Pi and Finite Measurements ```Name: Denis Status: student Age: N/A Location: N/A Country: N/A Date: N/A ``` Question: The value pi is an irrational number and never ending. If circles have areas and perimeters that are multiples of pi or just simply involve the value pi, doesn't a circle have a never ending length? Replies: There are two points here: Not only is pi irrational (its value cannot be expressed exactly as the ratio of two integers), it is also transcendental (it is not a solution to any algebraic equation that involves the arithmetic operations of addition, subtraction, multiplication, division, and taking "roots" of numbers). That doesn't mean that it is not finite, rather the ration of the circumference of a circle and its radius cannot be expressed as a number meeting the two negative stipulations above -- irrationality and "transcendentality". Sounds a little weird, but numbers are a lot weirder than they appear at first glance. Vince Calder Let us think about Pi, an irrational number, and the circle. An irrational number is a number that can not be written as one whole number divided by another whole number. A real number that is not a rational number is called an irrational number. The decimal expansion of an irrational number never repeats or terminates, unlike a rational number. You might think that if Pi is the ratio of the circumference to its diameter, then it must be able to be expressed as the ratio of two whole numbers; but it can not despite the fact that the circumference is of finite length. If you measure the diameter and express it as a whole number the circumference can not be a whole number. Therefore the ratio is an irrational number. If you do get a whole number, as you would for the perimeter of a polygon that approximates a circle, you are not dealing with a circle. Now consider this: The length of the perimeter of a polygon of N sides can be written as Perimeter=2*R*N*Sin(180 degrees/N) Where R is the distance from the center of the polygon to one corner. As N gets large, and approximates a circle, the Perimeter approaches 2*R*Pi, the equation for the circumference of a circle of radius R. For example For a 100 sided polygon The perimeter is 2R Sin (180/100)*100=2R*3.141 A circle is just a polygon with an infinite number of sides. Dave Kupperman Dear Dennis: Yes, pi is never ending. But extra digits are added after the decimal place. Every added place means we get closer to the real value of the ratio between a circle's radius and circumference. They do not affect the length as such. In other words, places added to pi make our answers that much closer to the real world numbers. At over a billion decimal places for pi, the difference between calculation and the real values is getting very small. R. Avakian Denis, Almost every measurement in the world is really a never ending number. Measure a stick. Your ruler may produce a length of 4.25cm. The stick might really be 4.249999678523987cm. You will never really know, but you can often see a certain "uncertainty" with your measurement. Was it right on the EXACT CENTER of the black line, or maybe just a little bit off? With digital devices, this uncertainty can be even harder to notice but is still there. A stopwatch is on 24.57 seconds. The watch reading indicates the watch was running for 24.57s, but it might have been 24.571234231199560782 seconds. The digital stopwatch will stay on 24.57 seconds until 24.58 seconds have passed. What this means is that we can never truly know exactly how big or small something is. We can be extremely close, but we will never know whether we are exact. Dr. Ken Mellendorf Physics Instructor Illinois Central College Denis, I am not sure if you are confusing an irrational number being 'never ending' with being infinite. Infinite would be a circle with a never ending length. But any circle you can draw or describe obviously HAS a limit to it's size, although trying to get a very precise measurement of that size might prove to be a little bit of a problem. That's where Pi being an irrational number comes in. Although the exact ratio of the circumference of a circle to it's diameter may be hard to define, it is always a constant value, slightly higher than 3.14. Okay, slightly higher than 3.14159. Make that a little higher than 3.14159265... By Irrational, it is simply meant that the precise value extends out an apparently infinite number of digits, but the value itself is by no means infinite. Ryan Belscamper Click here to return to the Mathematics Archives

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