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Existence of Shapes with Irrational Dimensions

Asked by Jon Cypryk, student, A.N. Myer on October 27, 1997:
We would like to pose a question which we have been unable to reach an agreement on. The question is "can a six-sided, 3-D shape exist with a combined surface area of 600 square units? Two opposite sides must have an area of 100 each, two opposite sides with an area of 150 each, and two opposite sides with an area of 50 each."

The tentative answer given is a shape with measurements of a=square root of 75, b=150 divided by the square root of 75, and c=50 divided by the square root of 75. Then a × b = 150 square units, b × c = 100 square units, and a × c = 50 square units.

We understand that this formula works out mathematically because the square root of 75 is rounded off for all intents and purposes because it would be insane not to since it can bring you infinitely close as possible to an area of 600 square units.

However, due to an intense desire to prove or disprove the above statement, we are being precise, exact and very stubborn to the point.

Is it correct that if the length of one of the sides cannot exist (eg., a square of 75), then we could not find an area of one sides equaling exactly and precisely 100 square units? And if this is so then can the shape described in the opening question exist with an exact area of 600 square units?

The item in question would seem to be the length of the a side equaling square of 75.

The two opposing beliefs are:

- the belief that there is no true or exact number for the square root of 75 since it is irrational and continues on forever with no repeating pattern. Therefore side a cannot exist if we choose to be stubborn and exact to the point, which we must in order to get the exact result of 600 without rounding off.

- the belief that even though we cannot get to or see the end of the square root of 75, it it plausible that it can still exist and therefore it is plausible that side a can exist with an exact measurement of square root of 75.

An answer to this would be greatly appreciated if possible to help us prove or disprove the original question.

Thank-you in advance for your help.

Jon Cipryk

The real question you are asking is "do irrational numbers exist?" They most certainly do. And so yes, the shape described above does exist.

Several issues need to be addressed to clear up the confusion. First, there are different kinds of numbers. One kind of number is concept of "natural number": the sort of number used to measure "how many". If you were to ask the question "does there exist a number between 1 and 2?", the answer would be "no" if you were referring to the kind of numbers used in counting. For example, it is not possible to press a computer key more than once but less than twice.

However, that does not mean that the number 3/2 does not exist! It just means that it isn't the sort of number used in counting. It exists as a "rational number": a ratio of two integers.

In the same way, a number like the square root of 75 does not exist in the context of rational numbers (just as "half of three" does not exist in the context of the integers). But it does exist in the context of a different number system called the "real numbers" (just as "half of three" does exist in the context of rational numbers).

There are several ways to rigorously define real numbers. One way is to define a real number to be a sequence of rational numbers. So, for example, the sequence of rational numbers 8, 8.6, 8.66, 8.660, 8.6602, . . . defines the real number  (IMAGE) . Note that no individual number in that sequence defines  (IMAGE) (which is what you were getting at when you said that no finite decimal exactly equals  (IMAGE) ); however, the entire sequence taken together defines  (IMAGE) .

Another way to define real numbers is to define a real number as a partitioning of the rational numbers into two sets, where everything in the first set is less than everything in the second set. (Intuitively, such a partition corresponds to a location on the number line: the place where the first set ends and the second set begins). Now,  (IMAGE) corresponds to a perfectly well-defined partition of the rational numbers: for each positive rational number r, either  (IMAGE) or  (IMAGE) , and this distinction lets us separate the rationals into two classes, thereby defining a real number if you interpret "real number" as meaning "partition of the rationals into two sets with the appropriate properties".

Each of these definitions is quite abstract. (If you completely understand the previous two paragraphs, you should consider yourself exceptionally gifted in mathematics and I'd encourage you to consider it as a career). Therefore, they are not usually taught until about the third year of an undergraduate program. However, the important thing is that there are such things as "real numbers", that can be rigorously defined (though the definition is abstract and difficult), and within this collection of real numbers there is one whose square is 75. Therefore, the square root of 75 exists.

It's important to realize that these "real numbers" are not just an artificial mathematical construction but are precisely the kind of number system relevant for length measurements (just as the natural numbers are the kind of number system relevant for counting). As a consequence, the square root of 75 exists not just as an abstract mathematical entity, but as a real geometrical length.

One good way to see that some real, physically existing lengths can only be measured by irrational numbers is to think of a square with side length 1. The diagonal of this square (a length that clearly "exists") has length  (IMAGE) , which is an irrational number.

One final confusion that arises is this: there's a temptation to forget the distinction between a number and a decimal representation of a number. The number 75, for example, is an abstract entity that exists in its own right quite independently of the fact that can be written as the sum  (IMAGE) so that we can write it down as a 7 followed by a 5. When you come across a number like  (IMAGE) and observe that it cannot be written down as a finite sum of the above form, meaning that there's no finite decimal representation for it, that doesn't mean the number itself fails to exist. It just means it's a number that happens not to have a finite decimal representation.

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