Date: 07/19/2002 at 04:51:41 From: Maxx Subject: Pi ratio Hi, I'm Maxx and I'll be taking Analysis 2 for math when I start 10th grade next year. I know that if a diameter of a circle is an integer then then the circumference cannot be an integer and vice versa, and that is why although pi is a ratio, it is an irrational. But then, why don't we just use the ratio (for example 2.4/4 - not really pi, just an example) when we calculate the area of a circle to get more accurate answers? Is it impossible to accurately measure circumferences?
Date: 07/19/2002 at 13:35:35 From: Doctor Ian Subject: Re: Pi ratio Hi Maxx, A subtle point, which a lot of people fail to appreciate, is that mathematical shapes are _idealizations_ of real shapes. Or to put that another way, real shapes are only _approximations_ to mathematical shapes. You can look at it either way (although philosophers would probably make a big deal out of which way you choose to phrase it). For example, you can't construct a perfect circle in the real world, if for no other reason than that a perfect circle contains an infinite number of points, and a real object contains a finite number of pieces. The formula circumference area pi = ------------- = -------- diameter radius^2 holds exactly _only_ for mathematical circles, not for real objects. If you try to apply it to real object, you run into several problems, which you're probably already aware of. One is that you can't make exact measurements! Another is that if the object isn't a perfect circle, the formula doesn't exactly describe the shape of the object. A third problem is that in many cases, you can't really measure the quantities you want to measure, because they don't really exist. For example, in a mathematical circle, every point on the circle is the same distance from the radius. That is, there is just the one radius, and it's the same everywhere. But now suppose you have an object that is approximately circular, and you want to find the 'radius'. Where should you measure it? Depending on how precisely you measure things, you'll get a slightly different value everywhere you look! So what _is_ the 'radius' of the object? (It might seem at first that 'circumference' would be immune to this problem, but it's not. Think about measuring the circumference of an island. If you use a yardstick, and measure points a yard apart, you get one value. If you use a foot-long ruler, you get another value. If you use an inch-long ruler, you get a third value. And so on. By determining the level of detail that you're willing to ignore, the length of your ruler determines the circumference of the object!) It's to get past problems like these that we agree to treat objects as if they corresponded to exact shapes like circles and squares. Then we can forget about the objects and just deal with the shapes to do our calculations. We give up a little accuracy this way, but we more than make up for it in convenience. Does this answer your question? Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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