Properties of Numbers


John A’s notebook begins with a number of definitions and properties of numbers. The inspiration seems to have been the opening of Book VII of Euclid’s Elements. This is a book on the fundamentals of number theory rather than the geometry of the first six books that form the basis of topics in geometry taught in schools today. Here is Macdonald’s list with similar items from Euclid’s Elements.



Macdonald’s notebook


Euclid’s Elements


A unit or unity is that by which every thing taken singly or by itself is considered as one.


A unit is that by virtue of which each of the things that exist is called one.


A number is that which is composed of one or more units as 1, 2, 3, 4 &c.


A number is a multitude composed of units.


The numbers which divide a number without remainder, are called divisors, measures or aliquot parts of it; thus 4 and 5 are divisors, measures or aliquot parts of 20.


A number is a part of a number, the less of the greater, when it measures the greater;



Note: A number which divides 2 or more numbers without remainder is called a common divisor or measure of these numbers.




The numbers, which when multiplied together, produce any number, are called factors of it; thus, 3 and 4 are factors of 12.

Note: Hence, every factor of a number is a divisor, measure or aliquot part of that number.




The product of a number by any multiplier is called a multiple of that number, and the products of several numbers by the same multiplier is called equimultiples of those numbers; thus when 4 is multiplied by 5 the product 20 is called the multiple of 4, and 12, 18, 27 being the products of 4, 6, 9 by 3 are called equimultiples of 4, 6, 9.


The greater number is a multiple of the less when it is measured by the less.



Note: Every multiple of a number can be divided without remainder by that number.




A number which can be divided without remainder by several numbers is called a common multiple of those numbers; thus, 12 is a common multiple of 2, 3, 4, 6.


Numbers relatively composite are those which are measured by some number as a common measure.



Note: A common divisor or measure of 2 numbers will divide their sum, their difference, or any multiple of either; hence 4 being the measure of 8 and 20 will divide 28, 12, 64, or 100.




Every number, which has divisors, or factors, is called a composite number, and every number which has no divisors or factors is called a prime number. Thus 6 being divisible by 2 is a composite number; but 7 not being divisible by any number except itself or unity, is a prime number.




A composite number is that which is measured by some number.

A prime number is that which is measured by a unit alone.



When 2 or more numbers have no common divisor, they are called prime to each other; thus 9, 11, 16 are prime to each other.


Numbers relatively prime are those which are measured by a unit alone as a common measure.



A perfect number is that which is equal to the sum of all its aliquot parts; thus, 28 being equal to 1 + 2 + 4 + 7 + 14 is a perfect number.


A perfect number is that which is equal to the sum its own parts.



The list of properties of numbers is not unique to John A’s notebook. A similar list appears in the standard British arithmetic textbook by John Bonnycastle, An Introduction to Arithmetic. The first two entries in Bonnycastle’s list are identical to Macdonald’s notebook. Other entries are similar to Macdonald’s, but not identical. Bonnycastle’s list appears near the end of his book, well after the mechanics of arithmetic are discussed. This makes it easier to understand the concepts given in his list.