Maths SolverToday is the fourteenth of March, which is $3/14$ in American date order, which is the first three digits of $\pi$, which is why every classroom in the United States has a pie on a desk and someone in a “$\pi r^2$” t-shirt. Pi Day is a fine excuse for a small celebration, but as it has spread, so have several pieces of folklore about pi that are either misleading or just wrong. This is the clean-up post.
Myth 1: pi was discovered recently
The number $\pi$ — or at least the relationship between a circle’s circumference and its diameter — has been known since antiquity. The Babylonians knew about it; the Egyptians used a value equivalent to about $3.16$ in the Rhind papyrus from around $1650$ BCE; Archimedes (around $250$ BCE) gave the first known rigorous bounds for $\pi$ by inscribing and circumscribing polygons in a circle.
What changed in 1988 was Pi Day itself, which was invented by a physicist named Larry Shaw at the San Francisco Exploratorium. That is the only thing that started in 1988. The number itself is older than writing.
Myth 2: pi is exactly 22/7
$22/7 \approx 3.142857$, which is close to $\pi$ but not equal to it. $\pi \approx 3.141593$. The two agree to two decimal places and disagree starting at the third, which is fine for ballpark arithmetic and useless for anything serious.
The reason this myth persists is that $22/7$ is a convenient fraction to remember and has been used as a teaching approximation for centuries. Archimedes proved that $\pi$ lies between $223/71$ and $22/7$. Some teachers truncated this to “$\pi$ is $22/7$,” which is the rough form most students encounter first.
The actual best simple-fraction approximations are $22/7$ (3 decimals), $333/106$ (5 decimals), and $355/113$ (6 decimals). The last one is genuinely impressive: it agrees with $\pi$ to seven significant figures using only a three-digit numerator and denominator. This is not a coincidence; it falls out of the continued-fraction expansion of $\pi$.
Myth 3: the digits of pi are random
The digits of $\pi$ are not random. They are the digits of $\pi$. There is exactly one infinite sequence of digits that constitutes the decimal expansion of $\pi$, and that sequence is fully determined — you cannot change a single digit and still have $\pi$.
What people usually mean by “the digits of $\pi$ are random” is something different and much more interesting: $\pi$ appears to be statistically random, meaning that if you sample its digits and run statistical tests on them, they pass the same tests as a sequence of digits generated by a fair die. Each digit appears with frequency about $1/10$. Each pair of digits appears with frequency about $1/100$. There is no detectable pattern in the digits.
Whether $\pi$ is “normal” in the technical sense (every finite digit sequence appears with the right frequency) is actually an open problem in mathematics. Numerical evidence strongly suggests yes, but no one has proved it. So the precise statement is “the digits of $\pi$ pass every statistical test of randomness anyone has tried.”
Myth 4: you need to memorise lots of digits
For working scientists and engineers, $3.14159$ is overkill in practice. NASA uses $\pi$ to about $15$ decimal places to navigate spacecraft to the outer solar system. The diameter of the observable universe in metres, computed using only $39$ digits of $\pi$, would be accurate to within the radius of a hydrogen atom. So unless you plan to compute the diameter of the universe, $\pi \approx 3.14159$ is enough.
Memorising more digits is fine as a party trick — the world record is over 70,000 digits, held by various competitors over the years — but it has nothing to do with maths. It is a memory sport that happens to use $\pi$ as the digit source.
What pi actually is
Definition: $\pi$ is the ratio of a circle’s circumference to its diameter. Pick any circle. Measure how far it is around (the circumference) and how far it is across (the diameter). Divide the first by the second. You get $\pi$. This works for every circle, which is the surprising part — the ratio is the same for all circles, large or small. That ratio is the number $\pi$.
This is also why $\pi$ shows up in formulas that don’t look like they have anything to do with circles. The area of a circle is $\pi r^2$. The circumference is $2 \pi r$. The volume of a sphere is $\tfrac{4}{3}\pi r^3$. The integral of any rotationally symmetric function in two dimensions involves $\pi$ somewhere. The Fourier transform involves $\pi$ because Fourier components are sinusoidal. The normal distribution in statistics has $\pi$ in its density formula because of an integral on the whole real line.
The pattern is: $\pi$ shows up whenever the underlying mathematics involves rotation, periodicity, or anything “circular” in a generalised sense. Anywhere those concepts appear, $\pi$ is basically guaranteed to be in the formula somewhere.
Pi is irrational, but not because of decimals
A common over-simplified explanation: “pi is irrational because its decimal expansion never repeats.” This is a correct consequence of irrationality, not a definition of it. “Irrational” means “cannot be written as a fraction $p/q$ for integers $p, q$.” The non-repeating decimal follows.
That $\pi$ is irrational was proved by Johann Lambert in 1761. The proof is non-trivial — it uses continued fractions and is beyond high school maths — but it has been checked and re-proved many times since. There is no possibility that $\pi$ is “really” some fraction we just haven’t found yet; it has been mathematically established that no such fraction exists.
$\pi$ is also transcendental, a stronger property: it is not the root of any polynomial with integer coefficients. This was proved by Lindemann in 1882 and famously settled the question of “squaring the circle” with compass and straightedge: because $\pi$ is transcendental, no such construction is possible. This was a centuries-old open problem and the proof is one of the genuinely beautiful results in 19th-century mathematics.
Pi vs tau
A small movement of mathematicians argues that the “real” constant should be $\tau = 2\pi$, the ratio of circumference to radius rather than to diameter. Their argument is that most formulas with $2\pi$ in them would be cleaner with a single symbol; a full revolution would be $\tau$ radians instead of $2\pi$ radians; the area formula $\pi r^2$ becomes $\tfrac{1}{2}\tau r^2$, which has the same shape as kinetic energy ($\tfrac{1}{2}mv^2$), revealing a structural connection.
This is a fun argument and there are reasonable points on both sides, but the practical situation is that essentially no one uses $\tau$ in real work. $\pi$ has 4,000 years of history and millions of textbooks; $\tau$ has been a serious proposal for about $20$ years. The replacement is unlikely to happen, but the debate is worth a few minutes if you have not heard about it. (Search for Michael Hartl’s “The Tau Manifesto” if you want the full pitch.)
So what is Pi Day actually for
Mostly it is for eating pie and making jokes about the irrational behaviour of mathematicians. Which, for a holiday celebrating a mathematical constant, is about the right level of seriousness. There’s also International Pi Approximation Day on $22$nd July, which is $22/7$ in day/month order, for parts of the world where Pi Day’s American date format does not work. Both are fine.
What I would not do is treat Pi Day as some kind of test. The point is not to memorise digits or wow your classmates with feats of recall. The point, if there has to be one, is to remember that mathematics has constants embedded in the structure of the universe that no one chose, that show up in every quantitative subject, and that are still being studied with new results every few years. $\pi$ is the most famous example. It earns the holiday.