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     The preceding example (Example 5) was a relatively prolonged problem but there is still one additional aspect to it that deserves further attention before leaving our 2-dimensional world for a while. The last computation by which the unit conversion was accomplished with our SOP does not reveal the ever-slightly more complex detail required for many higher-dimensional unit conversions.  We may close our discussion after tackling an extension to the previous example by asking ourselves, "what will be the equivalent answer in square kilometers?"
     Noticing the Area is equal to 235,000 m² is the most convenient place to begin.  We just need to get from meters to kilometers, or more precisely, convert from m² to km².  It probably appears quite reasonable to conjure up the conversion fraction...
1 km

1000 m
...however this quantity is insufficient to achieve our intended goal as the unit we need to cancel out is the unit of square meters (m²) and not just merely the unit of meters (m).  The first is a 2-dimensional unit a measure of Area; while the latter is 1-dimensional a measure of length or distance.  To see this, let's examine a naively proposed conversion which looks like something akin to the following one:
 
  235,000 m²     1 km  ?

1000 m
 
The unit of m in the denominator of the conversion fraction DOES NOT cancel out the unit of m² in the original quantity.  So instead, we adapt it as follows:
  235,000 m²   ( 1 km )²

1000 m
We are allowed to do this, as the conversion fraction is equal to "1" and since 1 = 1².  Continuing this conversion calculation, yields:
  235,000 m²    1² km²   =    235,000 m²  1 km² 


 1000² m²  1,000,000 m²  
=   cancel out a common factor of 1000 AND the unit of square meters (m²) present in both the numerator & denominator
=    235  1 km²  

1,000
=   0.235 km² 
That's it! Many readers initially expect the answer to be 235 km², but hopefully all are convinced that our derived answer is correct.  Similarily, can you justify why 1 yd² = __ ft², why 1 ft² = ____ in², and one last (metric) question, why 1 cm² ____ mm² ?
Answers are:  9, 144 & 100
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