Stop Memorizing Pi!
- MathBait
- Mar 14
- 10 min read
Happy Pi day! March 14th (3/14) has gained significant attention, inspiring students across the globe to learn more about the mysterious number π. However, as classes celebrate with circular treats, finding both joy and scrumptiousness in math, π day has become a bit of a novelty – missing the forest for the trees.
One such activity is challenging students to memorize as much of π as they can, with a young student holding the record of 111,700 digits! Computers are also working on learning more about π; they've beat humans, calculating its value to over 105 trillion digits.
And while we love all this enthusiasm over numbers – it is crucial to stop and ask ourselves: What's the point?
In this article, we explore who π really is, where we can find them, and provide some interesting π day activities to take us away from memorizing digits and into understanding and appreciating the power of numbers.
Who is π?
To answer this question, let's watch a short clip from the television show Person of Interest.
Pretty amazing, right? But is it true? Does π really contain every other number, every birthday, every book ever written? Not exactly.
The first problem we encounter is π is not the only irrational number, and as such, π cannot contain any of its friends. Some other famous unhinged numbers include Euler's number, e, and the golden ratio, φ. Since these are all just like π, never ending and never repeating, we can't find them within π's digits. We also cannot find π themselves within π. So, π doesn't contain every other number.
However, it is believed π contains every finite piece of information, although it has not been proven. Mathematicians believe this is the case because π seems pretty normal. In other words, if we look at what we do know, every digit (0-9) has about an equal probability of being placed as one of π's digits.
That means, your birthday is in there. But what might be more interesting is - what comes right after your birthday? And although we believe π is likely normal, we have no certainty that, say after 14 google digits, weird things start to happen.
Want to explore this idea more? Check out the Scientific American article: Do the Digits of Pi Actually Contain All of Shakespeare?
Now this certainly makes π seem quite interesting, so it makes sense why we have computers working overtime to find more and more digits. And since π is found in all sorts of intriguing places, the more we know about them, the more opportunities to better understand not just π, but irrational numbers in general. We can then take this understanding and use its powers in engineering and design.
Even if π contains all finite pieces of information, it still doesn't tell us who π is (apart from maybe everything!) or why we should stop trying to memorize their digits. Before we get to that, we need to explore another way to think about π.
Not a Number but a Ratio
When trying to figure out who π is, it makes sense to think about how π was discovered. We can understand the number 3, we've seen it. There are 3 pencils or 3 carrots or 3 math books, easy. But π? When have we ever seen that long-tailed monstrosity?
Honestly, you've probably seen π often, but just didn't notice them. And while π might have been out of reach, for thousands of years, humans did identify a strange ratio that seemed to pop up again and again. In fact, it even appears in the Bible (1 Kings 7:23).
This strange constant value seemed to love circles. They'd bask in the glory of the round and we could be sure to find them hiding there. So, in an attempt to pinpoint this mysterious ratio, Archimedes devised a clever way to catch them by inscribing and circumscribing polygons in circles.
For a fun fast-paced parody on Archimedes' process check out our previous Pi Day video.
While mathematicians had a good understanding of straight lines, curved lines were more challenging. To capture the mysterious ratio, Archimedes needed to come at it from both sides.
Noticing he could create polygons inside a circle (where each vertex lies on the circle - what is called inscribing), by increasing the number of sides he would arrive at shapes that looked more and more like a circle, but always less than a circle.
Similarly, if he made polygons around the circle (called circumscribing) he could attack the problem from the other direction; knowing the distance around the circle was always less than the distance around his polygon.
This led him to the rational approximation of 223/71< π < 22/7.
Ben Orlin, author of Math with Bad Drawings, argues we should take from Archimedes in celebration of π day. Since it is difficult to find 223/71 on a calendar, he suggests we instead celebrate π on July 22nd in honor of this 22/7 approximation. Seeing as 22/7≈3.142857...* it is closer to π than 3.14 alone.
Biblical authors, Archimedes, Zu Chongzhi, Ramanujan, and others, all studied how π relates to a circle, and using the circle, they were able to get a pretty good estimate as to who π is and where they like to hide. So while we might not have π LEGO, or π pairs of shoes, we actually do encounter this number quite often. In your pots and pans, in your water bottles, in your tires, and your cup holders . . . π is there hiding.
But is this all π is? No way! In fact, restricting π to the circle is a very dangerous idea . . .
*Want to meet 22/7? Check out Marco the Great and the Mystery of Phaseville where we meet this feisty rational number and learn just how many digits occur before they repeat.
Explore how quickly a polygon begins to look like a circle in the activity below!
©MathBait created with GeoGebra
When is π not π?
If that title made you scratch your head - good! While π is a dark and mysterious creature, it is just a number. So asking when π isn't π is like asking when 2 isn't 2. Yet, the way we describe π, and how it was first discovered, can allow for some devious twisting of reality.
As the first explorations of π concentrated on where it was found - the ratio comparing circumference (how far it is for an ant to walk around the circle) to diameter (the length across the circle passing through the center) - too often we end up reducing π to just this.
Discovering a tree in the forest might lead one to say, "Hey, trees are in the forest!" But extrapolating that to claim trees are only in the forest or defining a tree as "something in a forest", is risky business. Look outside - I bet you can find a non-forest tree!
And just like a tree, while π does like to hang out in a circle, we can find π without any circles at all!
Here's a new way to find π: take every unit fraction with an odd denominator, alternate between adding and subtracting your terms, and finally multiply the whole thing by 4. You'll find π!
π=4[1-(1/3)+(1/5)-(1/7)+(1/9)-(1/11)+...]
No circles needed.
Alas, some like to take advantage of finding π in a circle to make us think sometimes π isn't π. This isn't true. π is a number, just like 2, and it is always that long tortuous figure we are familiar with (3.14159...).
