# Proposition: 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction

### (Proposition 5 from Book 5 of Euclid's “Elements”)

If a magnitude is the same multiple of a magnitude that a (part) taken away (is) of a (part) taken away (respectively) then the remainder will also be the same multiple of the remainder as that which the whole (is) of the whole (respectively). * For let the magnitude $AB$ be the same multiple of the magnitude $CD$ that the (part) taken away $AE$ (is) of the (part) taken away $CF$ (respectively). * I say that the remainder $EB$ will also be the same multiple of the remainder $FD$ as that which the whole $AB$ (is) of the whole $CD$ (respectively). ### Modern Formulation

If we are given two positive real numbers $$\alpha$$, $$\beta$$,1 and the following multiples of aliquot parts $$m\ge 1$$, $$n\ge 1$$: $\alpha m=\beta m,\quad \alpha n=\beta n\quad\quad( * )$ then adding both equations gives us $\alpha(m-n)=\beta m-\beta n.$

### General Modern Formulation

Proofs: 1

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@Fitzpatrick

### References

1. From a geometrical point of view, $$\alpha,\beta$$ could e.g. mean the lengths of some segments, the areas of some plane figures or the volumes of some solids.