NUMBERS
The Blog Entry that Turned into a Monster
Win Corduan
Towards the end of October, 2014, June and I spent about a week in Nashville, TN, and went to various country music venues. I reported about those experiences in the blog entries of October 20, 22, and 27. All along I was reading some things, and I mentioned on the blog several times that I intended to post a "parallel" entry, providing some more conceptual remarks in addition to the reviews of musical events. However, I couldn't get that entry posted because it kept growing and growing. By the time we were home again, I realized that I probably couldn't upload it all at once. It had become a monster. In fact, when I tried to upload what I had by then thought of as the first part of two, I had exceeded the number of characters allowed in an entry. Finally it turned out to cover four installments. I'm also adding a relevant sidelight from a few years ago.
So while we were gone last week, what did I do when we weren’t out and about? For one thing, I was thinking about numbers, biblical and otherwise. "Numbers" is the common theme running through these various thoughts.
We’ll start with the “otherwise.” In this case, as you will see further below, “otherwise” consists of various disconnected trains of thought, presented for your entertainment purposes onlyor at least that's the impression I wish to create.
I. Concession Stand Math. The Grand Ole Opry building has concession stands in the lobby, much like a sports facility. On Tuesday evening of last week, I had brought a small bottle of water along; June had forgotten hers. I asked her if she wanted anything from the concession, and she said that a bottle of water would be good. (I would have gladly shared mine, but it really was small, and there wasn’t that much left.) So I stood in line watching the preshow inanities on the screen, waiting for it to be my turn. While standing there I realized that I was actually a little hungry. Oftentimes on trips we eat a big meal late in the afternoon (you can call it “Tea,” if you like), which pretty much carries us through, except for the occasional snack. Well, I decided I wanted a hot dog (technically an “Opry Dog”). I noticed on the price list that one of those cost $4.25, a confiscatory price even for a fairly large wiener, but in keeping with the timehonored marketing approach of fleecing tourists. Large drinks were $4.00. The real bargain came into play If you were to buy both, a hot dog and a drink you could get them at the astounding rock bottom price of $8.00, saving 25 cents. So, I got $8.00 ready, but had a quarter easily available as well, just in case I was missing the fine print. When it was my turn I asked for a hot dog and a bottle of water.
“That’ll be $8.25,” the young lady stated immediately.
“Oh, the special price does not apply when you get water?” I asked, placing the quarter on the pile of notes I had set on the counter.
“Hm, let me check; I think that’s only when you get chips as well.” (For my readers from the Commonwealth, in American English “chips” are potato chips, not French Fries, as we call les pommes frites here.
“Okay, then, I’ll have a bag of chips as well.”
She went to the other side of the stall, talked to someone, and returned with a small bag of potato chips. Somewhat apologetically she said, “I’m afraid that’s actually $9.00 now.”
What’s the use of arguing? I was there to have fun, and she was obviously not caught up or not trained or—most likely—neither. I removed the quarter, added a dollar bill, and thanked her for her service as I walked away with the imitations of nutrition, trying to figure out the equation behind this transaction.
II. “Friends don’t let friends clap on one or three.” Someone posted that slogan on Facebook a little while ago, and I was glad to see it. Over the years I’ve tried to get the occasional audience to see the point. Here is what it means: When most American audiences clap along to a catchy song, they usually clap on the first beat of a bar along this pattern:
clap2 / clap2 / clap2 ...
To be honest, it can drive me crazy at times. Clapping on the second beat actually gets you much more into the song.
1clap / 1clap / 1clap ...
I remember once wondering for a moment why I found it so much easier to stand and clap along with the music in an AfricanAmerican church; then I realized that they were clapping on 2, not on 1. Just this last Saturday (actually, now week before last), at Cowboy Church, I reminded the good folks there to follow this pattern. 1clap, 1clap, 1clap, as I was singing “Just a Little Talk with Jesus” in a somewhat jazzedup style, and I think they caught on to the feeling.
But there is one strange exception, which is provided by oldfashioned rock and roll. For example, I noticed it on one of the song Keith Urban was singing the other night. For good old rock, the emphasis goes often goes on 3, i. e. the third beat. Now that may sound really strange, if not impossible, to you, but there’s a trick. If you were to look at the music written out, you would see that you’re still clapping on the first beat of a measure. However, it would be preceded by two beats in the previous measure. To really simplify the thing, a stereotypical rock number begins with two prior to the four beats to the first full measure.
Let’s put it this way and distinguish between pauses (  ), light beats (L), and heavy beats (H). Then the pattern looks like this:
[   L L] [H L L L] [H L L L] [H L L L] [H L ...
The brackets separate the bars, and the color highlights indicate the rhythmic feel that the musician actually conveys to the audience.
III. Binary Numbers. I’ve been reading the very colorful Raymond Smullyan’s Beginner’s Guide to Mathematical Logic. I’m sure that, when (or if) he reads this, he will get a kick of my posting a blackand white picture of him after describing him as “colorful.”
At one point in his exposition, which “beginners” will find somewhat challenging as a whole, he brings up the idea of dyadic numbers. They are very similar to binary numbers, but carry an interesting twist, and that's one part of this monster entry I'm postponing at this moment. First let’s take a look at or review, as the case may be, the binary system. The contrast will hopefully also highlight what’s so good about our decimal system. Presumably, one could count without any system, but then every number on the infinite number line would have to have its distinctive symbols with no repetitions and no intrinsic ordering in what we write down.
As wearisome as Roman numerals can be, even they have a pattern. (The Romans were extremely intelligent, by the way; even their youngest children learned Latin at a very early age.) To get back to the point, as an example, the decimal numbers 4, 5, and 6 could theoretically be written as iiii, iiiii, iiiiii, but instead, the Romans wrote iv, v, and vi. Still, the setup was quite cumbersome, particularly for anyone who would actually do math with them, which was the case right through the Middle ages.
*****
So, the ancient Hindus came up with a system in which a number consists of serials of numerals (the symbolic representations), each of whose value is partially determined by its place in the series. The function of 0 (zero) as a placeholder greatly aided the construction of this system. Imagine a set of boxes representing place values. Conceivably they could stretch out toward infinity toward the left, but I don’t want to use up all my bandwidth for that purpose. With no other values indicated, theoretically there are implicit 0s in all places.
0 
0 
0 
0 
0 
Now, take the numeral 7; put it into the first box on the right, you have the number 7.
0 
0 
0 
0 
7 
Stick it into the second box, and it means “70.”
0 
0 
0 
7 
0 
I know that you don't need me to teach you how to write numbers; I'm only setting this up in order to clarify the contrast with the binaries. Each box can contain any other number from 0 to 9. Let's leave out the unnecessary 0s from now on.
Can we put a 10 or anything larger into any box, such as the first one on the right?




