CHAPTER 1: Introduction, Measurement, Estimating Answers to Questions 1. (a) Fundamental standards should be accessible, invariable, indestructible, and reproducible. A particular person’s foot would not be very accessible, since the person could not be at more than one place at a time. The standard would be somewhat invariable if the person were an adult, but even then, due to swelling or injury, the length of the standard foot could change. The standard would not be indestructible – the foot would not last forever. The standard could be reproducible – tracings or plaster casts could be made as secondary standards. (b) If any person’s foot were to be used as a standard, “standard” would vary significantly depending on the person whose foot happened to be used most recently for a measurement. The standard would be very accessible, because wherever a measurement was needed, it would be very easy to find someone with feet. The standard would be extremely variable – perhaps by a factor of 2. That also renders the standard as not reproducible, because there could be many reproductions that were quite different from each other. The standard would be almost indestructible in that there is essentially a limitless supply of feet to be used. 2. There are various ways to alter the signs. The number of meters could be expressed in one significant figure, as “900 m (3000 ft)”. Or, the number of feet could be expressed with the same precision as the number of meters, as “914 m (2999 ft)”. The signs could also be moved to different locations, where the number of meters was more exact. For example, if a sign was placed where the elevation was really 1000 m to the nearest meter, then the sign could read “1000 m (3280 ft)”. Including more digits in an answer does not necessarily increase its accuracy. The accuracy of an answer is determined by the accuracy of the physical measurement on which the answer is based. If you draw a circle, measure its diameter to be 168 mm and its circumference to be 527 mm, their quotient, representing , is 3.136904762. The last seven digits are meaningless – they imply a greater accuracy than is possible with the measurements. The problem is that the precision of the two measurements are quite different. It would be more appropriate to give the metric distance as 11 km, so that the numbers are given to about the same precision (nearest mile or nearest km). A measurement must be measured against a scale, and the units provide that scale. Units must be specified or the answer is meaningless – the answer could mean a variety of quantities, and could be interpreted in a variety of ways. Some units are understood, such as when you ask someone how old they are. You assume their answer is in years. But if you ask someone how long it will be until they are done with their task, and they answer “five”, does that mean five minutes or five hours or five days? If you are in an international airport, and you ask the price of some object, what does the answer “ten” mean? Ten dollars, or ten pounds, or ten marks, or ten euros? If the jar is rectangular, for example, you could count the number of marbles along each dimension, and then multiply those three numbers together for an estimate of the total number of marbles. If the jar is cylindrical, you could count the marbles in one cross section, and then multiply by the number of layers of marbles. Another approach would be to estimate the volume of one marble. If we assume that the marbles are stacked such that their centers are all on vertical and horizontal lines, then each marble would require a cube of edge 2R, or a volume of 8R3, where R is the radius of a marble. The number of marbles would then be the volume of the container divided by 8R3.
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