Imaginary Number Day

This article is a break from our “regularly scheduled programming" into a delightful coincidence that I stumbled across the other day. Some cursory googling on the internet revealed this isn’t a well-communicated fact, so I thought I would go ahead and share it with whoever picks it up here. If you’re the type who likes to learn random-but-interesting things, or you happen to be a math nerd like I was/am, you’ll probably enjoy this. If not, I’ll get back to more typical topics of writing next.

This little discovery came about because I was having a conversation with my wife Brittany and the kids the other day at dinner. Somehow the topic of “pi day” - 3/14 came up. Brittany was explaining that pi, the mathematical constant connected to a number of properties of circles - was celebrated on March 14th because its decimal form can be (rounded to) 3.14. Doubtless you’ve seen whatever math-nerd(s) you happen to know post about this on social media every March.

As she finished up her point, I remarked to her that it’s funny to me that in my opinion there is a strong case that could be made that if you really understand math, pi day should not be March 14th. For example, in Putty’s opinion there are at least two days that make a lot more sense to be pi day than March 14th (the next two sections are a diversion, feel free to skip lower if you want to skip the mathemagical details):

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March 4th

The first better option for Pi day is based on this observation: the 14th is a really weird way to represent .14 as a date. Pi day is selected on the fact that the digits on the month & day happen to correspond to the digits in pi - but this overlooks an important fact: the digits on the calendar are strange and arbitrary and not the way decimal digits are set up. Decimal digits are constructed on each digit representing a factor of 10 (or a tenth), each filled up with one of the ten options 0-9 to indicate how “full” that digit is on the way to the next factor of ten.

This isn’t the only way to write down numbers though, there are many ways numbers can be recorded using different representations. Two of the more commonly used schemes in our digital era are binary and hexadecimal. Binary represents a number using digits that only have two options instead of ten. Consequently, binary numbers are written using only 0s and 1s and might look like this:

       1001101 (= 77 in decimal numbers)

This is the same number, its just written differently. Numbers exist separate of the way we represent them. We can choose to write down the same abstract object with the decimal representation “77” or the binary representation “1001101”, but these are just handles for the same object.

If this is a new idea or you’re not mathematically inclined, a good illustration of this same idea comes up in foreign language. Anyone who has taken a foreign language in middle or high school has had the experience of choosing a name in that other language. When I was in high school I took Spanish and had to choose a Spanish name (oddly enough I don't remember what it was right now, but it wasn't related to Putty and it wasn't Roberto). The names “Putty” and whatever my Spanish name was were really just labels for the same person though; me. Neither name is actually who I am, they are both handles used to point to a person, and in different languages that handle may change, but they still point to the same person in reality.

This is what’s happening in these different representation systems. There is a number that exists separate of it’s written representation which is written as 77 in the system we call ”decimal”, 1001101 in the system we call “binary”, and be written in other ways in other systems as well (for example 4D in hexadecimal, a system that has 16 different possibilities in each placeholder).

So back to the calendar: the calendar is really wonky in that the month-day counting system is oddly arbitrary. There are 12 options in the month column that add up to one year, and there are a varying number of days that add up to one month. Consequently, the number 0.14 (the fractional part of pi) doesn’t have any real mathematical correlation with the 14th day in a given month. They happen to be expressed using the same numbers, but they’re numbers in different representational systems. Going back to the language illustration above, this is kind of assuming that because a word is spelled the same in two different languages that it has the same meaning in both languages. There are plenty of examples where that’s not true. Just for fun–the word spelled "gift" in German means poison, and it means married in Swedish.

What would be a more fair way to determine Pi day? Actually select the day that is 0.14 of the way through the 3rd month. This isn’t terribly hard to compute. Because there are 31 days in March, 0.14 of the way through the month is 0.14 * 31 = 4.34 days. So, if you want to pick the day that is 3.14 months into the year, March 4th is your day. But, it turns out there is probably even a better way to go than that.

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July 1st

What makes Pi a special number is its link to the behavior of circles. We first meet this as we learn to compute the area or circumference of a circle. Related to all of this is a natural method of measuring angles that is called radians. When we first meet angles we usually use degrees to measure them (another unit system that has 360 elements in one cycle). If your study of circular-like objects continues it often makes sense to switch the way you measure angles to a different unit which measures the circumference of any given angle tracked compared with the radius of that angle. This is a unit that is based on the actual ratio of physical elements of the circle, so it has a lot more natural properties than degrees, which are basically made up (probably to try and mimic the days of the year). It turns out when you do that ratio there are exactly 2pi radians in a full cycle (which is why the circumference of a circle is 2pi times the radius of the circle).

