height

In graphs, generally represented as the y axis. In three dimensional space, generally represented as distance from the ground, or from sea level. Measurements in the metric system are generally used for altitude, while ones for stature are most often noted in the imperial system, which has nothing to do with Star Wars.

Property

height

Values

<length>, auto

Initial

auto

Inherited

no

This property can be applied to text, but it is most useful with replaced elements such as images. The height is to be enforced by scaling the image if necessary. When scaling, the aspect ratio of the image is preserved if the 'width' property is 'auto'.

Example:

      IMG.icon { height: 100px }

If the 'width' and 'height' of a replaced element are both 'auto', these properties will be set to the intrinsic dimensions of the element.

If applied to a textual element, the height can be enforced with e.g. a scrollbar.

Negative values are not allowed.

CSS1 core: UAs may ignore the 'height' property (i.e., treat it as 'auto') if the element is not a replaced element.

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+...+a_nxⁿ is

H(P)=max { |a₀| , |a₁| , ... , |a_n| }

whilst for an algebraic integer ζ such that the polynomial of smallest degree with ζ as a root is P=a₀+a₁x+...+a_nxⁿ, we define the height of ζ by

h(ζ)=n+ |a₀| + |a₁| + ... + |a_n|

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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Height (?), n. [Written also hight.] [OE. heighte, heght, heighthe, AS. he�xa0;hu, fr. heah high; akin to D. hoogte, Sw. hojd, Dan. hoide, Icel. hae, Goth. hauhipa. See High.]

1.

The condition of being high; elevated position.

Behold the height of the stars, how high they are! Job xxii. 12.

2.

The distance to which anything rises above its foot, above that on which in stands, above the earth, or above the level of the sea; altitude; the measure upward from a surface, as the floor or the ground, of animal, especially of a man; stature.

Bacon.

[Goliath's] height was six cubits and a span. 1 Sam. xvii. 4.

3.

Degree of latitude either north or south.

Guinea lieth to the north sea, in the same height as Peru to the south. Abp. Abbot.

4.

That which is elevated; an eminence; a hill or mountain; as, Alpine heights.

Dryden.

5.

Elevation in excellence of any kind, as in power, learning, arts; also, an advanced degree of social rank; preeminence or distinction in society; prominence.

Measure your mind's height by the shade it casts. R. Browning.

All would in his power hold, all make his subjects. Chapman.

6.

Progress toward eminence; grade; degree.

Social duties are carried to greater heights, and enforced with stronger motives by the principles of our religion. Addison.

7.

Utmost degree in extent; extreme limit of energy or condition; as, the height of a fever, of passion, of madness, of folly; the height of a tempest.

My grief was at the height before thou camest. Shak.

On height, aloud. [Obs.]

[He] spake these same words, all on hight. Chaucer.

© Webster 1913.