# Definitions 6–10

6. An *even number* is that which is divisible into two equal parts.

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.

8. An *even-times-even number* is that which is measured by an even number according to an even number.

9. An *even-times-odd number* is that which is measured by an even number according to an odd number.

10. An *odd-times-odd number* is that which is measured by an odd number according to an odd number.

## Guide

#### On Definition 6

The definition even number is clear: the number *a* is even if it is of the form *b* + *b.* The first few even numbers are 2, 4, 6, 8, 10.

#### On Definition 7

The definition for odd number has two statements. The first can be taken as a definition of odd number, a number which is not divisible into two equal parts, that is to say not an even number. The first few odd numbers are 3, 5, 7, 9, 11. Euclid did not treat 1 as a number, but now 1 is also considered an odd number.

The other statement is not a definition for odd number, since one has already been given, but an unproved statement. It is easy to recognize that something has to be proved, since if we make the analogous definitions for another number, say 10, then analogous statement is false. Suppose we say a “decade number” is one divisible by 10, and and “undecade number” is one not divisible by 10. Then it is not the case that an undecade number differs by a unit from a decade number; the number 13, for instance, is not within 1 of a decade number.

The unproved statement that a number differing from an even number by 1 is an odd number ought to be proved. That statement is used in proposition IX.22 and several propositions that follow it. It could be proved using, for instance, a principle that any decreasing sequence of numbers is finite.

#### Definitions 8–10

These are also clear. A product of two even numbers is an even-times-even number. The first few are 4, 8, 12, 16, 20. They are all multiples of 4.
A product of an even and an odd number is an even-times-odd number. The first few are 6, 10, 12, 14, 18, 20. Note that a number like 12 is both even-times-even and even-times-odd being at the same time 2 times 6 and 4 times 3.

The numbers which are even-times-even but not even-times-odd are just the powers of 2, namely, 4, 8, 16, 32, etc. These are the numbers which are even-times-even only, and they occur in proposition IX.32.

A product of of two odd numbers is an odd-times-odd number. The first few are 9, 15, 21, 25, 27.