No, as soon as we get up to 10 or more we need to put the excess multiple of 10 into the appropriate box on the left.



1 
0 
So, if we add 7 + 7, we don’t get 





but



1 
4 
Each time the value of a place exceeds 9, we need to push the number one space further to the left. I apologize for this obvious stuff, but I like to set up the easy things so that the more difficult ones won't be as confusing.
*****
B. Flashback to Babylonian Numbers in Cuneiform
It occurred to me that somewhere on this blog I had posted a brief description of numbers in Babylonian cuneiform script, and I thought I might as well resurrect it since it seems to fit the pattern. Obviously the day on which I wrote this was the 13th of that month.
Today is the 13th. 13 is supposedly an unlucky number, a superstition that is allegedly derived from the Babylonian system of numbers. I know that this explanation passes as conventional wisdom, but I'm just a little skeptical.
The Babylonian number system was based on the number 60. Depending on its position, a single downstroke could mean either 1, 60, or 3,600, as the picture shows. So, everything from 1 to 59 goes into the first box, everything from 60 to 3,599 into the second one, and so forth.
So, how would you write the number 13? This is where it gets funny because there is a special symbol for 10; consequently there must be some kind of decimal hunch underlying the otherwise sexagesimal system. The depiction for 10 reminds me of an angel fish. If you've ever had an aquarium, you may have had one of them swallow up all your neons.
So, if you want to write 13, you write the symbol for 10, link three 1's together, and combine them.
However, you must be sure to keep all your symbols in the far right box. Otherwise, you get the wrong number. Just by itself, the sign could mean either 13 or 780 or something even higher, so it needs to be in the right position. Location is everything.
There's only one problem. To the best of my knowledge, when the Babylonians scratched with their styli into the wax tablets or cylinders, they didn't have any boxes. So, then, how did they indicate position for their numerals? They invented a placeholder sign, which eventually took the place of our zero. It looked a little like a slanted pi. And thus, the happy outcome was that you could avoid having your calculations be accidentally off by 767.
None of which says anything about why 13 should be an unlucky number. In Chinese culture, the number 8 is considered to be lucky because its pronunciation is a lot like the word for "good fortune," and number 4 has the misfortune of sounding somewhat like "misfortune." I know of no such explanation for 13 in Babylonian, though my Akkadian is extremely weak. Still, I had fun learning about Babylonian numerals, and I hope you did, too.
*****
C. Binary Numbers: Construction
2^{0} =  1 
2^{1} =  2 
2^{2} =  4 
2^{3} =  8 
2^{4} =  16 
2^{5 }=  32 
2^{6} =  64 
2^{7} 
2^{6} 
2^{5} 
2^{4} 
2^{3} 
2^{2} 
2^{1} 
2^{0} 
2^{n} 
128 
64 
32 
16 
8 
4 
2 
1 
VALUE 
So, let’s set out to make the comparison.
The number written out as 0 in a decimal system would be written out in binary as 0.

