You Cannot Square a Circle. “…so, they spend all this money, but they’re not doing it right,” my friend Mitch said. “I know there’s a big business there. All we have to do is square the circle.”
“Huh? I interrupted. “What does square the circle mean?” You can’t turn a circle into a square!”
David was waving his hand like that obnoxious kid in your sixth-grade class who always had the answer. “I know, I know,” David said, barely able to control himself.
“Okay, David, tell us… but please spare us the long mathematical explanation” I begged.
Even though David’s a math savant, he doesn’t get to show off his knowledge very often. He could barely control himself.
“That a circle cannot be squared was not known to the ancients because the Greeks didn’t grasp the concept of irrational numbers,” he started. “Indeed, one of the Pythagoreans was killed for attempting to explain that the square root of two is indeed an irrational number. But before considering irrational numbers like the square root of two or pi, let’s consider rational numbers. Rational numbers are the RATIO of two integers.”
I already regretted asking David to explain. But it was too late.
“Rational numbers come in two flavors, those that repeat and those that terminate. 3/4 is the ratio of the integers three and four. Its decimal equivalent is .75. 3/4 terminates. 2/3 is a rational number that repeats. 2/3 equals .66666… The sixes repeat forever. 2/3 does not terminate.
But irrational numbers neither terminate nor repeat. The square root of two is equal to precisely 1.414213562.”
“I’ll bet you didn’t have to look that up,” I smirked.
David nodded and continued: “Pi is also an irrational number. The infinite decimal expansion of pi neither terminates nor repeats. The first few digits of pi are 3.14159265358979323846264338327950288. I could go on.”
“Please don’t” we said in unison.
“But you get the point. Unlike 3/4 which terminates, unlike 2/3 which repeats, irrational numbers neither terminate nor repeat.”
We didn’t get the point but he was on a roll.
“Now that we have distinguished between rational and irrational numbers, we have to distinguish between two kinds of irrational numbers. The square root of two is an irrational number. Pi is an irrational number that is also transcendental. Irrational numbers, like the square root of two, can be the answer to quadratic equations. X squared -2 equals zero has an answer. The answer is the square root of two. But there are no quadratic equations that have pi as an answer. Pi is not only irrational but transcendental.
The area of a circle is pi times the radius squared. A circle with a radius five has an area of 25 pi.
The area of a square is the length of a side times itself. The area of a square with a side of eight is equal to eight squared or 64.
Squaring the circle means finding a circle whose area is exactly equal to the area of a square using only a finite number of steps. Since the area of the circle will always be a transcendental number and the area of a square has to be an integer, this can never happen in a finite number of steps. Therefore, you cannot square a circle. It’s a metaphor for that which cannot be done.”
“You mean it’s impossible,” said Mitch. “Why didn’t you just say that in the first place?”
David was knowledgeable. Mitch was erudite.
Which are you? Are you presenting your brand, your business, and yourself in simple terms that your customers can understand? Or are you wrapping yourself in acres of tiresome talk? Are you using big words when small ones will do? Are you using long rationalizations when simple examples offer more clarity? Are you writing an SAT essay when you should be tweeting a competitive advantage?