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Quick Reciprocal Calculations

If you ever use a compass to pilot a vessel or an aircraft, chances are you'll need to calculate reciprocal headings from time to time. Now everyone knows that a heading's reciprocal can be determined by adding or subtracting 180 to or from the heading.

Problem is, sometimes you don't have time to find a calculator; And adding 180 to a number in your head is a lot harder than, say, adding 200 to a number.

Enter "The Rule of Twos"
The Rule of Twos takes advantage of the fact that adding 180 is equivalent to adding 200 and subtracting 20. Similarly, subtracting 180 is the same as subtracting 200, then adding 20. What's more, since 180 is a factor of ten, we can actually treat the problem as one of adding (or subtracting) 18 to the first two digits of the heading. Consequently, the problem becomes one of adding 20 and subtracting two. If this sounds confusing, just read on. It gets easier from here.


Use Three-digit Headings/Bearings
The key to using this method is to treat all headings as three-digit numbers. This should be second-nature since good piloting dictates that we record and call out all headings in this format, anyway.

So a heading of "12" is represented as "012". "45" becomes "045". etc.

For our purposes, we'll label the digits of our numbers as follows:



The Method
1Always leave the third digit alone. (We'll be adding/subtracting 18 instead of 180.)
2AIf the heading is less than 180, then
a) Add 2 to the first digit
b) Subtract 2 from the second digit
BIf the heading is greater than (or equal to) 180, then
a) Subtract 2 from the first digit
b) Add 2 to the second digit

The results of applying A or B in step 2 are shown in the table below.

000102030405060708091011121314151617
181920212223242526272829303132333435

Compass Heading "Decades" (leading two digits) and their Reciprocals

From Steps 1 & 2 (or from the table), we see that:
   Reciprocal of 244 = 064
   Reciprocal of 312 = 132
   Reciprocal of 148 = 328
etc.

The Heading "Decades" 00, 18, 01, 19, 10, 28, 11, and 29 are shown in yellow in the table because they require "carrying" and "borrowing" to perform the simple addition and subtraction that is required.

It might be well to memorize these 4 pairs to avoid any confusion. Just remember that we're adding 18 to the numbers in the top row to obtain the numbers in the bottom row.

Incidentally, these pairs will look familiar to anyone who pilots a plane since runways are numbered by the first two digits of their heading. If a runway can be approached from either direction, the two ends of the runway will be labeled with one of the pairs of reciprocals from the table above. For example:


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