Before we get into pi, we need to explore a bit of other mathematics first. Let’s start with a couple of sets of numbers, beginning with rational numbers.

Rational numbers

Rational numbers are numbers that can be expressed as a fraction, and this group most famously includes the counting numbers (also known as whole numbers) that we first learn as toddlers. Wait, you might say, the counting numbers are not fractions. You would be correct; however, they can be expressed as fractions. For example, our first counting number, 1, can be expressed as 1/1. Remember that fractions consist of a numerator, which is the number on top of the line, and a denominator, which is the number at the bottom of the line. The word denominator gives insight into another way to think of fractions: division. Indeed, we know that 1 divided by 1 is 1. Your toddler at preschool might even become aware that rational numbers can not only be expressed as fractions but they can also be expressed as infinitely many fractions. For example, 8 snacks for 8 preschoolers will luckily work out as 1 snack for each child. Of course, the laws of mathematics do not always apply to children, so the picky kid who refuses to eat their snack might mean the other 7 kids each get a bit extra for the day. Or perhaps a lucky sibling at pickup will get a nice surprise.

At any rate, these numbers make a lot of sense to us from a very young age. That also just happens to be a perfect definition of the word rational --- it just makes sense. It isn’t long after 1 snack for each of us that other numbers start making sense too. A pizza with 8 slices means 4 slices each for 2 friends. The sticky blob vending machine at the grocery store costs a quarter, which is 25 cents, which is ¼ of a dollar, which can also be written as 0.25. Ironically, the cost of getting the cheap goo out of your car seat later is much greater.

Later, if we write a check for more money than we have in the bank, we might really start to understand the concept of negative numbers (the whole/counting numbers and their opposites, the negatives, are also referred to as integers). However, as Mr. Escalante famously told us, when we dig a hole at the beach, we have just created a negative number. Pour back in what we took out, and of course, we have filled the hole, which is why whole numbers (positive counting numbers) and their opposites (negative numbers) are always equal to 0. That makes sense. It is all very rational.

Irrational numbers

Enter the world of irrational numbers. After our introduction to rational numbers, one might conclude that irrational numbers do not make sense, and that is indeed a very accurate description of this set of numbers. Irrational numbers cannot be expressed as a fraction, and this creates a mystery for our human minds. Having grown up dealing with rational numbers embroiled in our daily lives, it is hard to imagine what exactly might make an irrational number.

Along comes pi

So, what exactly is pi? Interestingly, while pi cannot be expressed as a fraction, it does derive its circular origins in division. Sometime between 287 and 212 BC, Archimedes of Syracuse is the first known mathematician (we will save the concept of more ancient knowledge for another time) to have stumbled upon pi and some of its mysterious properties. When he measured the circumference of a circle (the distance around a circle’s perimeter) and divided it by the diameter of the same circle (the distance across the middle of a circle, from one side to the other), he got a most unusual number. A number that he could only approximate to being somewhere between 3 1/7 and 3 10/71. Imagine his surprise when he performed this simple act of division for a circle of a different size, then another, and another, and another, and each circle always yielded the exact same result! For most of us, when rounded to the nearest hundredth, pi becomes its most recognizable decimal equivalent: 3.14 (let’s revisit the decimal equivalent of pi in a moment).


One of pi’s mysterious properties lies in its dependability. Archimedes must’ve been shocked at seeing this odd number (odd in a strange way, not in the way of non-even whole numbers)be calculated every time he performed the division of circumference and diameter. It also must have dawned on him that he had stumbled upon a universal constant. The notion of constants is a very important part of science, and this would later be understood by the general population through the popularity of Einstein’s theory of special relativity: E = MC2. The universal applicability of pi is one of the reasons why many scientists believe that if we ever communicate with extraterrestrials, it will be through mathematics. Should Plutonians, Uranians, or extraterrestrials travel to Earth, it is doubtful they will be proficient in English, French, or Spanish. However, one language they almost certainly will grasp to a far higher level than us is mathematics. You see, if an alien engineer 10,000 light years away is designing a fleet of various-sized flying saucer UAPs (UFOs) for interstellar travel, then the ratio of all their circumferences to respective diameters will, quite magically, be the same irrational number that one Archimedes of Syracuse stumbled upon some 2,200 years ago.

Beyond being a universal constant, pi also has other mysterious properties. Let’s revisit our friends, the rational numbers for a second. When we start dividing numbers, we realize that basically, one of two things happens.

  1. The resulting decimal number terminates when the remainder is 0. For example, 1 divided by 2 equals 0.5, precisely. We know this makes sense because the notion of splitting something in half is so easy for us to comprehend at some point (no pun intended).
  2. The resulting decimal number goes on forever in a repeating pattern. For example, 1/3 (1 divided by 3) is equal to 0.333333… if we care to do the division, we will always get another 3 added on to the end of the number. It doesn’t take humans very long to comprehend the rationality of repeating numbers that go on forever.

What makes much less sense is the notion of a number that goes on for infinity (∞) … yet never repeats itself in any discernable pattern! Yet that is exactly what pi does. When Archimedes began exploring pi, he was able to divide it out into 15 decimal places: 3.141592653589793. So far as we know, it was three years later that another Dutch mathematician, Ludolph van Ceulen, was able to calculate pi to 20 decimal places: 3.14159265358979323846. As of this writing, pi has been successfully calculated to over 62 trillion digits. Despite going on for more digits than is comprehensible by our minds and knowing that it goes on for infinitely more (again, an astounding concept) the next number in the sequence of pi cannot be accurately predicted. Because there is no pattern, predicting the next number can never be done without calculating it exactly, and any other prediction would simply be a random guess. Thus, pi’s exact value can never be known. It goes on forever with no pattern.

Knowledge of everything

Albert Einstein and Stephen Hawking both sought to unify all of existence within a singular equation of everything. Although they both made incredible breakthroughs in explaining our universe with equations that are beautiful and elegant, neither was able to come close to uniting everything with a solitary equation.

I can’t help but wonder:

  • Given that there are primordial mathematical constants in the universe such as pi that defy all attempts at rational sense, and…
  • Given the concept of pi remains far beyond our grasp, it could have begun to be derived from a Neanderthal’s circular stick sketch in the clay.

Will we ever be meant to comprehend more than an infinitesimal portion of this universe? Pi tells us that no matter how much we know, no matter how much we discover, there will always be one more digit that remains unknown.