The Field Axioms of ℝ Notes

2.3. Axioms of addition

These axioms define the structure of addition in the real numbers:

1. Associativity of Addition:

   $a + (b + c) = (a + b) + c$

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

2. Commutativity of Addition:

   $a + b = b + a$

   a+b=b+a

3. Additive Identity:
   There exists a

   $0 \in ℝ$

   0∈R such that

   $a + 0 = a$

   a+0=a

4. Additive Inverses:
   For each

   $a \in ℝ$

   a∈R, there exists

   $-a \in ℝ$

   −a∈R such that

   $a + (-a) = 0$

   a+(−a)=0

These form the abelian group structure of

$(ℝ, +)$

(R,+).