The Number Dirf

November 23, 2010

I've just watched an enjoyable episode of a television series on the Nickelodeon channel. It's a teen show, so maybe I'm entering my second childhood; but I'll claim that I'm doing research for a children's book. This was the iTwins episode of iCarly that first aired on July 11, 2009. Watch the listings for the next airing of this episode. One advantage(?) of cable television is that if you miss something, you just need to wait a few days and it'll be aired once again. This has allowed me to see every James Bond 007 movie multiple times.

The reason I mention this episode is that it involves numbers in base-11. In the plot, the main character is tutoring a boy who just assaulted her brother. In this comedy world, her brother is twice the size and twice as old as the boy, but that's called a "willing suspension of disbelief." To subtly punish the boy by having him fail his next math test, he's told that the UN had just passed some new international math laws. One of these concerns putting a new number, "dirf," between five and six. A dirf is written like this:

The number, dirf.

Since I was a member of the "New Math" generation, I was introduced to various number bases in eighth grade. We worked a lot with Base-12, simply because it was easy to represent ten by t and eleven by e and have students remember what they meant. Nowadays, the proper way to do this is using a for ten and b for eleven. Base-11, also called undenary or undecimal, isn't very useful, although it did figure into the plot of Carl Sagan's novel, "Contact." In standard base-11, you would count 1-2-3-4-5-6-7-8-9-a. In the dirf number system (using the Greek letter delta for dirf), you would count 1-2-3-4-5-δ-6-7-8-9. Here are the first few digits of pi in standard base-11 and dirf-base:[1]
3.16150702865a48523521525977752941838668848853163
a1a54213004658065227350533715271781a6563715781334
9288852819129920634252707812755482692769781806403
86187079590752454659a8876a29287267aa9575416475428

3.1δ1506027δ5947523521525866652841737δδ77477531δ3
91954213004δ570δ52263505336152616719δ5δ3615671334
8277752718128820δ34252606712655472δ826δ867170δ403
7δ176068580652454δ589776δ9282762δ699856541δ465427

Number bases have some interesting properties. First, it isn't required that the base be a non-zero integer.[2,4] In fact, the value of pi in base-pi is 10.[3,4] Making a base like this perverts the normal meanings of numbers, since representing the decimal number four in base-pi requires an infinite number of digits.

Donald Christensen, former editor of the IEEE Spectrum, wrote an opinion piece several years ago in an issue of "Today's Engineer,"[5] that poses the following problem. "The square of 24 in base b equals 554 in base b. What is base b?" I'll leave you with the following as a starter. The number 237 is represented in base 10 as
(2)(102) + (3)(101) + (7)(100), or
(2)(b2) + (3)(b1) + (7)(b0), where b = 10.

If you're stuck, you can find the answer in one of my earlier articles (All Your Base (Continued), December 11, 2006).

I can't write an article about bases without a mention of the most famous Internet base; namely, the one from the opening scene of the 1989 Japanese video game, Zero Wing. The English used in the game was likely just a placeholder for a better translation that never happened, and a particular phrase used in the game, "All your base are belong to us," became an Internet meme. You can view a popular video remix of AYBABTU on YouTube.

References:

1. I used the The GNU Multiple Precision Arithmetic Library (GMP) in a simple C-language program for this calculation.
2. George Bergman, "A Number System with an Irrational Base," Mathematics Magazine, vol. 31, no. 2 (November.December, 1957), pp. 98-110.
3. Donald E. Knuth, "Art of Computer Programming, Volume 2: Seminumerical Algorithms," Addison-Wesley Professional (Third Edition, November 14, 1997), 784 pages.
4. Base entry on MathWorld.
5. Donald Christiansen, "Math... What Good Is It?", IEEE-USA Today's Engineer, December, 2006, p. 4.
6. David A. Wheeler, "Way Off Base," August 23, 2003.

Linked Keywords: Nickelodeon; iTwins; iCarly; James Bond 007; willing suspension of disbelief; United Nations; UN; New Math; number bases; undenary; undecimal; Carl Sagan; Contact; pi; integer; IEEE Spectrum; Zero Wing; All your base are belong to us; Internet meme; GNU Multiple Precision Arithmetic Library (GMP).