Lab1_Quantitative Reasoning 1_3.pdf - QUANTITATIVE REASONING Understanding the Mathematical Patterns in Nature Second Edition Frederick P Greenleaf

Lab1_Quantitative Reasoning 1_3.pdf - QUANTITATIVE...

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QUANTITATIVE REASONING Understanding the Mathematical Patterns in Nature Second Edition Frederick P. Greenleaf Department of Mathematics New York University The McGraw-Hill Companies, Inc. Primis Custom Publishing New York St. Louis San Francisco Auckland Bogota Caracas Lisbon London Madrid Mexico Milan Montreal New Delhi Paris San Juan Singapore Sydney Tokyo Toronto
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1.3 Dealing with Units. 41 3 Real World Measurements: Dealing with Units. I am often asked "Why do we have to deal with units? They're such a pain!" One answer is to refer you to the NY Times article reproduced on the previous page, which shows that bad things can happen when people fumble with units. Perhaps someday soon we will all have inexpensive calculators that handle units for us, 1 but until that millenium arrives there is no way to _ avoid this issue. Numbers in the real world often come with units attached. For instance, should you travel to France you would face a new language, and unfamiliar units such as francs, kilograms, liters, etc. To function in that new enviornment you must know how to translate these into familiar units such as dollars, pounds, gallons. Without that ability you are helpless. If we restrict our attention to pure mathematics, without reference to the world around us, we could work with pure numbers; then statements like 3 + 4 = 7 3 • 4 = 12 would have an unambiguous meaning. However, if we want to talk about the outside world we have to go out and make measurements, and these come 1;ifith "units" attached. For example, suppose I tell you that the Greeks knew the Earth is round and that its diameter is 1580. Do you have any idea what the Greeks thought, just knowing the number 1580 without its units? Even if I tell you that they meant "1580 stadia", you're still in trouble. How does the Greek unit of length, the "stadium", relate to our miles or kilometers? For better or worse, dealing with units is inescapable, and we are going to take the time to face this issue squarely. We will always indicate units in square brackets following the number. Examples: The speed of light is about 186,321 miles per second: 1.863 x 10 5 P s ti . The area of that roof is 40,000 square feet: 4 x 10 4 [ft2]. The concentration of salt in sea water is about 0.04 {-5] = 4 x 10 -2 [5] . The price of steak in Paris is about 101 [Francs] kg 1 Units cause little trouble if you are aware of a few simple principles. The first thing to keep in mind is that 1 Many PCs or Macs are already equipped with software that does this.
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42 1. Measuring Things in the Real World. You can only add quantities expressed in the same units. Thus 4 [mi] + 3 [mi] = 7 [mi], but a statement like 4 [mi] + 3 [km] = 7 [???] makes no sense at all. However, the last "calculation" works if you first convert both lengths into miles, or both into kilometers, which is easy if you have a table of conversion factors. Such information can be found in Appendix B, where you will find that 1 [mi] = 1.609 [km] Multiplying both sides of this identity by 4, we get 4 [ml] = 4 x 1.609 [km] = 6.436 [km] and this yields a result that makes sense: 4 [mi] + 3 [km] = 6.436 [km] + 3 [km] = 9.436 [km] .
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