Complex Numbers cannot be Extended to Algebra in Three Dimensions with Real Scalars

Theorem
It is not possible to extend the complex numbers to an algebra of $3$ dimensions with real scalars.

Proof
that $\set {1, i, j}$ forms a basis for an algebra of $3$ dimensions with real scalars.

Let $1$ and $i$ have their usual properties as they do as complex numbers:
 * $\forall a: 1 a = a 1 = a$
 * $i \cdot i = -1$

Then:
 * $i j = a_1 + a_2 i + a_3 j$

for some $a_1, a_2, a_3 \in \R$.

Multiplying through by $i$:


 * $(1): \quad i \paren {i j} = \paren {i i} j = -j$

and:

But this implies that ${a_3}^2 = -1$, which contradicts our supposition that $a_3 \in \R$.

Hence the result by Proof by Contradiction.