Given a set of objects, one uses a number to describe how many objects are in the set. Zero is the count of "no objects"; in more formal terms, it is the number of objects in the empty set. The concept of parity is used for making groups of two objects. If the objects in a set can be marked off into groups of two, with none left over, then the number of objects is even. If an object is left over, then the number of objects is odd. The empty set contains zero groups of two, and no object is left over from this grouping, so zero is even. These ideas can be illustrated by drawing objects in pairs. It is difficult to depict zero groups of two, or to emphasize the nonexistence of a leftover object, so it helps to draw other groupings and to compare them with zero. For example, in the group of five objects, there are two pairs. More importantly, there is a leftover object, so 5 is odd. In the group of four objects, there is no leftover object, so 4 is even. In the group of just one object, there are no pairs, and there is a leftover object, so 1 is odd. In the group of zero objects, there is no leftover object, so 0 is even. There is another concrete definition of evenness: if the objects in a set can be placed into two groups of equal size, then the number of objects is even. This definition is equivalent to the first one. Again, zero is even because the empty set can be divided into two groups of zero items each. Numbers can also be visualized as points on a number line. When even and odd numbers are distinguished from each other, their pattern becomes obvious, especially if negative numbers are included:  The even and odd numbers alternate. Starting at any even number, counting up or down by twos reaches the other even numbers, and there is no reason to skip over zero. With the introduction of multiplication, parity can be approached in a more formal way using arithmetic expressions. Every integer is either of the form or the former numbers are even and the latter are odd. For example, 1 is odd because and 0 is even because Making a table of these facts then reinforces the number line picture above. The precise definition of a mathematical term, such as "even" meaning "integer multiple of two", is ultimately a convention. Unlike "even", some mathematical terms are purposefully constructed to exclude trivial or degenerate cases. Prime numbers are a famous example. Before the 20th century, definitions of primality were inconsistent, and significant mathematicians such as Goldbach, Lambert, Legendre, Cayley, and Kronecker wrote that 1 was prime.