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What is meant by reasonableness in math?

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In math, reasonableness refers to the results of a calculation or problem-solving operation reflecting what is reasonable within the context of the given factors or values. Two qualifiers of an answer's reasonableness are the order of magnitude within the framework of the problem and whether the results are either positive or negative. An answer can also be determined to be reasonable based on an estimate.

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In less complicated math problems, a rough and mentally-calculated estimate, sometimes referred to as a "ballpark estimate," can help to predict the numerical range the final answer should fall within. For example, if the given problem is to determine the sum of 23 plus 76, it can be estimated that the final and precise answer will be somewhere close to 100 because the known calculation 25 plus 75 equals 100 is close to the given problem 23 plus 76. Younger students are often taught to check the reasonableness of their final answers by comparing them to a ballpark estimate.

Another method of establishing the reasonableness of an answer is to compare it to a Fermi estimate. Named after the Italian physicist Enrico Fermi, these are estimates based on an understanding of the nature of the parameters involved and the known orders of magnitude relevant to them. Provided that the estimated answer is based on quantities that are known to be reasonable for the specific parameters involved, the estimated upper and lower variation limits will also be reasonable.

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