Negative Behaviour: Solutions

Depending on how you prefer to think, separating these multiplications out like this might not seem that practical and some of the expansions given here are certainly superfluous but they do show us how the product of two negative numbers is forced to be positive. It is likely that you began using the tables further right to inform your leftmost answer. While we might consider this method beneath us now, the multiplications are undeniably made easier by it. In the final row, the square of a large 2-digit prime can be found simply by multiplying powers of 10, a small child could do it. Simple but effective.  

We can observe in these examples, especially the algebraic fractions, how cancelling (or introuducing depending on how you prefer to think about it) a common factor of negative one has a 'swapping of signs' effect.

 

In the last example especially we can see that the value of the fraction is maintained when we swap the signs in a bracketed factor of the numerator provided we also swap the signs in a factor of the denominator. Or we can swap the signs in both factors of the numerator without touching the denominator, or vice versa. As long as a swaps in the numerator are balanced by those in the denominator we can carry on as long as we like. All because -1 divided or multiplied by -1 is 1.