🕒 5 min
Now, now. Lay down the pitchforks. The title of this post might seem all the more blasphemous to some of you due to its date of publication, but it allows us to introduce a very important topic – the power of approximation, and when it should and should not be employed.
It is important to note here that we’re not talking about the extreme, commonly joked about version of stating this post’s title, which would warrant picking those pitchforks of yours back up: π = 3. Not all “equals” signs are born, well, equal. In fact, there is only one equals sign and that is the one used above, incorrectly. What we’re talking about is equality’s cousin, approximation, and why there really are sensible ways to use π ≈ 3.
Approximation is a valuable skill that is, arguably, often neglected in traditional schooling. It is an extension of your natural feel for quantity. Humans aren’t naturally all that great at telling large quantities apart, however. You may have heard that we perceive quantity logarithmically, meaning that a difference of one item doesn’t always carry the same weight, but instead gets drowned out more and more as its neighbors grow more numerous. We also don’t have a good sense of just how large very big numbers actually are. Here, too, you might be aware of how shocking the difference between one million and one billion seconds is when you convert them to more workable units of time (if not, try to guess and then check). We don’t need to rely on guesswork, though – we’d like to try and make educated guesses instead. Approximating, and approximating well, is a skill that can be learned.
The value of an approximation depends on how close is acceptably close. This depends on what it is you’re doing and how much time you have. After all, every experimental measurement is fundamentally an approximation, constrained by the precision of the instrument. There is some room for debate because certain experimental results are so precise that they may well not be treated as approximations at all, but it is always good to keep their true nature in the back of your mind.
When it comes to pi, any decimal representation of it that you’ve ever seen obviously must have been an approximation. 3.14 is closer to the real value of pi than 3 is – by quite a bit, too – but it still undershoots. Even if you took all of the dozens of trillions of digits of pi that we have calculated so far, they would not be pi. However, they don’t have to be. There is a time and place for approximation. It is very important to keep formal, abstract expressions clear and precise, but science deals with the real world, and that means we’re always going to run into rough edges. No real-life circle is truly circular, no measurement can be infinitely precise and no finite amount of digits of pi can measure up to the accuracy of writing the Greek letter, π.
If you’re working with something in your daily life and need to perform calculations involving pi, plugging in 3.14159 is more than likely to be enough. If you ever have to deal with data analysis – say, the results of an experiment of some sort – and a numerical value is your output, approximating it beforehand helps you keep your work tangible. Even if whatever analysis you’re performing seems like it’s coming out of a black box, you have to know you can trust it, and knowing what to expect is a must.
And then there is the case of needing to quickly understand some amount with minimal requirements for precision. This is where orders of magnitude come into play. You can think of an order of magnitude as the smallest power of some particularly useful, round number that can be used to approximate some value. In scientific notation, this corresponds to the power of 10. You don’t necessarily have to think in terms of powers of 10, though. If you need to get a rough idea of just how much of something there is – say, if you’re buying something in bulk – before committing to more precise (and more time-consuming) calculations, you can use whatever base amount fits you best. All this also means that, if you’re using powers of 10, you don’t even need to think of pi as 3 – you can think of it as 1.
This is the true benefit of π ≈ 3 or π² ≈ g or whatever best suits your particular problem – approximations aren’t necessarily better if they’re more precise, they are better if they help you reach a useful conclusion. They improve upon purely estimating a quantity based on a gut feeling (which is, however, also useful for those things that can only be estimated, but that’s a story for another time). Just make sure not to claim false equalities. Or else – pitchforks.
Did you enjoy? Learn something new? Or maybe you just wanna brag about how many digits of pi you can name off the top of your head? Whatever it is, feel free to let us know!
Finally, if you didn’t get the opportunity to celebrate pi’s most widely known approximation with a homophonous dessert this March 14, perhaps you can relish in the knowledge that July 22 might be a good excuse to make up for the experience.
Here are some common (and some less so) problems for you to build your approximation muscle!
How many digits of π would you need to get the length of the equator accurate down to the diameter of a hydrogen atom?
How many carbon atoms fit onto the diameter of a human hair?
How much heavier are you than a coronavirus particle?
If all people alive today started running at the same speed, how fast would they have to run for their kinetic energy to equal that of the annual output of the Sun?
And finally, a must-feature: