Do you need to be a mathematician to fly? No.
Most of us first encountered reciprocals in the traffic pattern when we were trying to figure out the correct heading for flying the downwind leg. And complements-90 degrees-can be used for all turns after entering a traffic pattern.
How good is your math? Let's start with the 18 possible reciprocal pairs for runways:
010 and 190*
020 and 200
030 and 210
040 and 220
050 and 230
060 and 240
070 and 250
080 and 260
090 and 270
100 and 280*
110 and 290*
120 and 300
130 and 310
140 and 320
150 and 330
160 and 340
170 and 350
180 and 360
Of course, you can use memory, and if that works for you perhaps you don't need the rest of this article. Also, if you happen to be a good mental mathematician and can add and subtract 180 effortlessly, the following may be superfluous.
After just a couple of lessons, students begin to recognize the reciprocal pairs 180 and 360, and 090 and 270. So you will recognize those without calculation. While 000 and 360 are mathematically identical, in aviation North is always referred to as 360 degrees.
I learned to fly at an airport with runways numbered 06 and 24, so, after many, many touch and goes, I knew that reciprocal pair by heart. Runways are typically numbered to the nearest 10 degrees of their magnetic bearing. For example, a runway bearing 286 degrees is usually designated Runway 29. You may have already memorized your home airport's runway numbers.
Now how do you calculate other reciprocal pairs? Obviously you need to add or subtract 180. However, mentally it is easier to add 200 and then subtract 20; or add 20 and then subtract 200. So try this:
What is the reciprocal of 340 degrees? You don't want to add 200. That would be too big. So subtract 200 (340 - 200 = 140), and then add 20 (140 + 20) to get 160.
Here's another example. To find the reciprocal of 070, add 200 to get 270, and then subtract 20. The reciprocal is 250 degrees.
However, there is an even easier way. You simply mentally move a "2" either to the right or left as appropriate. Take away a two from one place, and add it to the other. For example, with 120 degrees, move the two from right to left-subtract it from the 2 and add it to the 1-to get the reciprocal of 300 degrees. By the way, never change the units digit on the far right; it remains the same when calculating reciprocals or complements. For the reciprocal of 330 degrees, move a two from the left to the right to get 150. Of course, do not move a 2 so as to make the hundreds place greater than 3.
Refer to the list above and notice how "moving the 2" works in either direction. Let me say that again-in either direction. However, this is not true for the three pairs on p. 41 marked with an asterisk: 010 and 190 degrees, 100 and 280 degrees, and 110 and 290 degrees. Three cases-010, 100, and 110-do not have a 2 to move, but it is easy to mentally add 180 to each and obtain 190, 280, and 290, respectively. However, these three pairs do work in reverse, with a little adjustment. Moving the two in 190 results in 370, which is 10 degrees more than 360, or 010. Moving the 2 in 280 and 290 (2 + 8 = 10, and 2 + 9 = 11) provides the correct results of 100 and 110.
The trick is to visualize taking away a 2 from one digit and adding to the other. If you understand the above trick, you can easily determine reciprocals "on the fly."
Here is a variation. If the first digit is a 0 or 1, add a 2 and then subtract a 2 from the second digit. If the first digit is a 2 or 3, subtract 2 and add 2 to the second digit. (This is because the result cannot be less then 0 or more than 3.) For example: What is the reciprocal of 040 degrees? As the first number is 0, add 2; and then subtract 2 from the second digit. The result is 220 degrees.
Let's look at some more examples. What is the reciprocal of 140 degrees? As the first digit is a 1, add 2 and then subtract 2 from the second digit-the result is 320 degrees. What is the reciprocal of 297 degrees? As the first digit is 2, subtract 2 and add 2 to the second digit so the result is 2 + 9 = 11, and the 7 on the right remains the same. The answer is 117. (Did I fool you, or did you just forget that the last digit always remains the same?)
As the first digit of any compass direction must be 0, 1, 2, or 3, it is immediately apparent whether a 2 should be added or subtracted. For a check, note the sum of the first two digits of any reciprocal pair is the same; 140 and 320 1+4 = 5 and 3+2 = 5. And the last or units digit remains the same.
If you understand the trick of "moving the 2," then finding most complements is almost as easy. There are two differences: you must "move a 1," and recall there are two complements: 90 degrees counterclockwise and 90 degrees clockwise. So here is the drill. Consider 250 degrees-one complement is 160 (moving a 1 from left to right), and the other is 340 (moving a 1 right to left). However, this trick does not work if the complement crosses north, or if there is no "1" to move. While I will not belabor the method, you can calculate most of these other complements by moving a "3." For example, consider 340 degrees. The clockwise complement is 070 degrees. Another example: The counterclockwise complement of 052 degrees is 322 degrees.
There is also a trick you can use with a heading indicator. Open-face heading indicators typically have peripheral ticks every 45 degrees. Thus, a glance at the correct tick will provide the reciprocal and the complements of your current course.
If you have begun instrument training, you will find reciprocals useful when you are flying on an airway between two VORs. You will be tracking outbound on one radial, usually halfway, and then tracking inbound on the reciprocal course-most of the time. If there is a difference in local magnetic variation between the two VORs, then the outbound and inbounds are not exact reciprocals.
Not all Victor airways are true reciprocals; some airways are affected by magnetic variation. The effect of magnetic variation is more pronounced in the east/west direction and over longer distances between VORs.
The differences from true reciprocals are small, only a degree or so. A glance at the units digit will tell the story. Reciprocals and complements always have the same units digit. Reciprocals often appear when flying:
"Cessna Three-One-Golf, make a right downwind for Runway 23." You reply, "Roger, Three-One-Golf," as you turn to a heading of 050 degrees.
And for the instrument pilot-"Learjet Three-Juliet-Echo, hold northwest on the 317-degree radial of Lost Nation VOR..." and you know the inbound course to the VOR will be 137 degrees. Boy, are you quick!
Dr. Ian Blair Fries is a CFI, senior aviation medical examiner, and ATP, and holds a Lear 35 type rating. He serves on the AOPA Air Safety Foundation Board of visitors and is cochairman of the AOPA Board of Medical Advisors.
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