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$\bf History$

It is speculated that the seemingly arbitrary number $360$ used to indicate a full revolution in degrees was chosen because the Babylonians counted in base $60$ and $60 \times 6 = 360$.

Alternatively, if $\pi$ were roughly approximated as $3$, one radian would be equal to $60$ degrees, which could be what the Babylonians did in order to arrive at the number $60$.

$\bf Alternative\,1$

Gradians. Each quadrant is assigned a range of $100$ gradians. A full revolution is therefore $400$ gradians. Much more intuitive for users of a base 10 numerical system. Well-established unit of measurement used in surveying across many parts of the world.

$\bf Alternative\,2$

If the Babylonians chose the number $360$ to represent a full turn because they counted in base $60$ and $6 \times 60 = 360$, it would seem logical that we choose the number $100$ to represent a full turn because $10 \times 10 = 100$.

Not only would this make arithmetic involving perpendicular and opposite angles easier, it would facilitate teaching the concept of angles to primary school children because of everybody's familiarity with providing a quantity out of $100$ such as percentages, or school grades in many countries.

$\bf Practicality$

The nation of Turkey switched from using an Arabic script into using a Latin script smoothly over a period of 4 years, and haven't looked back every since. Changing from degrees to another angle measure is a small change in comparison to changing a whole script.

$\bf Question$

So, with a strong argument for a change, is there a reason that we stick to using degrees?

$\bf Edit$

Sure there's also radians but we still use degrees quite a lot, in education for example. People who do not end up taking more advanced mathematics (most people) never learn what radians are. Also protractors work with degrees as opposed to radians.

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  • $\begingroup$ In Alternative 2, $60 \times 60 \neq 360$ $\endgroup$ – Noble Mushtak Jun 21 '16 at 19:00
  • $\begingroup$ @NobleMushtak Oops, fixed! $\endgroup$ – hb20007 Jun 21 '16 at 19:01
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    $\begingroup$ For any "serious" use, mathematicians use radians. So, degrees have been replaced. Degrees are convenient because 360 has a lot of factors. And 400 gradians in a circle is even more arbitrary in my mind than 360 degrees. $\endgroup$ – Doug M Jun 21 '16 at 19:02
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    $\begingroup$ Degrees have largely been decimalized already, minute and second are less and less used. $\endgroup$ – André Nicolas Jun 21 '16 at 19:08
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    $\begingroup$ Why weren't inches and ounces replaced with more intuitive measures? $\endgroup$ – Greg Martin Jun 21 '16 at 19:38
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History aside, you would like all the angles that commonly occur in elementary plane geometry (and are classically constructible and are rational multiples of $\pi$) to be integer numbers of degreess. Given that we need to put $90^\circ$, $60^\circ$ (in an equilateral triangle) and $45^\circ$ (in a right isosceles triangle) and $72^\circ$ (in a regular pentagon, which is constructible) to be integer numbers of degrees, we want to divide the circle into a number of degrees that is divisible by $60$. That rejects $400$.

In fact, if we also impose that a one degree angle difference should be something that can be discerned but not easily, the choices that make sense are that a circle is $300, 360, or 420$ degrees.

The choice of $360$ is somewhat historical and arbitrary among those.

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For high schoolers, degrees would definitely be easier than gradians. This is because $360$ is divisible by $2$, $3$, and $5$, so many angles of interest are integers in degrees while they are not in gradians: $$\frac{\pi}{3}=60^\circ=\frac{200}{3}^\text{g}$$ $$\frac{\pi}{4}=45^\circ=50^\text{g} \ \text{(OK, this works out for both.)}$$ $$\frac{\pi}{10}=18^\circ=20^\text{g} \ \text{(This works out, too; also, really only used in pentagons)}$$ $$\frac{\pi}{6}=30^\circ=\frac{100}{3}^\text{g}$$ $$\frac{\pi}{12}=15^\circ=\frac{100}{6}^\text{g} \ \text{(This is used a lot in pre-calc for half-angle identities.)}$$ Thus, even if gradians are useful in fields like surveying where angles are measured and used more numerically, in school, while angles are often given to us as an arbitrary decimal in trigonometry, we also often had to deal with these very specific angles, so it's really more helpful for those angles to be integers than for all of the decimals to be changed a little by one quadrant being $100^\text{g}$ instead of $90^\text{g}$.

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