|
Math PagesEstimating ETAFun Stuff
|
ReciprocalsIn aviation a reciprocal is a heading, bearing or track in the opposite direction. So the reciprocal of north is south, and the reciprocal of east is west, and so on. We use reciprocals in many situations as IFR pilots. In holds for example the outbound heading is the reciprocal of the inbound. On approaches the outbound course is the reciprocal of the inbound. Also, when we look at our airway charts the airway radials are marked but, we need the reciprocal of this when we are on the inbound leg of an airway. So, suffice it to say that you will need to be able to calculate the reciprocal of any heading or course. In this article we will go over the most common methods of determining reciprocals and point out a few tips and tricks. The tips are mostly aimed at catching ourselves if we make a mistake - because mistakes are what we humans are best at. Remember - to err is human - to hit a mountain hurts. 200/20 systemThe reciprocal of any number is always 180 degrees more (or less) than that number. If we humans could just add 180 to any number less than 180, or subtract 180 from any number more than 180 we would be reciprocal machines. But, it is quite difficult to do that, so the first trick is to add 200 and subtract 20 - for example: The reciprocal of 30 is - 30 plus 200 is 230 - then 230 minus 20 is 210. So, the answer is 210. The reciprocal of 300 is - 300 minus 200 is 100 - then 100 plus 20 is 120. So, the answer is 120 Many people do no more than use the above formula. However, it is easy to make a mistake so it is valuable to have a couple of tricks up your sleeve to check your answer. The first thing you MUST know is that the last digit of three digit reciprocals is always the same. In other words the reciprocal of xx7 is yy7. and so on. So, when you glance at the map and realize you need the reciprocal of 354, just forget about the 4 for a minute. Just find the reciprocal of 350, which is 170 (200/20 system.) So, the reciprocal of 354 is 174 (and the reciprocal of 352 is 172, etc.) Reincarnation - or "I've been here before"The above realization means that there are really only 18 pairs of reciprocals, NOT 360 as you may have feared. Therefore a viable strategy is to just memorize the 18 pairs. One of the things I have noticed is that most pilots do have numerous reciprocal pairs memorized but they don't recognize that the information is in their brain when the circumstances are not right. For example my airport has runways 33 and 15. I.E. 150 is the reciprocal of 330. What runways does your airport have? Commit these to memory. Once you have been flying for even a short while you will know quite a few airports and therefore quite a few runway pairs. Of course everyone knows the pairs 360/180 and 270/090 so you likely have at least five or six of the eighteen pairs already memorized. Use this information to your advantage to avoid calculations when flying. Bearing it
Sum it upThere is an interesting mathematical oddity caused by the very symmetry that makes the 200/20 system so easy for we humans. If you take any number and its reciprocal (except 290 and 110) and add up the digits within each number you get the same total. For example 150 and 330 are reciprocals - in 150 the 1 plus the 5 is 6 - in 330 the 3 plus the 3 is also 6. You can try this for any pair of reciprocals you want and you will see that it works - the only exception is 29x/11x. The Second Century has Such SymmetryThe sum it up technique is specifically useful when figuring the reciprocal of any course that is between 200 and 299. All you have to do is add up the first two numbers and that is the reciprocal. For example what is the reciprocal of 237. Start by forgetting the 7 - then add 2+3 to get 5. The answer is 57. Try it yourself, What is the reciprocal of 289. 2+8=10, So, the answer is 109. Try as many as you want it works from 200 up to 299. On OddityIt is very important to note that the reciprocal of an odd number is an odd number and the reciprocal of an even number is an even number. But, unlike normal rules for determining odd and even the "odd" thing here is that what matters is the middle number. So for example if I am trying to think of the reciprocal of 130 I note that the middle number, 3, is odd. Therefore, the middle number of the reciprocal must also be odd. The reciprocal is 310, and sure enough 1 is odd. What is the reciprocal of 245? Because 4 is even I know the middle number must be even. The answer is 065 and sure enough the 6 is even. The odd / even analysis is one of the MOST IMPORTANT rules to use because, let's face it, you are most often going to figure reciprocals by using the "bearing it" system described above. But, when you do that the most likely problem is "crooked eye disease." In other words your eye wanders a bit and you think the reciprocal of 249 is 059 (an easy mistake to make.) You catch yourself on this mistake by realizing that 4 is even and 5 is odd, so something is wrong. Usually people make mistakes of only 10 degrees when "looking across the compass card." So, the odd / even test is very valuable. Combine it with a few of the other ideas above and reciprocals should be no problem for you. SummaryDon't rely on just one system of determining reciprocals. Do some memorizing (runway numbers) and a tiny bit of math (200/20 and sum it up, symmetry) and mostly just look "bearing it" across the HI. But, then use a system such as the "Oddity" test to confirm you did not make a mistake. While care in not making mistakes is a worthy goal. An even more important attitude for the professional pilot is to accept our human limitations and build cross checks into our procedures to catch errors. Other crew members are helpful in this regard when available. But, as a lone IFR pilot we must be our own cross checkers. So, develop an appropriate risk reduction strategy from the beginning of your piloting career.
|