The whole one yard

Mathematician John Allen Paulos offers us the following food for thought:

What's wrong with the following not quite impeccable logic? We know that 36 inches = 1 yard. Therefore, 9 inches = 1/4 of a yard. Since the square root of 9 is 3 and the square root of 1/4 = 1/2, we conclude that 3 inches = 1/2 yard!

Think about that before scrolling down. (By the way Paulos does not answer his own question, leaving it to the reader to figure this one out.)
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The problem of course lies in the formulation of the conversion equation itself. We can perform various operations on both sides of a mathematical equation and still preserve the equality (e.g. multiply/divide both sides by the same value, or take the square root of both sides), but we must have a legitimate equation in the first place.

"36 inches = 1 yard" is a true statement, for indeed 36 inches make a yard. But it is not a mathematical equation. In that statement "inches" and "yard" are not variables; they are units of length. In order to write a valid conversion equation for this particular problem we have to spell out the variables and what these variables stand for, and know which variable is to be multiplied by the constant--36.

If we let
i = number of inches
y = number of yards

then the correct conversion equation is

i = 36y

As you can see this is--at least visually/graphically--the "opposite" of the statement "36 inches = 1 yard."

Let's now see what happens when we perform the mathematical operations that Paulos suggested.

We divide both sides of the equation by 4:

i/4 = 36y/4

i/4 = 9y

Then we raise both sides of the equation to the power of 1/2, i.e., we take their square roots :

(i/4)^0.5 = (9y)^0.5

Now that looks like rocket science so let's simplify it and see what new species we end up with:

i^0.5/2 = 3y^0.5

i^0.5 = 6y^0.5

Finally, let's pull out those dang exponents by squaring both sides:

i = 36y

Hey, we're back to square one! As it should be.

Now plug in 36 for every instance of i and 1 for every occurrence of y into all of the equations above and you should see that both sides of every equation will have the same value, meaning of course the equations are correct.

One lesson from this is, unless we can translate the problem into its algebraic form correctly, we can perform as much valid math operations as we want to but we will, sooner or later, end up with trash.


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John Allen Paulos. 1988. Innumeracy: Mathematical Illiteracy and Its Consequences. New York: Vintage. p. 95.