Additive Group of Real Numbers

Theorem

Let $\R$ be the set of real numbers.

The structure $\left({\R, +}\right)$ is an infinite abelian group.

Proof

Taking the group axioms in turn:

G0: Closure

Real Addition is Closed.

$\Box$

G1: Associativity

Real Addition is Associative.

$\Box$

G2: Identity

From Real Addition Identity is Zero, we have that the identity element of $\left({\R, +}\right)$ is the real number $0$.

$\Box$

G3: Inverses

From Inverses for Real Addition, we have that the inverse of $x \in \left({\R, +}\right)$ is $-x$.

$\Box$

C: Commutativity

Real Addition is Commutative.

$\Box$

Infinite

Real Numbers are Uncountably Infinite.

$\blacksquare$

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 4.5$: Example $81$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 7$: Example $7.1$
• Ian D. Macdonald: The Theory of Groups (1968): $\S 1$: Example $1.03$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 34 \ (1)$
• John F. Humphreys: A Course in Group Theory (1996): $\S 1$: Example $1.4$