In algebra minus 1 multiplied by minus 1 gives plus 1. This is conventional but is there a better reason?

1 - 1 = 0

mult by -1

-1*1 - 1*(-1) = 0

add 1*1

-1*(-1) = 1*1

Here's a decent explanation from the website I will list as the source:

Imagine a number line on which you walk. Multiplying x*y is taking x steps, each of size y. Negative steps require you to face the negative end of the line before you start walking and negative step sizes are backward (i.e., heel first) steps. So, -x*-y means to stand on zero, face in the negative direction, and then take x backward steps, each of size y.

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Here's a proof from the same website

Let a and b be any two real numbers. Consider the number x defined by

x = ab + (-a)(b) + (-a)(-b).

We can write

x = ab + (-a)[ (b) + (-b) ] (factor out -a)

= ab + (-a)(0)

= ab + 0

= ab.

Also,

x = [ a + (-a) ]b + (-a)(-b) (factor out b)

= 0 * b + (-a)(-b)

= 0 + (-a)(-b)

= (-a)(-b).

So we have

x = ab

and

x = (-a)(-b)

Hence, by the transitivity of equality, we have

ab = (-a)(-b).