Box 6-2 The Logic of Approximations

Many problems are difficult to solve without judicious approximations. For example, rather than solving the equation 

But how can we be sure that our solution fits the original problem  

When we use an approximation, we assume it is true. If the assumption is true, it does not create a contradiction. If the assumption is false, it leads to a contradiction. You can test an assumption by using it and seeing whether you are right or wrong afterward. 

You may object to this reasoning, feeling How can the truth of an assumption be tested by using the assumption Suppose you wish to test the statement Gail can swim 100 meters. To see whether or not the statement is true, you can assume it is true. If Gail can swim 100 m, then you could dump her in the middle of a lake with a radius of 100 m and expect her to swim to shore. If she comes ashore alive, then your assumption was correct and no contradiction is created. If she does not make it to shore, then there is a contradiction. Either the assumption is correct and using it is correct or the assumption is wrong and leads to a contradiction. (Another possibility in this case is that there are freshwater sharks in the lake.) 

In Example 2, the assumption leads to a contradiction, so the assumption cannot be correct. When this happens, you must solve the cubic equation (x)(2x + 0.030) = 7.9 X 10  

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