 is “the opposite of (owing 3 dollars) done three times.”
4. Result: Owing “Owing $3” Three Times If you “remove” a debt of $3 three separate times, you actually gain $9. This lines up with the arithmetic fact that ?3 × ?3 = +9.

A Quick Breakdown

1. ?3 = owe $3
2. Multiplying by ?3 = do the “opposite” of owing $3 three times.
3. If you’re no longer on the hook for $3 on three different occasions, you come out $9 ahead overall.

That’s exactly what the math statement ?3 × ?3 = +9 captures: “two negatives” (two instances of turning something around) yields a positive result.