Table of contents
Definitions
- Definition 1
- A unit is that by virtue of which each of the things that exist is called one.
- Definition 2
- A number is a multitude composed of units.
- Definition 3
- A number is a part of a number, the less of the greater, when it measures the greater;
- Definition 4
- But parts when it does not measure it.
- Definition 5
- The greater number is a multiple of the less when it is measured by the less.
- Definition 6
- An even number is that which is divisible into two equal parts.
- Definition 7
- An odd number is that which is not divisible into two equal parts, or that which differs by a unit from an even number.
- Definition 8
- An even-times-even number is that which is measured by an even number according to an even number.
- Definition 9
- An even-times-odd number is that which is measured by an even number according to an odd number.
- Definition 10
- An odd-times-odd number is that which is measured by an odd number according to an odd number.
- Definition 11
- A prime number is that which is measured by a unit alone.
- Definition 12
- Numbers relatively prime are those which are measured by a unit alone as a common measure.
- Definition 13
- A composite number is that which is measured by some number.
- Definition 14
- Numbers relatively composite are those which are measured by some number as a common measure.
- Definition 15
- A number is said to multiply a number when the latter is added as many times as there are units in the former.
- Definition 16
- And, when two numbers having multiplied one another make some number, the number so produced be called plane, and its sides are the numbers which have multiplied one another.
- Definition 17
- And, when three numbers having multiplied one another make some number, the number so produced be called solid, and its sides are the numbers which have multiplied one another.
- Definition 18
- A square number is equal multiplied by equal, or a number which is contained by two equal numbers.
- Definition 19
- And a cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.
- Definition 20
- Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.
- Definition 21
- Similar plane and solid numbers are those which have their sides proportional.
- Definition 22
- A perfect number is that which is equal to the sum its own parts.
- Proposition 1
- When two unequal numbers are set out, and the less is continually subtracted in turn from the greater, if the number which is left never measures the one before it until a unit is left, then the original numbers are relatively prime.
- Proposition 2
- To find the greatest common measure of two given numbers not relatively prime.
Corollary. If a number measures two numbers, then it also measures their greatest common measure.
- Proposition 3
- To find the greatest common measure of three given numbers not relatively prime.
- Proposition 4
- Any number is either a part or parts of any number, the less of the greater.
- Proposition 5
- If a number is part of a number, and another is the same part of another, then the sum is also the same part of the sum that the one is of the one.
- Proposition 6
- If a number is parts of a number, and another is the same parts of another, then the sum is also the same parts of the sum that the one is of the one.
- Proposition 7
- If a number is that part of a number which a subtracted number is of a subtracted number, then the remainder is also the same part of the remainder that the whole is of the whole.
- Proposition 8
- If a number is the same parts of a number that a subtracted number is of a subtracted number, then the remainder is also the same parts of the remainder that the whole is of the whole.
- Proposition 9
- If a number is a part of a number, and another is the same part of another, then alternately, whatever part or parts the first is of the third, the same part, or the same parts, the second is of the fourth.
- Proposition 10
- If a number is a parts of a number, and another is the same parts of another, then alternately, whatever part of parts the first is of the third, the same part, or the same parts, the second is of the fourth.
- Proposition 11
- If a whole is to a whole as a subtracted number is to a subtracted number, then the remainder is to the remainder as the whole is to the whole.
- Proposition 12
- If any number of numbers are proportional, then one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents.
- Proposition 13
- If four numbers are proportional, then they are also proportional alternately.
- Proposition 14
- If there are any number of numbers, and others equal to them in multitude, which taken two and two together are in the same ratio, then they are also in the same ratio ex aequali.
- Proposition 15
- If a unit number measures any number, and another number measures any other number the same number of times, then alternately, the unit measures the third number the same number of times that the second measures the fourth.
- Proposition 16
- If two numbers multiplied by one another make certain numbers, then the numbers so produced equal one another.
- Proposition 17
- If a number multiplied by two numbers makes certain numbers, then the numbers so produced have the same ratio as the numbers multiplied.
- Proposition 18
- If two number multiplied by any number make certain numbers, then the numbers so produced have the same ratio as the multipliers.
- Proposition 19
- If four numbers are proportional, then the number produced from the first and fourth equals the number produced from the second and third; and, if the number produced from the first and fourth equals that produced from the second and third, then the four numbers are proportional.
- Proposition 20
- The least numbers of those which have the same ratio with them measure those which have the same ratio with them the same number of times; the greater the greater; and the less the less.
