/sci/ - Science & Math

>>11380431
Why (-)/(-)=(-)(-)?

Because 1/(-1) = (-1)*1/(-1)*(-1) = -1/1 = -1

Why (-)(-)=(+)?

Because:

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

(-1)*(-1) = 1

Why i can multiply by (-1) both sides?

Because:

a=a

Obviously -a=-a


but if a=b, replacing:

-a=-b

Exercise: Show that for a in a field:
(-1)a=-a where -a is the additive inverse of a.

This proof is valid for rational, real and complex numbers. For finite fields too, you can check which field axioms i used (exercise for the reader).

Is valid for integers too because the only number in this numeric structure with multiplicative inverse is 1.

What you need to accept is those axioms and the properties of equality (valid not only for numbers in a field but also for any mathematical object like sets, vectors, functions, etc).

You mention "logic", maybe you would like to read a book on mathematical logic. Seek recomendations.