# Multiplication of Real Numbers Distributes over Subtraction

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## Theorem

### Multiplication of Real Numbers is Left Distributive over Subtraction

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 $m$:

- $m a - m b = m \paren {a - b}$

### Multiplication of Real Numbers is Right Distributive over Subtraction

In the words of Euclid:

*If two magnitudes be equimultiples of two magnitudes, and any magnitudes subtracted from them be equimultiples of the same, the remainders also are either equal to the same or equimultiples of them.*

(*The Elements*: Book $\text{V}$: Proposition $6$)

That is, for any number $a$ and for any integers $m, n$:

- $ma - na = \left({m - n}\right) a$