Gradian

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This article is about a unit of plane angle. For the grade of an inclined surface, i.e. gradient, slope, or pitch, see grade (slope). For the gradient in vector calculus, see gradient. For the radian unit, see radian.

The gradian is a unit of plane angle, equivalent to 1400 of a turn.[1]

It is also known as gon, grad, or grade. One grad equals 910 of a degree or π200 of a radian. In continental Europe, the French term centigrade was in use for one hundredth of a grad, and the term myriograde was in use for one ten-thousandth of a grad. This was one reason for the adoption of the term Celsius to replace centigrade as the name of the temperature scale.[2][3]

History

The unit originated in France as the grade, along with the metric system, hence it is occasionally referred to a "metric degree." Due to confusion with existing grad(e) units of northern Europe, the name gon was later adopted, first in those regions, later as the international standard. In German, the unit was formerly also called Neugrad (new degree), likewise Nygrad in Swedish, Danish and Norwegian (also Gradian), and Nýgráða in Icelandic.

Although attempts at a general introduction were made, the unit was only adopted in some countries and for specialised areas such as surveying. The French artillery has used the grad for decades. The degree, 1360 of a circle, or the mathematically more convenient radian, 1(2π) of a circle (used in the SI system of units) are generally used instead. In the 1970s and 80s most scientific calculators offered the grad as well as radians and degrees for their trigonometric functions,[4] but in recent years most offer degrees and radians only.[dubious ][5]

The international standard symbol for this unit today is "gon" (see ISO 31-1). Other symbols used in the past include "gr", "grd", and "g", the last sometimes written as a superscript, similarly to a degree sign: 50g = 45°. "Grad" was commonly used on calculators with LCD displays, as "DEG", "GRAD", and "RAD" could all be represented as a subsection of a panel with the letters "DEGRAD".

Benefits

Each quadrant is assigned a range of 100 gon, which eases recognition of the four quadrants, as well as arithmetic involving perpendicular or opposite angles.

 0° = 0 gon 90° = 100 gon 180° = 200 gon 270° = 300 gon 360° = 400 gon

One advantage of this unit is that right angles are easy to add and subtract in mental arithmetic. If one is traveling on a course of 117 gon (clockwise from due North), say, then the direction from one's left is instantly convertible into 17 gon; while the direction from one's right is 217 gon; and the direction from behind one is 317 gon. A disadvantage is that the common angles of 30° and 60° in geometry must be expressed in fractions (3313 gon and 6623 gon, respectively). Similarly, in one hour (124 day), Earth rotates by 15° or 1623 gon.

Originally, 1 gon of arc along the Earth's surface was equal to 100 kilometers of distance at the equator; therefore 1 centigrad of arc equals 1 kilometer.[6]

Gradians are also convenient when working with vectors on the complex plane. The exponent of the imaginary unit on any given vector is equal to its angle (argument) in hectogradians (100 gon) from the positive x-axis: $i^n$ has an argument of $100n$ gradians.

Use in surveying

In surveying, the gradian is the default unit of angles in many parts of the world.[7] Subdivisions of gradian used in surveying can be referred to as c and cc (1 c = 0.01 grad) and cc's (1 cc = 0.0001 grad).

Conversion of some common angles

Units Values
Turns   0 1/24 1/12 1/10 1/8 1/(2π) 1/6 1/5 1/4 1/3 2/5 1/2 3/4 1
Radians 0 $\tfrac{\pi}{12}$ $\tfrac{\pi}{6}$ $\tfrac{\pi}{5}$ $\tfrac{\pi}{4}$ 1 $\tfrac{\pi}{3}$ $\tfrac{2\pi}{5}$ $\tfrac{\pi}{2}$ $\tfrac{2\pi}{3}$ $\tfrac{4\pi}{5}$ $\pi\,$ $\tfrac{3\pi}{2}$ $2\pi\,$
Degrees   15° 30° 36° 45° 57.3° 60° 72° 90° 120° 144° 180° 270° 360°
Grads 0g 16⅔g 33⅓g 40g 50g 63.7g 66⅔g 80g 100g 133⅓g 160g 200g 300g 400g

References

1. ^ Patrick Bouron (2005). Cartographie: Lecture de Carte. Institut Géographique National. p. 12. Retrieved 2011-07-07.
2. ^ Frasier, E. Lewis (February 1974), Improving an imperfect metric system, Bulletin of the Atomic Scientists: 9ff. On p. 42 Frasier argues for using grads instead of radians as a standard unit of angle, but for renaming grads to "radials" instead of renaming the temperature scale.
3. ^ Mahaffey, Charles T. (1976), Metrication problems in the construction codes and standards sector, NBS Technical Note 915, U.S. Department of Commerce, National Bureau of Commerce, Institute for Applied Technology, Center for Building Technology, "The term "Celsius" was adopted instead of the more familiar "centigrade" because in France the word centigrade has customarily been applied to angles."
4. ^ Maloney, Timothy J. (1992), Electricity: Fundamental Concepts and Applications, Delmar Publishers, p. 453, ISBN 9780827346758, "On most scientific calculators, this [the unit for angles] is set by the DRG key"
5. ^ Cooke, Heather (2007), Mathematics for Primary and Early Years: Developing Subject Knowledge, SAGE, p. 53, ISBN 9781847876287, "Scientific calculators commonly have two modes for working with angles – degrees and radians"
6. ^ Cartographie – lecture de carte – Partie H Quelques exemples à retenir
7. ^ Lindeburg, Michael R. (2012), Civil Engineering Reference Manual for the PE Exam, Professional Publications, Inc., p. 78-7, ISBN 9781591263807.