A degree (in full, a degree of arc, arc degree, or arcdegree),
usually denoted by ° (the degree symbol), is a measurement of plane
angle, representing 1/360 of a full rotation; one degree is
equivalent to pi/180 radians. When that angle is with respect to a
reference meridian, it indicates a location along a great circle of
a sphere, such as Earth, Mars, or the celestial sphere. A circle
with an equilateral Chord. One sixtieth of this arc is a degree.
Six such chords complete the circle
The selection of 360 as the number of degrees (i.e., smallest
practical sub-arcs) in a circle was probably based on the fact that
360 is approximately the number of days in a year. Its use is often
said to originate from the methods of the ancient Babylonians.
Ancient astronomers noticed that the stars in the sky, which circle
the celestial pole every day, seem to advance in that circle by
approximately one-360th of a circle, i.e., one degree, each day.
(Ancient calendars, such as the Persian Calendar, used 360 days for
a year.) Its application to measuring angles in geometry can
possibly be traced to Thales who popularized geometry among the
Greeks and lived in Anatolia (modern western Turkey) among people
who had dealings with Egypt and Babylon.
The earliest trigonometry, used by the Babylonian astronomers
and their Greek successors, was based on chords of a circle. A
chord of length equal to the radius made a natural base quantity.
One sixtieth of this, using their standard sexagesimal divisions,
was a degree; while six such chords completed the full circle.
Another motivation for choosing the number 360 is that it is
readily divisible: 360 has 24 divisors (including 1 and 360),
including every number from 1 to 10 except 7. For the number of
degrees in a circle to be divisible by every number from 1 to 10,
there would need to be 2520 degrees in a circle, which is a much
less convenient number.
Divisors of 360 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20,
24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360.
Subdivisions
For many practical purposes, a degree is a small enough angle
that whole degrees provide sufficient precision. When this is not
the case, as in astronomy or for latitudes and longitudes on the
Earth, degree measurements may be written with decimal places, but
the traditional sexagesimal unit subdivision is commonly seen. One
degree is divided into 60 minutes (of arc), and one minute into 60
seconds (of arc). These units, also called the arcminute and
arcsecond, are respectively represented as a single and double
prime, or if necessary by a single and double quotation mark: for
example, 40.1875° = 40° 11' 15? (or 40° 11' 15").
If still more accuracy is required, decimal divisions of the
second are normally used, rather than thirds of 1/60 second,
fourths of 1/60 of a third, and so on. These (rarely used)
subdivisions were noted[citation needed] by writing the Roman
numeral for the number of sixtieths in superscript: 1I for a
"prime" (minute of arc), 1II for a second, 1III for a third, 1IV
for a fourth, etc. Hence the modern symbols for the minute and
second of arc.
Alternative units
In most mathematical work beyond practical geometry, angles are
typically measured in radians rather than degrees. This is for a
variety of reasons; for example, the trigonometric functions have
simpler and more "natural" properties when their arguments are
expressed in radians. These considerations outweigh the convenient
divisibility of the number 360. One complete turn (360°) is equal
to 2p radians, so 180° is equal to p radians, or equivalently, the
degree is a mathematical constant ° = p/180.
With the invention of the metric system, based on powers of ten,
there was an attempt to define a "decimal degree" (grad or gon), so
that the number of decimal degrees in a right angle would be 100
gon, and there would be 400 gon in a circle. Although this idea did
not gain much momentum, most scientific calculators used to support
it.
The turn (or revolution, full circle, full rotation, cycle) is
used in technology and science. 1 rev = 360°.
An angular mil which is most used in military applications has
at least three specific variants.
In computer games which depict a three-dimensional virtual
world, the need for very fast computations resulted in the adoption
of a binary, 256 degree system. In this system, a right angle is 64
degrees, angles can be represented in a single byte, and all
trigonometric functions are implemented as small lookup tables.
These units are sometimes called "binary radians" ("brads") or
"binary degrees"
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