Ordinal numbers refer to position relative to an ordering, as first, second, third, etc.
To the natural numbers one adjoins their negatives and zero to form the integers. Real numbers that have a nonrepeating decimal expansion are called irrational, i.e., they cannot be represented by any ratio of integers. The proof that 2 is unable to be represented by such a ratio was the first proof of the existence of irrational numbers, and it caused tremendous upheaval in the mathematical thinking of that time. Mathematicians have extended this concept even further, as in quaternions .