Now, given that the year is defined as one cycle around the sun, it would make a tremendous amount of sense to describe pi day as the day that we have traveled pi radians around the sun. This would be using pi to describe where we are in the (approximate) circle we travel in the annual cycle. Given this is exactly what pi is supposed to do; help describe elements of circles, it seems like it would make sense to use that property when determining pi day.

So what day is pi day then? It’s easy to compute; pi day is the day that we hit pi radians in our rotation around the sun for the year. Given there are 365 days in most years, this happens halfway through the 182nd day or on July 1st (at least not on a leap year).

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Imaginary Number Day

Okay, so all of that was just for fun and to warm us up to the idea. Back to the conversation with my wife. The fact that pi day is selected by arbitrary means (when if ever there is a day that should be selected using mathematical precision this is it) made me reflect on the fact that it’s also odd that pi gets selected as the special mathematical constant that gets its own day. There are other mathematical constants that are at least as interesting. The Euler's constant (e) comes to mind: the number that is naturally related to exponents and logarithms (e = 2.718...) Or another less known but incredibly interesting number, Feigenbaum’s constant; a number that has to do with the breakdown of orderly patterns into chaotic instability (δ = 4.669...).

If you dig through the nooks and crannies of mathematics, there are a number of fascinating constants. So far as I can tell the only reason pi gets selected is because it’s the one you run into first.

Musing on this made me realize the day I would pick if I was given the choice to make any day a holiday: imaginary number day. The imaginary number, i, is one of those things you run into that makes people want to entirely give up on mathematics. The imaginary number is described as the square root of negative one:

Now you probably learned when you first learned about squares and then square roots that there is no such thing as the square root of a negative number. You can multiply two by itself to get four, so the square root of four is 2 (or -2 as well), but there is no number you can multiply itself by and wind up with a negative number because a positive times a positive is always positive, and a negative times a negative is always positive as well.

Well, just because the square root of one doesn’t physically exist doesn’t mean you can't define it and learn to work with it, and that’s exactly the kind of thing mathematicians love to do. Once you go through the work of creating that abstraction it winds up being incredibly useful! i shows up everywhere and makes a lot of things much easier to figure out. One of the more remarkable relationships in mathematics is Euler’s Identity:

This fascinating relationship ties together a number of the most important numbers in mathematics into one relationship that is so beautiful it gives most people who can wrap their head around it the shivers. That’s the kind of mathematical beauty that will make people believe there may be a maker behind all of this after all.

Anyways, if I could choose a day to create, it would be imaginary number day, because that’s the exact kind of impishness that I love to partake in. Why shouldn’t we have a date that celebrates a mathemagical abstraction that can’t be described with any obvious direct physical parallel? I mean forget circles, that’s cool!

So I began to think a bit about what imaginary number day would be. How could one determine what day represents this non-physical entity? It wasn’t at all obvious at first. You can’t do anything with the digits of i, because i is mostly its own kind of digit that is different than any other one.

But then it came to me in a flash because I was thinking about how pi day could more properly be celebrated on pi radians around the sun. It turns out that imaginary numbers are also really useful when it comes to describing cycles and that made a particular selection obviously make sense.

Once you wrap your head around the idea of imaginary numbers, it makes a lot of sense to package them together with real numbers in a new idea called complex numbers. Complex numbers have a real component and an imaginary component. You can define rules for adding them and multiplying them and so on, and one thing that becomes very helpful to do is to put them on a chart and look at circles in the complex plane:

The points on the unit (radius = 1) circle in the complex plane fits within the box  ranging from -1 to 1 on the real axis and from -i to i on the complex axis. Strange as this might seem, this is standard fare complex number analysis. (We work with this all the time in physics).

Anyways, what all this means is that the unit circle in the complex contains the point i in exactly one well-defined place: at precisely one quarter turn of the way through the rotation. Since our annual rotation around the sun is not far from a circle, it makes perfect sense to define imaginary number day as the day we’ve traveled a quarter turn (pi/2 radians) around the sun since the beginning of our calendar year.

So what day is that? 

Well, with 365 days in the year, ¼ of the rotation happens a quarter of the way through the 91st day. What is the 91st day of the year?

April 1st.

That’s right, imaginary number day would be April fools day. April fools day!! The day we celebrate practical jokes and tom-foolery is the day we could be celebrating a type of number so ridiculous the best mathematicians could do is name it, "we made this one up."

And I’m not even joking. How perfect is that?

Forward this to a friend and help me get the word out there. Doesn’t this need to become a thing? I think this needs to become a thing. Math-nerds unite and let’s make it happen!