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Precise calculations are often impossible due to mathematical or informational limitations. Estimates provide useful approximate answers, acting as a preliminary check for exact calculations; a significant discrepancy between an estimated and calculated result indicates an error. Order-of-magnitude estimates, approximating values to the nearest power of 10, are sufficient for many problems. Even crude estimates (within a factor of 10) offer valuable guidance; for example, estimating the number of people with a disease to within an order of magnitude can highlight significant public health concerns. In making these estimates, rounding numbers significantly (e.g., π ≈ 1) is acceptable; for better accuracy, systematically balancing over- and underestimation is helpful, even when some quantities are unknown.

Getting an exact answer to a calculation may often be difficult or impossible, either
for mathematical reasons or because limited information is available. In these
cases, estimates can yield useful approximate answers that can determine whether
a more precise calculation is necessary. Estimates also serve as a partial check if
the exact calculations are actually carried out. If a large answer is expected but a
small exact answer is obtained, there’s an error somewhere.
For many problems, knowing the approximate value of a quantity—within
a factor of 10 or so—is sufficient. This approximate value is called an order-ofmagnitude estimate, and requires finding the power of 10 that is closest to the
actual value of the quantity. For example, 75 kg , 102 kg, where the symbol ,
means “is on the order of” or “is approximately.” Increasing a quantity by three
orders of magnitude means that its value increases by a factor of 103 5 1 000.
Occasionally the process of making such estimates results in fairly crude
answers, but answers ten times or more too large or small are still useful. For
example, suppose you’re interested in how many people have contracted a certain
disease. Any estimates under ten thousand are small compared with Earth’s total
population, but a million or more would be alarming. So even relatively imprecise
information can provide valuable guidance.
In developing these estimates, you can take considerable liberties with the numbers. For example, p , 1, 27 , 10, and 65 , 100. To get a less crude estimate, it’s
permissible to use slightly more accurate numbers (e.g., p , 3, 27 , 30, 65 , 70).
Better accuracy can also be obtained by systematically underestimating as many
numbers as you overestimate. Some quantities may be completely unknown, b