Chapter 12: HABITS OF MIND [12-25] & [12-26]




  • 日常的な長さ・重さ・時間を概算.
  • 地図から距離と移動時間を概算.
  • 縮小図をもとにして物体の実寸を概算.
  • 日常的な状況の生起確率を,過去の記録(たとえばあるフットボールチームはこの10年間に開始試合で8勝しているなど)やありうる場合の数(たとえばサイコロには6つの面があるなど)から概算.


  • 和・差・積・商・比率・パーセンテージの大まかな概算を求めること.
  • 概算と計算結果が大きく食い違う要因を把握すること.
  • 数の大きさがおおよそ10の何乗くらいにあたるのか特定すること.たとえば世界の人口は「おおよそ」10の9乗の桁(10億)か10の10乗の桁(100億)である.数字が「1桁ぶん」大きくなったという場合,じっさいの大きさはもとの数字におおよそ10ぐらいの因数をかけたものとなっている──すなわち,4〜5倍から 20〜30倍の大きさになる.これがたとえば40や200〜300の因数なら2桁の違いになるだろう.




There are many circumstances in which an approximate answer is as useful as a precise one. Indeed, this may be the rule rather than the exception. Estimating approximate answers can often take the place of making a precise measurement or a careful calculation but in most cases will serve as a check of calculations made using electronic calculators or paper and pencil. Skill in estimation is based on a sense of what an adequate degree of precision is in a particular situation, which in turn depends on understanding the context of the problem and the purpose of the calculation. Among particular estimation skills, everyone should be able to estimate the following:

  • Familiar lengths and weights, and also time periods.
  • Distances and travel times from maps.
  • The actual sizes of objects, based on the use of scale drawings.
  • Probabilities of outcomes of familiar situations, either on the basis of history (such as the fact that a certain football team has won its opening game eight times in the last 10 years) or on the basis of the number of possible outcomes (for example, there are six sides on a die).

It often happens that an answer shown on a calculator is wrong because the information entered was wrong, the information was entered incorrectly, or the wrong sequence of operations was used. In situations where there is no basis for judging the appropriateness of an answer arrived at by calculation, everyone should be able to figure out a rough estimate of what the answer ought to be before accepting it. This involves being able to do three things:

  • Carry out rough estimates of sums, differences, products, quotients, fractions, and percentages.
  • Trace the source of any large disparity between the estimate and the calculated answer.
  • Specify a magnitude only to the nearest power of 10. Thus, the population of the world is "on the order" of 109 (a billion) or 1010 (ten billion). Something that is improved by "an order of magnitude" changes by a factor of about 10—that is, anything from 4 or 5 times to 20 or 30 times larger (or smaller). A factor of 40 or a few hundred, for instance, would be more like two orders of magnitude.