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A class assignment made a comment on the conversion of some angle $A$ into degrees, minutes, and seconds. Why is this information useful? How is it done?

Here is the specific text:

Angles in degrees may also be measured using degrees, minutes, and seconds. For example, $42°20'43''$ represents $42$ degrees, $20$ minutes, and $45$ seconds.

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  • $\begingroup$ See Sexagesimal system : " is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form—for measuring time, angles, and geographic coordinates." $\endgroup$ – Mauro ALLEGRANZA Oct 3 '18 at 14:33
  • $\begingroup$ As an example, I downloaded some GPS data from my camera the other day in which I found numbers like $4215.983.$ This turned out to represent $42$ degrees and $15.983$ minutes. If you go to a particular latitude and longitude on Google Maps it will show the latitude and longitude both in degrees with a decimal fraction and also in degrees, minutes, and seconds with a decimal fraction. Navigational software sometimes uses radians. So there are at least four ways angles are commonly measured, and that's just for latitude and longitude! $\endgroup$ – David K Oct 3 '18 at 15:46
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It is one traditional way of representing angles. You can think of it as similar to converting between metric and English units. The point is to be able to read sources that use that format.

Given an angle $A$ in decimal degrees, you are basically converting the fraction into base $60$. Take the fractional part of a degree, multiply by $60$, and take the integer part of the result as the minutes. Take the fractional part of the minutes, multiply by $60$, and you have the seconds.

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This is linked by tradition to the numbering system of the Babylonians (circa $2000$ BC), whose base is $60$ and not $10$ as our decimal system. The ancient Babylonians had $59$ "digits" because they ignored the great mathematical discovery that consisted in inventing the zero.

This tradition is also given in the measurement of hours, minutes and seconds in today's day.

It should be borne in mind that this measurement of the angles lacks the mathematical value that has the most "natural" measure in radians which comes from the extraordinary number $\pi$ which, as is well known, is equal to the relationship between any arbitrary circumference and its diameter.

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