Pure NumbersForget units for the moment. We will concentrate on pure numbers. In the above example, the four groups of three apples becomes simply 4(3) = 12. Suppose, instead of 4(3) we make on of the numbers negative. Let’s use –3. We now have
4(–3) = ?We could be a dead horse, but it seems pretty intuitive that four groups of –3 should add up to –12.
But what if both numbers are negative. Do we wind up with a positive number? We write
(–4)(–3) = ?
Modifying the EquationOur goal is not to give a rigorous mathematical proof, but to provide intuitive insight concerning negative numbers. Let’s modify the equation to read
–4(–3 + 3) = ?This equation suggests the equivalence of subtracting a positive number to adding a negative one. That being the case, we see
–4(–3 + 3) = –4(–3) + –4(3) = 0
–4(–3) + (–12) = 0Or, moving the –12 to the other side of the equals sign,
–4(–3) = +12
Negative times negative makes positiveWe see from the above that negative four times negative three makes positive twelve. In using pure numbers instead of numbers with units, we took away the stumbling block of preconceived mental prejudice.
Note: You might also enjoy What is a Negative Number?
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