**Meditating on Transcendental Numbers**

When I was first started creating comics to post to my blog (which actually started a year and a half before I scribbled those Co-Medians in **MS-Paint**), one of my earliest attempts at *Punnery* was this comic about **Transcendental Numbers**

Link to comic page.

One thing that I never answered in the blog -- because I really didn't use the comic as an opportunity for learning at the time -- was just **what is a transcendental number**? (I could make another joke right now, but I'm writing on a notepad to use at a later date.)

Transcendental numbers are a type of Irrational number, but they aren't the same. In fact, when I create that "What Kind of Numbers Are There?" Venn diagram on the first day of *Algebra* class, I could actually include a subset circle on the Irrational side. (But, I don't for reasons similar to why I don't have an Imaginary oval next to the Real one. It's not going to matter any time soon.)

There are a lot of Irrational numbers (as we discussed a few days ago in a different post), including square roots of non-perfect squares. Those numbers aren't transcendental. Which numbers "transcend" these ordinary irrationals?

The definition of a Transcendental number is any number that is NOT Algebraic. Don't you just love negative definitions? Anyway, an Algebraic number is one that can be the root of a polynomial equation (such as a quadratic equation) which has only rational coefficients. Since **x ^{2} - 2 = 0** has rational coefficients, its roots -- plus or minus radical two -- are Algebraic. But some numbers cannot be roots of these equations. Which ones?

As the cartoon suggests, pi and e are two examples. The third one, gamma, is the *Euler–Mascheroni constant* (also called Euler's constant), which is found in Number Theory, and which I freely admit I was completely unaware of before 2008 when I looked up transcendental numbers to make the comic (and earlier this evening when I wanted to write this entry).

The last is phi (pronounced "fee"), which I've mentioned before in the blog and in the comics, which is related to the **Golden Ratio**. This has the distinction of being incorrect, but I'm not updating it eight years after the fact. I read some information back then that some considered it to be one for whatever properties, so I used it because I had a comic to make. However, according to the definition I used above, it can't be transcendental because it is the root of **x ^{2} - x - 1 = 0**. (What other definitions or properties are out there? That is

*left an exercise for the reader*to ascertain, if you really want to.)

Finally, closing out the joke with a further pun, because it is "Transcendental Meditation", each of the characters (with their little legs are cross as they could go) are saying "Ommmm....", which sounds like "Ohm", which is a unit of **Resistance**, which uses the letter **omega**. What I actually knew at the time because I had just read a book about it the previous summer, was that omega was actually the symbol for **Chaitlin's constant**, which -- drumroll -- is irrational and transcendental.

Seriously, I really did know about this, even if I was, at the time, a little fuzzy on what a transcendental number was. But, at least, now you're not.

## No comments:

Post a Comment