Does Pi Equal 3.14?Date: 01/27/98 at 15:39:39 From: Adam Glover Subject: We use pi as 3.14. How can this be right? In my math class pi has always been 3.14. We use this to find area, volume, and any other equations using pi. If pi has an infinite number of digits, how can 3.14 be exact? I am very concerned about our society using 3.14 as pi when it has an infinite number of digits. Anyway, I was just wondering about this. Adam Glover Date: 01/27/98 at 16:08:32 From: Doctor Rob Subject: Re: We use pi as 3.14. How can this be right? You only need to use a value of Pi with as many significant figures as the other measurements you used for other values in the same equation. For example, if you wanted to measure the area of a circle, and the radius was measured as 36.2 feet (to three significant figures), the actual radius might be anywhere in the interval 36.15 < r < 36.25. Then the area would be Pi*1306.8225 < A < Pi*1314.0625. Even if we used 15 decimal digits for Pi, all we would know about the area is that 4105.50396554585 < A < 4128.24909635784. If we use just 3.14 for Pi, we get 4103.42265 < A < 4126.15625. In either case, we can't tell what the fourth significant digit of A is, and the third digit is probably a 1, but maybe a 0, 2, or 3. The extra digits of Pi don't help unless you make more accurate measurements. The rule of thumb is to use the same accuracy for your constants as for your least accurate measurement, and the answer will have the same accuracy. In this example, 3.1400000*36.200000^2 = 4114.7816 (exactly), but if we round off to three significant figures, we get 3.14*36.2^2 = 4110 as the best thing we can write for the area. Don't worry about society - engineers and mathematicians are well aware of this situation! If you want to use more decimals, you can do so, but your answers will still only have the accuracy of the least accurate quantity in the formula. We usually used 22/7 when I was in school, which is about the same accuracy as 3.14. If you want a better value, you can use 3.1416, or 3.14159, or 355/113, or 3.14159265, depending on the accuracy of your inputs. If the measurements are ideal, and hence exact to infinitely many decimal places, I would leave the answer in the form 100*Pi (if r = 10, say). The context of the problem should make it clear whether the quantities are exact or only approximate. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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