Multiplication of Real Numbers is Left Distributive over Subtraction
Theorem
In the words of Euclid:
- If a magnitude be the same multiple of a magnitude that a part subtracted is of a part subtracted, the remainder will also be the same multiple of the remainder that the whole is of the whole.
(The Elements: Book $\text{V}$: Proposition $5$)
That is, for any numbers $a, b$ and for any integer $m$:
- $m a - m b = m \paren {a - b}$
Proof 1
Let the magnitude $AB$ be the same multiple of the magnitude $CD$ that the part $AE$ subtracted is of the part $CF$ subtracted.
We need to show that the remainder $EB$ is also the same multiple of the remainder $FD$ that the whole $AB$ is of the whole $CD$.
Whatever multiple $AE$ is of $CF$, let $EB$ be made that multiple of $CG$.
We have that $AE$ is the same multiple of $CF$ that $AB$ is of $GC$.
So from Distributive Property, $AE$ is the same multiple of $CF$ that $AB$ is of $GF$.
By by assumption, $AE$ is the same multiple of $CF$ that $AB$ is of $CD$.
Therefore $AB$ is the same multiple of each of the magnitudes $GF, CD$.
Therefore $GF = CD$.
Let $CF$ be subtracted from each.
Then the remainder $GC$ is equal to the remainder $FD$.
Since:
- $AE$ is the same multiple of $CF$ that $EB$ is of $GC$
- $GC = DF$
it follows that $AE$ is the same multiple of $CF$ that $EB$ is of $CD$.
That is, the remainder $EB$ will be the same multiple of the remainder $FD$ that the whole $AB$ is of the whole $CD$.
$\blacksquare$
Proof 2
\(\ds m \times \paren {a - b}\) | \(=\) | \(\ds m \times \paren {a + \paren {- b} }\) | Definition of Real Subtraction | |||||||||||
\(\ds \) | \(=\) | \(\ds m \times a + m \times \paren {- b}\) | Real Number Axiom $\R \text D$: Distributivity of Multiplication over Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds m \times a + \paren {- m \times b}\) | Multiplication by Negative Real Number | |||||||||||
\(\ds \) | \(=\) | \(\ds m \times a - m \times b\) | Definition of Real Subtraction |
$\blacksquare$
Also see
Historical Note
This proof is Proposition $5$ of Book $\text{V}$ of Euclid's The Elements.