The statement 0.x = 0 is not an axiom and can thus be proved.

The axioms for the integers are, for addition

(A1) a + (b + c) = (a + b) + c

(A2) a + 0 = a

(A3) a + (-a) = 0

(A4) a + b = b + a

for multiplication

(M1) a(bc) = (ab)c

(M2) 1a = a

(M3) ab = ba

and for addition and multiplication ('distributive laws')

(D1) a(b+c)=ab+ac

(D2) (a+b)c=ac+ab.

So how do mathematicians prove that 0.a=0?

They do something like this.

a = a

a = (1+0).a (by A2 and M2)

a = 1.a + 0.a (by D2)

a = a + 0.a (by M2)

0.a = 0 by (A2).

2-2018 Teaching by misleading

2 months ago

I've forgotten what the correct word is for this type of nit-picking.

ReplyDeleteYou should have had A3 in the last statement:)

Ray

Or: (0 + 0)a = 0a; so 0a + 0a = 0a, hence 0a = 0.

ReplyDelete