040dms

toc = Degrees, Minutes, Seconds =

Up to this point, to provide more accuracy than the nearest degree, you have used decimal parts of a degree.

For mathematicians, until recently, it was much more common to express parts of a degree in terms of minutes and seconds.
 * 1 degree = 60 minutes (1º = 60')
 * 1 minute = 60 seconds (1' = 60")

**For example:**
 * 23.5º = 23º 30' {half a degree is 30 minutes}
 * 18.25º = 18º 15' {one quarter of a degree is 15 minutes}
 * 31.329º = 31º 19' 45"

math \\ . \qquad 19^\circ \; 12' = \left( 19 + \dfrac{12}{60} \right)^\circ \\ \\ \\ . \qquad 57^\circ \; 32' \; 10'' = \left( 57 + \dfrac{32}{60} + \dfrac{10}{3600} \right)^\circ math
 * Note: **

Since hand held calculators became available, decimals have became more commonly used, but degrees, minutes & seconds are still used in mapping and navigation (latitude & longitude), astronomy and some parts of surveying.

Calculator Use
Most scientific calculator have a button for entering angles as degrees, minutes & seconds.

My calculator has a button like this: **[DºM'S]** Some calculators have a button like this: **[º ' "]** Some calculators have a button like this: ** [DMS] **

To enter 18º 15' : type
 * 18 **[DºM'S]** 15 **[=]**

To enter 26º 20' 13" : type
 * 26 ** [DºM'S] ** 20 ** [DºM'S] ** 13 ** [=] **

To convert between decimals and DºM'S", you have to press ** 2nd Function ** and then the ** [DºM'S] ** button {Notice the **<—> DEG** in yellow above the [DºM'S"] button!}

To change 12.1º to DºM'S" : type
 * 12.1 **[2nd] [DºM'S]**

To change 49º 17' to decimals : type
 * 49 ** [DºM'S] ** 17 ** [2nd] [DºM'S] **

DMS on the Casio Classpad
First check that your Classpad is set to Degrees Mode (It should say DEG in the mode bar along the bottom of the screen.)
 * If it says either RAD or GRA, tap on that spot until DEG appears.

You may prefer to have your Classpad in Decimal Mode (It should say DECIMAL in the mode bar along the bottom of the screen)
 * It if says STANDARD, tap on that spot until it says DECIMAL


 * To change from Decimal Degrees to Degrees, Minutes, Seconds, **
 * Go to the ACTION menu, TRANSFORMATION submenu
 * Select **toDMS** from the very bottom of the menu
 * Type the angle in degrees (no need for the symbol)
 * Press EXE
 * Your answer will appear as DMS(degrees, minutes, seconds)[[image:040classpad1.gif align="right"]]


 * Example: **
 * toDMS(46.78) {gives}
 * dms(46,46,48)
 * {This means}
 * 46.78º = 46º 46' 48"


 * To change from Degrees, Minutes, Seconds to Decimal Degrees **
 * Go to the ACTION menu, TRANSFORMATION submenu
 * Select dms from the very bottom of the menu
 * type the degrees, minutes and seconds with a comma between them (no other symbol)
 * Press EXE
 * Your answer will appear as a decimal (if in DECIMAL mode) or a fraction (if in STANDARD mode)


 * Example **
 * dms(57, 24, 45) {gives}
 * 57.4125
 * {this means}
 * 57º 24' 45" = 57.4125º

For more instructions on how to use Degrees, Minutes & Seconds on the Casio Classpad, go here.

Rounding off
You are used to rounding off decimals by rounding up when the next digit is 5 or higher.

If asked to round an angle in degrees & minutes to the nearest degree, round up if the amount of minutes is 30 or more (ie half a degree)

Round the following to the nearest degree
 * For example: **
 * 26º 38' = 27º
 * 63º 19' = 63º

History
We can trace the way we divide time into 24 hours, 60 minutes, 60 seconds and the similar way we divide angles into 360 degrees, 60 minutes, 60 seconds, all the way back to the Sumerians who lived in Mesopotomia (modern day Iraq) from around 4,000 BC to around 2,000 BC.

They were the first to develop writing (Cuneiform) and they had a number system that was base 60. Clay tablets (baked hard in ovens or in the sun) have been found dating back as far as 2600 BC with multiplication tables, algebra, fractions, quadratic equations and even Pythagoras' Theorem (but not called that).

Unlike the later Romans, they used a place value system like we do, so
 * 100 would be written as 1, 40 (1 lot of 60 + 40).
 * 200 would be written as 3, 20 (3 lots of 60 + 20).

Hours in the day: It is speculated that they were the ones who started dividing the morning (sunrise to noon) into 6 hours and similarly dividing the afternoon (noon to sunset) into another 6 hours. (Given that sunset and sunrise change with the seasons, this means that the length of an hour was not fixed). They could then talk about fractions of time and given that their number system was base 60, they logically divided the hour up into 60 parts (minutes).

Notice that 24 hours is 4 × 60 and 360º is 6 × 60.

The Babylonians took over the Mesopotamian region (2, 000 BC to 540 BC) and adopted the Sumerian writing system and their number system and their system of measuring time in hours and minutes. These ideas then spread west to the Mediterranean region (Israel, Egypt and Greece) and east to India and China. The tablet to the right ( approx. 1800 BC - 1600 BC) shows a value for the square root of 2 in cuneiform drawn over a square with the diagonals drawn in.

math \\ 1, \; 24, \; 51, \; 10 = 1 + \dfrac{24}{60} + \dfrac{51}{60^2} + \dfrac{10}{60^3} \\ \\ . \qquad \qquad \quad = 1.414 \; 212 \; 96 ... math

compare this with the actual value math . \qquad \qquad \sqrt{2} = 1.414 \; 213 \; 56 ... math

This is an error of only 0.000 000 602 ...

Further to this: The square is shown with a side length of 30.

Pythagoras' Theorem tells us: math \text{the diagonal would therefore have a length of: } 30\sqrt{2} math

Using their value for the square root of 2: math \\ . \qquad 30 \sqrt{2} = 30 \times \left( 1 + \dfrac{24}{60} + \dfrac{51}{60^2} + \dfrac{10}{60^3} \right) \\ \\ \\ . \qquad \qquad = \big( 30 \times 1 \big) + \left( \dfrac{1}{2} \times \dfrac{24}{1} \right) + \left( \dfrac{1}{2} \times \dfrac{51}{60} \right) + \left( \dfrac{1}{2} \times \dfrac{10}{60^2} \right) math

math \\ . \qquad \qquad = 30 + 12 + \left( \dfrac{1}{2} \times \dfrac{50}{60} + \dfrac{1}{2}\times\dfrac{1}{60} \right) + \dfrac{5}{60^2} \\ \\ \\ . \qquad \qquad = 42 + \dfrac{25}{60} + \dfrac{30}{60^2} + \dfrac{5}{60^2} \\ \\ \\ . \qquad \qquad = 42 + \dfrac{25}{60} + \dfrac{35}{60^2} math

This value would be written as: 42, 25, 35 which we can see written under the diagonal on the tablet.

The person who created the tablet (possibly a student) approximately 3,500 years ago has drawn a square with a side length of 30 and correctly calculated the length of the diagonal. {I think this is awesome!!. RB}

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