Real Projective Line - Algebraic Properties

Algebraic Properties

The following equalities mean: Either both sides are undefined, or both sides are defined and equal. This is true for any .

\begin{align} (a + b) + c & = a + (b + c) \\ a + b & = b + a \\ (a \cdot b) \cdot c & = a \cdot (b \cdot c) \\ a \cdot b & = b \cdot a \\ a \cdot \infty & = \frac{a}{0} \\ \end{align}

The following is true whenever the right-hand side is defined, for any .

\begin{align} a \cdot (b + c) & = a \cdot b + a \cdot c \\ a & = (\frac{a}{b}) \cdot b & = \,\,& \frac{(a \cdot b)}{b} \\ a & = (a + b) - b & = \,\,& (a - b) + b \end{align}

In general, all laws of arithmetic are valid as long as all the occurring expressions are defined.

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