In mathematics, a 'height' is sometimes used to describe the akwardness of an object, where the usual notions of size or boundedness are unhelpful in doing so. For instance, the number 1999/2000 is very close in absolute size to 1, but is much harder to manipulate (quickly- what is 1999/2000 cubed?); whereas the number 100 is much larger than 1, yet easier to, say, multiply by than 1999/2000. Thus rather than thinking in terms of size, we may consider the height of a rational number, defined as the maximum of |p| and |q| for a number of the form p/q (in lowest terms). Then the height of 1 is 1, and that of 100 is 100- significantly bigger, but far less than the height of our awkward 1999/2000, which clocks in at 2000.
This approach has a number of uses. For instance, the height of a rational number as defined above is a non-negative integer, and thus well-ordered: the technique of proof by infinite descent can be used. Another application is in a proof of the countability of the rational numbers: for each natural number n, the set of rational numbers of height n is finite (neither the numerator nor denominator, each an integer, can fall outside the range -n...n) and any rational number falls into one of those sets, ensuring that the rationals are a countable union of countable (since finite) sets and thus themselves countable.
Whilst rational numbers provide a gentle introduction to the notion of height, more complicated objects can be tackled in this way. For instance, the height of a polynomial P=a0+a1x+...+anxn is
H(P)=max { |a0| , |a1| , ... , |an| }
whilst for an algebraic integer ζ such that the polynomial of smallest degree with ζ as a root is P=a0+a1x+...+anxn, we define the height of ζ by
h(ζ)=n+ |a0| + |a1| + ... + |an|
Similarly to the rationals, organising the algebraic numbers via height proves that they are countable, and hence that there are transcendental numbers. The height of a polynomial is of interest when considering computational aspects of polynomial algebra. For instance, in the greatest common divisor node I gave an example of two polynomials of height 1 whose gcd had height 2- that is, such a computation can create output which is more complicated than the input, and algorithms cannot therefore depend on a decrease in height despite a probable decrease in degree.
Probably the most powerful application of height is in the study of algebraic number theory, and as a special case elliptic curves. It is possible to give a description of a height function on an abelian group, giving rise to the Mordell-Weil theorem.
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