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Proposition 4

Any number is either a part or parts of any number, the less of the greater.
Let A and BC be two numbers, and let BC be the less. I say that BC is either a part, or parts, of A.
| Either A and BC are relatively prime or they are not. First, let A and BC be relatively prime.
Then, if BC is divided into the units in it, then each unit of those in BC is some part of A, so that BC is parts of A. | VII.Def.4
Next let A and BC not be relatively prime, then BC either measures, or does not measure, A.
Now if BC measures A, then BC is a part of A. But, if not, take the greatest common measure D of A and BC, and divide BC into the numbers equal to D, namely BE, EF, and FC. | VII.Def.3 VII.2
Now, since D measures A, therefore D is a part of A. But D equals each of the numbers BE, EF, and FC, therefore each of the numbers BE, EF, and FC is also a part of A, so that BC is parts of A.
Therefore, any number is either a part or parts of any number, the less of the greater.
Q.E.D.

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This proposition says that if a is a larger number than b, then b is a proper fraction of a, perhaps a unit fraction.

The proof is simplified if 1 is considered to be a number. Let a and b be two numbers with a > b. Using VII.2, let d = GCD(a, b). Then d divides both a and b, that is, d is a part of a and a part of b. Let e be the number of times that d measures b, that is, e = b/d. Then b is e parts of a. (If e = 1, then b is just one part of a.) Q.E.D.

This proposition is used in VII.20.

Book VII Introduction - Proposition VII.3 - Proposition VII.5.

© 1996
D.E.Joyce
Clark University