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Interpreting Integer Multiplication as Repeated Addition

Integer multiplication can be understood through the lens of repeated addition or repeated subtraction, depending on the sign of the second factor. The expression aba \cdot b means "add aa, bb times":

  • Positive second factor (add repeatedly): When bb is positive, multiplication means adding aa a total of bb times. For example, 535 \cdot 3 means "add 55 three times," yielding 5+5+5=155 + 5 + 5 = 15. Likewise, 5(3)-5(3) means "add 5-5 three times," giving (5)+(5)+(5)=15(-5) + (-5) + (-5) = -15.

  • Negative second factor (take away repeatedly): When bb is negative, multiplication is interpreted as subtraction — "take away aa, b|b| times." For example, 5(3)5(-3) means "take away 55 three times." Since there is nothing on the workspace to remove, neutral pairs must first be introduced; after removing the positives, only negatives remain, giving 15-15. Similarly, (5)(3)(-5)(-3) means "take away 5-5 three times"; after introducing neutral pairs and removing the negatives, only positives remain, giving 1515.

This repeated-addition interpretation provides an intuitive explanation for why two negatives multiply to a positive: removing negative quantities leaves behind their positive counterparts.

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Updated 2026-04-21

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