1.9 Approximations

In one of the first atomic bomb tests, a major question concerned the magnitude of the explosion. The exact answer required complex calculations involving parameters such as air temperature, air pressure and the pressure gradient at various points from the bomb. Scientists installed sensors and monitoring equipment around the bomb to obtain the numbers they needed to answer the yield question.

At the moment of the blast, one of the physicists threw a few scraps of paper in the air and noted how fast the wind blew the paper. Using a slide rule and a handful of approximations, he calculated the approximate yield of the bomb. A few weeks later, when the yield of the bomb was calculated with more exact methods, it was found that the two answers differed by only 50%.

The reason this type of calculation works is that the errors in simplifying assumptions are often independent of each other. In other words, if we guess a little high on one number and a little low on another number, the errors will tend to average out. Exact answers to questions such as What are the sidelobe levels of this antenna?, or Will my system work in this crowded spectral environment? frequently require expensive measurements or a precise knowledge of quantities that are difficult to measure. We can often provide answers which are approximate by making some simplifying assumptions and quick calculations. Approximations can also provide an understanding of what is reasonable.