Real Multiplication Distributes over Addition/Geometric Proof

Theorem

The operation of multiplication on the set of real numbers $\R$ is distributive over the operation of addition:

$\forall x, y, z \in \R:$

$x \times \paren {y + z} = x \times y + x \times z$

$\paren {y + z} \times x = y \times x + z \times x$

Proof

In the words of Euclid:

If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.

(The Elements: Book $\text{II}$: Proposition $1$)

Let $A$ and $BC$ be two straight lines.

Let $BC$ be cut at random at points $D$ and $E$.

Then the rectangle contained by $A$ and $BC$ is equal to the sum of the rectangles contained by $A$ and $BD$, by $A$ and $DE$, and by $A$ and $EC$, as follows:

Construct $BF$ perpendicular to $BC$.

Construct $BG$ on $BF$ equal to $A$.

Construct $GH$ through $G$ parallel to $BC$.

Construct $DK, EL, CH$ through $D, E, C$ parallel to $BG$.

Then we have that:

$\Box BCHG = \Box BDKG + \Box DELK + \Box ECHL$

Now $\Box BCHG$ is the rectangle contained by $A$ and $BC$, because it is contained by $BC$ and $BG$, and $BG = A$.

Similarly, from Opposite Sides and Angles of Parallelogram are Equal:

$\Box BDKG$ is the rectangle contained by $A$ and $BD$, because it is contained by $BD$ and $BG = A$

$\Box DEKL$ is the rectangle contained by $A$ and $DE$, because it is contained by $DE$ and $DK$, and $DK = A$

$\Box ECHL$ is the rectangle contained by $A$ and $EC$, because it is contained by $EC$ and $EL$, and $EL = A$.

Hence the result.

$\blacksquare$

Historical Note

This proof is Proposition $1$ of Book $\text{II}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 1 (2nd ed.) ... (previous) ... (next): Book $\text{II}$. Propositions