- Proposition 21
- Numbers relatively prime are the least of those which have the same ratio with them.
- Proposition 22
- The least numbers of those which have the same ratio with them are relatively prime.
- Proposition 23
- If two numbers are relatively prime, then any number which measures one of them is relatively prime to the remaining number.
- Proposition 24
- If two numbers are relatively prime to any number, then their product is also relatively prime to the same.
- Proposition 25
- If two numbers are relatively prime, then the product of one of them with itself is relatively prime to the remaining one.
- Proposition 26
- If two numbers are relatively prime to two numbers, both to each, then their products are also relatively prime.
- Proposition 27
- If two numbers are relatively prime, and each multiplied by itself makes a certain number, then the products are relatively prime; and, if the original numbers multiplied by the products make certain numbers, then the latter are also relatively prime.
- Proposition 28
- If two numbers are relatively prime, then their sum is also prime to each of them; and, if the sum of two numbers is relatively prime to either of them, then the original numbers are also relatively prime.
- Proposition 29
- Any prime number is relatively prime to any number which it does not measure.
- Proposition 30
- If two numbers, multiplied by one another make some number, and any prime number measures the product, then it also measures one of the original numbers.
- Proposition 31
- Any composite number is measured by some prime number.
- Proposition 32
- Any number is either prime or is measured by some prime number.
- Proposition 33
- Given as many numbers as we please, to find the least of those which have the same ratio with them.
- Proposition 34
- To find the least number which two given numbers measure.
- Proposition 35
- If two numbers measure any number, then the least number measured by them also measures the same.
- Proposition 36
- To find the least number which three given numbers measure.
- Proposition 37
- If a number is measured by any number, then the number which is measured has a part called by the same name as the measuring number.
- Proposition 38
- If a number has any part whatever, then it is measured by a number called by the same name as the part.
- Proposition 39
- To find the number which is the least that has given parts.
Guide
Book VII is the first of the three books on number theory. It begins with the 22 definitions used
throughout these books. The important definitions are those for unit and number, part and
multiple, even and odd, prime and relatively prime, proportion, and perfect number. The topics
in Book VII are antenaresis and the greatest common divisor, proportions of numbers, relatively
prime numbers and prime numbers, and the least common multiple.
The basic construction for Book VII is antenaresis, also called the Euclidean algorithm, a kind of reciprocal subtraction. Beginning with two numbers, the smaller, whichever it is, is repeatedly subtracted from the larger until a single number is left. This algorithm, studied in propositions VII.1 through VII.3, results in the greatest common divisor of two or more numbers.
Propositions V.5 through V.10 develop properties of fractions, that is,
they study how many parts one number is of another in preparation for ratios and proportions.
The next group of propositions VII.11 through VII.19 develop the
theory of proportions for numbers.
VII.11. a : b = c : d implies
both equal the ratio a – c : b – d.
VII.12. a : b = c : d implies
both equal the ratio a + c : b + d.
VII.13. a : b = c : d implies
a : c = b : d.
VII.14. a : b = d : e and
b : c = e : f
imply a : c = d : f.
VII.17. a : b = ca : cb.
VII.19. a : b = c : d if an only if
ad = bc.
Propositions VII.20 through VII.29 discuss representing ratios in lowest terms as relatively prime numbers and properties of relatively prime numbers. Properties of prime numbers are presented in propositions VII.30 through VII.32. Book VII finishes with least common multiples in propositions VII.33 through VII.39.
Postulates for numbers
Postulates are as necessary for numbers as they are for geometry. Euclid, however, supplies none. Missing postulates occurs as early as proposition VII.2. In its proof, Euclid constructs a decreasing sequence of whole positive numbers, and, apparently, uses a principle to conclude that the sequence must stop, that is, there cannot be an infinite decreasing sequence of numbers. If that is the principle he uses, then it ought to be stated as a postulate for numbers.
Numbers are so familiar that it hardly occurs to us that the theory of numbers needs axioms, too. In fact, that field was one of the last to receive a careful scrutiny, and axioms for numbers weren’t developed until the late 19th century by Dedekind and others. By that time foundations for the rest of mathematics were laid upon either geometry or number theory or both, and only geometry had axioms. About the same time that foundations for number theory were developed, a new subject, set theory, was created by Dedekind and Cantor, and mathematics was refounded in terms of set theory.
The foundations of number theory will be discussed in the Guides to the various definitions and propositions.