Finite and Irrational Numbers
Date: Fall 2013
Was reading the Pi and Finite questions and answers in the Mathematics Archive. Interesting as I have always pondered the same question. Most answers were leaning toward no number is finite. Here is another perspective. Let us say you have on paper a given whole number which corresponds to perimeter of a square.You make the square out of yarn, the exact length is a whole number, and is known. You take the same yarn and make a perfect circle. Now you know the circle's circumference is an exact whole number. Why is it not possible to mathematically come up with an exact whole number value as in the square? Or, if you make a perfect circle out of yarn, then open it up straight and measure it, why can math not achieve the measured exact whole number?
Thanks for the question. Let N be a whole number which corresponds to the perimeter of a square. Then N/4 is is the length of one side of the square. N/4 may or may not be a whole number, but such a number exists. The length N can be the circumference of a circle. In terms of the radius of said circle, the relation is N = 2*pi*R. Since pi is irrational, R must be irrational.
Let me state the equation for the circumference, C, of a circle in terms of the radius R: C = 2*pi*R. If R is irrational, say 1/pi, then C could be rational. You cannot have both C and R being rational.
I hope this helps.
I may be misinterpreting your question, you seem to be arguing that a circle cannot be an exact or whole number. That's not what pi is suggesting. Since pi is the ratio of the circumference of a circle to its diameter, and pi being an irrational number, then either the circumference (C) or diameter (d), in the ratio of C/d = pi, can be exact numbers -- but not both at the same time. So while C = 22 (exact), d is nearly exact at 7.002.... In your examples, you can get the circumference to be an exact number, but the diameter will turn out to be inexact.
Greg (Roberto Gregorius)
This is a problem of relating numbers to measurements. If you define a finite number as that which can be measured, then no number can truly be finite. A yardstick is composed of atoms, and an atom does not have an exact size. In fact, the yardstick will be at a certain temperature, and the atoms within the yardstick will be randomly moving to some extent so that those at the very edges will not be even fixed in location. You could conceivably get a very accurate length, down to the rough size of an atom (even below), but mathematics requires perfect accuracy. Such accuracy does not exist in the physical world, and as such, represents the flaw in using the concept of a physical measurement for an abstract concept. The Greeks recognized the flaw in relating numbers to lengths, but they could not resolve it. Consider, though, the problem like you stated it. Assume a rope has a finite length, say 5 meters long. We have to assume that it actually can be exactly 5 meters long. I have made the argument for physical reasons that it cannot, but let us consider a “perfect” rope exactly of this length. The problem is not that we join the ends together into a circle; we would still assert that the circumference is a finite length. But the diameter of the circle will be circumference/pi. If we take another rope and cut it to the length of the diameter of the circle, it cannot be of a finite length. We assert, though, that it must be because we cut the rope to that length. Thus the paradox. The same problem happens with a right triangle. A right triangle, as you recall, has a 90 degree angle, and if its two sides are exactly 1 meter long, its hypotenuse (diagonal) has a length of the square root of 2. Thus the two sides would be defined as finite, but the diagonal would not. Our perfect ropes we could cut to each of these lengths and thus be finite in length.
Again, this problem came from the Greek notion of relating the physical concept of measurement to the abstract concepts of both geometry and numbers. Mathematicians have since realized that numbers are truly abstract, and when we relate numbers to the physical world, we can additionally have numbers that do not correspond to the physical world in the exact sense. If we drop the concept of “finite” as being “measurable,” however, we can define finite as “bounded or limited in magnitude.” In other words, “finite” is the opposite of “infinite.” In this sense, a circle’s circumference and diameter are finite but not both measurable at the same time.
Kyle Bunch, PhD, PE
You need to distinguish between ?measuring? and ?geometry?. In measuring, there is no such thing as an ?exact number?. Putting the shape of the object aside, the distance between the beginning and the end of the distance between two points (which are also not ?points? if you microscope is powerful enough) is at best the size of a few nanometers. Remember that the sheet of paper you use to draw the line is a mountain range on the microscopic scale. So as an object of ?measurement? there is no such thing as a ?perfect? anything.
The counterpoint is ?geometric shape?. Geometrically a square (or rhombus) has four sides that are ?exactly? the same length. This is a mental construction, not an engineering construction. The ratio of the opposite sides of a square or rhombus is EXACTLY UNITY (the mental construct).
In the case of a circle, the ratio of the circumference, C, and the diameter, D, (C/D), is the irrational number ?pi?. It does not matter how accurate your ?measuring? device is, ?pi? can never be written as an ?exact? number.
These are fundamentally different concepts ?physical measurement? and ?mathematical concept?.
Thanks for your question.
In a circle, the circumference = pi*D where D is the diameter. In the usual examples, the diameter might be a whole number and the circumference pi*D would be irrational.
But in your example, you are starting with a circumference as a whole number. There is nothing wrong with that. But now the diameter = cirumference/pi so it will be an irrational number. In a circle, either the diameter or the circumference must be an irrational number. It is also possible that they can both be irrational.
I hope this helps.
The circumference in that case can be an exact whole number, but the diameter and radius cannot. If the diameter is finite, then the circumference is not. If the circumference is finite, then the diameter is not. This is because pi, which is not a finite decimal, is circumference divided by diameter.
Dr. Ken Mellendorf
Illinois Central College
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