Constructing the negative numbers as equivalence classes

Professor Zap sketches the construction of the negative numbers as equivalence classes of pairs of positive integers

Category:

Education

Tags:

• math
• equivalence relations
• mathematics
• University of South Alabama

[VeryEvilPettingZoo]

Nice. I hope kids who are interested in math pay attention, because this kind of trick is ubiquitous to modern math.

In the closing seconds, there were some errors on the board. You wrote -1 = [(a-1,1)], -2 = [(a-2,2)], which are both incorrect for the same two reasons. First [(a-2,2)] corresponds to a-4, not -2. Second, the point of this is to define negative numbers, so [(a-2,2)] is meaningless when a=0 or a=1.

I think you meant -1 corresponds to [(a,a+1)] and -2 corresponds to [(a,a+2)].

[ProfessorElvisZap]

@VeryEvilPettingZoo Thanks, I'll fix the typos in an over screen!

[VeryEvilPettingZoo]

(cont)

In general, [(a,b)] corresponds to a-b.

(My post had an error too, as you've excluded 0 from N. Thus my comment using a=0 wasn't permissible, though the point I made is still correct for other values of a.)

The generalization of this trick defines (a1,b1) ~ (a2,b2) when there exists some c (here, in N) such that a1+b2+c = a2+b1+c. (That then defines the "Grothendieck group" of a commutative monoid.)

If you use that definition, then you don't need to use the cancellation property of N.