But what is true, is that geometry is amazing and can push the way we think about things.
A metric is a way we select to measure distance. In school, we learn the Euclidean metric and this is what allows us to find π in circles. But just as we can measure in inches or centimeters, we too can think of different ways to interpret distance. There are many metrics that are fun to explore, and in each one we have the chance to discover if π (the number) is hiding in circles over there too.
This video from PBS explores this idea. While it is interesting - remember π is always π! Although slightly deceiving, the video investigates different metrics and looks to find the ratio of the circumference to diameter in each. In doing so, they discover π only hides in Euclidean circles.
(BTW π is never 3.14)
Want to play with the idea of the Taxi Cab Metric? In this activity, you are the taxi cab driver. Collect fares and try to take each customer to their destination in the quickest way possible! Calculate your taxi cab distance then find the Euclidean distance.
How Do I find Euclidean Distance?
©MathBait created with GeoGebra
Why Stop Memorizing Digits?
Now that we understand a little more about π, the number, the ratio, and how it relates to Euclidean circles, we are finally ready to tackle why we should stop encouraging memorization of its digits.
One of the most interesting things about irrational numbers is that they go on and on forever, never stopping, and never repeating. This is a very hard concept for humans to grasp. When things get big, humans often group all things "large" into one category. One way to help us get an idea of just how big something is, is with time.
For instance, how long do you think it will take you to count to 1 million? If you do the math, it ends up being about 11 and a half days.
Now, how long do you think it will take you to count to 1 billion?
Tell Me!
Let's take this same approach with π. Suppose you were to memorize the first 100 digits. Think of each digit as 1 second. That means, you are getting a tiny glimpse into π's life. And it's only the first minute and 40 seconds!
What if someone saw the first minute and 40 seconds of your life - would they know much about you? Who you are? What you like? Your talents, your skills, everything that makes you you? Probably not. Even with 100 digits, all we are seeing with π is the nurses sucking out the fluid from their mouth and nose and handing the little bundle of joy back to their parents!
While memorizing 100 digits of anything is an impressive feat - it doesn't help us at all to know π, to understand who π is, or to find fun and clever ways to work with π!
Does Memorizing π Make Calculations More Precise?
We asked a student who has been memorizing digits when this long string would come in handy. They smartly replied, "in finding a more accurate circumference". But is this true?
Not really. If you have a circle with radius 3, the circumference (how far around the outside) is 6π. That is the absolute most accurate you could be! Because in that little symbol, we hold all of π, not just the first 100 seconds of their life.
But in reality, we can't work with π. It's too long. If I want to build a Ferris wheel, I can't have my workers up all night calculating digits. I need an approximation, a good one that will minimize my error.
But be careful!
Error compounds.
If I decide to use 3.14 instead of π, when I multiply by 6 I am multiplying my error by 6 too! With 3.14, 6π is approximated as 18.84. The actual value is closer to 18.85. This doesn't seem too far off. However, this 1/8 of an inch error could be disastrous for my Ferris wheel!
Wait... doesn't that mean typing the 100 digits I memorized into the calculator would in fact give me a better approximation? Yes it would! But you still shouldn't memorize it. In most real world scenarios we only need a few decimal places. Even if we used a 100 digit approximation, we don't have precise measuring tools that would even allow us to find 0.849555922 of a foot.
There's a better way!
Calculators generally have a π button. This little punchy guy takes only a second to press and already has π programmed in to the maximum precision your calculator will hold! So while typing the 100 digits you memorized would give you a better approximation, the π button gives you the absolute best approximation your calculator can perform in a fraction of the time!
If you find yourself in a pinch, without a calculator handy, 22/7 is a great approximation as well and also easier than remembering lots of digits. Don't forget, 22/7 was Archimedes' upper bound, making it a bit more than π. This means the value you find using 22/7 will be a little more than if you used the real π.
How much of π does your calculator know?
Every calculator is different. You might be interested in figuring out just how good yours is at approximating π. Here's how:
Hit the π key.
Since the first digit of π is 3, subtract 3. (π-3).
Make a tally mark to indicate you've tested 1 digit.
Multiply by 10. (You started with 3.14159.... Subtracting 3 gives you 0.14159. This step now gives all the digits a promotion to 1.4159...)
Subtract the next digit of π (in this case 1). Now you are at 0.41459...
Make a mark to indicate you've tested another digit.
Keep going!
Multiply by 10
Subtract the whole number
Make a mark
Repeat step 7
Eventually you will run out of digits! Count up your marks to see just how many digits of π your calculator can hold. We tested on an iPhone's standard calculator and found it stores 34 digits! Not bad Apple...not bad at all.
Conclusion and Extra Resources
π is a fascinating number. Its infinite digits are mesmerizing and its properties are fun to explore. And although π was first discovered in a circle, describing the ratio of the shape's circumference to its diameter, π is much more than just this! Finding π outside the circle is even more perplexing, interesting, and even surprising.
While we fully encourage celebrating π day (on March 14th, July 22nd, or both!) find a creative way other than memorizing its digits to explore this masterpiece. Memorizing digits only gives us the tiniest of a glimpse into who this fantastical creature is. To get a good taste of π, you'd want to know about 2,428,272,000 digits, and that's a lot more than the current world record!
We also challenge you to look outside the circle for π. π isn't just a ratio in a Euclidean circle, but a truly intriguing number that hides in the most unexpected of places! For more on this, consider Wired's article, Pi is Hiding Everywhere.
Here are some fun activities to try out this π day.
Want to meet π? Get the chance in Marco the Great and the Shape of Everything coming 2025.
Interested in learning more about numbers? Meet the Numberfolk in our award-winning math adventure series!
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