Location, Trigonometry, and Measurement of the Sphere - Spatial Mathematics: Theory and Practice through Mapping - page 42

pattern is unfortunate because multiplying fractions is straightforward and
natural; further, it is critical to know how to multiply fractions in order to
add fractions, systematically, with different denominators. To add 1/3 and
1/4 multiply the first fraction by 4/4 and the second one by 3/3 yielding
4/12 + 3/12 = 7/12. For those wishing to think of fractions in terms of pieces
of pie, cut one pie into thirds and split each third into quarters so that the
entire pie is split into 12 pieces. Split another pie into quarters and split each
quarter into thirds so that the entire pie is split into 12 pieces. Now the two
pies are commensurable, each having 12 pieces of equal size. From the first,
take four pieces and from the second take three pieces; so the sum is 4 + 3
or 7 of the pieces that are each 1/12th of the pie. Again, the answer is 7/12.
The idea of using the number 1 expressed in creative formats is a power-
ful numerical transformation. Look for it throughout mathematics from the
simple to the advanced.
2.5 Other common coordinate systems
We have discussed two different ways of representing latitude and longi-
tude: As degrees, minutes, and seconds, and as decimal degrees. As we noted
above, these are commonly abbreviated as DMS and DD, respectively. A third
common way of representing latitude and longitude is called decimal minutes
(DM). As the name implies, this method uses degrees, minutes, and then
fractions of a minute as its notation, such as 105 degrees 30.5 minutes. Recall
our work earlier in this chapter using the value of 42 degrees, 31 minutes, 47
seconds in DMS notation. In DD format, it is represented as 42.529756. In DM,
this can be written in fractional form as 42 degrees, 31 + 47/60, or 42 degrees,
31.7833 minutes. DM format is used by specific groups such as the geocaching
community to locate hidden treasures with GPS technology, and it is a com-
mon format that most GPS receivers can display.
Five other coordinate systems are worth discussing here because they are
other manifestations of absolute location. Any coordinate system requires
three things to represent absolute location: An origin point, two sets of axes,
and a unit of measurement. Latitude and longitude are referenced from the
origin point of (0, 0), and its two sets of axes are the Equator (the X -axis)
and the Prime Meridian (the Y -axis). Their unit of measurements can be in
any of three formats—degrees minutes seconds, decimal degree, and decimal
minutes.
At the beginning of this chapter, street addressing was discussed. Street
addressing in many places around the world is a manifestation of a coordinate
system. In most cities, an east-west street or an imaginary line divides the
north from the south sectors of the city, and a north-south street or imaginary
line divides the east from the west sectors. The place where these lines inter-
sect does not have to be in the center of the city, and oftentimes is not, due to
the historical pattern of the way the city grew after the coordinate system was