0 
The number 1 is represented as 1.

1 
But it’s not time to yawn. From here on out it gets a little more interesting.
The decimal number 2 cannot be written as "2" in the binary system.
The second box from the right is reserved for 2s, so we mark it by placing a 1 there, meaning that there is one 2^{1} in that place. The first box is now empty and marked with a 0.

1 
0 
Of course, as mentioned above, people don't usually write their numbers in nicely prepared boxes. So, the binary representation of the decimal 2 is actully written "10." It would make quite a difference should you forget which system you're working in! Don't read that number as "ten"! Read it as "onezero"!
We need to come back to the farright place and include it in order to write the number 3 which comes out as 11. That means that there is a 2 in the 2^{1} slot and a 1 in the 2^{0} slot.

1 
1 
In fact, all odd numbers will need a 1 in the first box on the right since all of the other boxes contain multiples of 2, giving us only even numbers when they are occupied
Decimal Binary 0 = 0 6 = 110 11 = 1011 16 = 10000 1 = 1 7 = 111 12 = 1100 17 = 10001 2 = 10 8 = 1000 13 = 1101 18 = 10010 3 = 11 9 = 1001 14 = 1110 19 = 10011 4 = 100 10 = 1010 15 = 1111 20 = 10100 5 = 101
Decimal
Binary
Decimal
Binary
Decimal
Binary
So, as long as we keep in mind which multiple of 2 is represented in a particular place, we can convert binary numbers into decimal numbers just by adding. Sum up the 2^{n}s in keeping with their proper location, don’t forget about the 1 on the furthest end on the right, and you get the decimal equivalent.
*****
D. My Age in Binary Notation
In binary form, I’m 1000001 years old.
To convert this number into a decimal, just count up the powers of 2 from right to left and then add up the value of the places where a 1 has marked the spot.
128  64  32  16  8  4  2  1  2^{n} 
1  0  0  0  0  0  1  My Age 
There is a 1 in the space for 64 and a 1 in the spot for 1s. Thus, 64 + 1 = 65, and that's my age according to our common decimal notation.
[It is right about here that the character limit intervened when I tried to post this part of the blog entry, which was much larger than the above. I decided not to try to get a second entry out of what was left right then and there, but to pick up where I had left off on a later occasion.]
E. Binary Numbers Continued
Let’s now go in the other direction. Say you have won 37 games of "Words with Friends,"and we want to express that number in binary code. We need to reserve enough places that there is room for a 32 as our largest number. We could put a whole bunch of 0’s to the left, but that’s unnecessary because clearly we don't need a 64 or anything higher.
32 
16 
8 
4 
2 
1 
? 
? 
? 
? 
? 
? 
37 is larger than 32, so we can mark a 1 in the 32 box, subtract 32 from 37, and have 5 left to allocate.
32 
16 
8 
4 
2 
1 
1 
? 
? 
? 
? 
? 
Then, if we were to check off either the 16 or 8 boxes, we would have to add those values to our number, which would in either case be much larger than 37, so they each need to take a 0.
32 
16 
8 
4 
2 
1 
1 
0 
0 
? 
? 
? 
Is there room for a 4? Yes, we can put a 1 into the place for 4s.
32 
16 
8 
4 
2 
1 
1 
0 
0 
1 
? 
? 
We have now created our binary number up to 36 (32 + 4), just 1 short of the goal. If we were to put a 1 into the very next spot to the right, that would mean that we were adding a 2, which would give us 38, again too large. So the location for 2’s gets another 0. Now we are looking at
32 
16 
8 
4 
2 
1 
1 
0 
0 
1 
0 
? 
and clearly the last place gets a 1, representing the additional value of 1.
32 
16 
8 
4 
2 
1 
1 
0 
0 
1 
0 
1 
Thus we have the full binary notation for the decimal number 37: 100101.
F. 2014 in Binary
One other example. This is the year AD 2014.
To write this number in binary style, we need to draw a few more boxes, all the way up to 1024. The next higher multiple of 2, 2048, cannot be of any use to us since it exceeds the decimal number we want to express.
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 
? 
? 
? 
? 
? 
? 
? 
? 
? 
? 
? 
We put a 1 into the 1024 place and subtract 1024 from 2014,
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 
1 
? 
? 
? 
? 
? 
? 
? 
? 
? 
? 
which leaves us with 990 to fit in. There is plenty of room for 512, so that box gets a 1 as well,
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 
1 
1 
? 
? 
? 
? 
? 
? 
? 
? 
? 
and we still have 478. We can also place a 1 into the box for 256 and subtract that amount from 478.
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 
1 
1 
1 
? 
? 
? 
? 
? 
? 
? 
? 
Then we still have 222 to allocate. 222 can accommodate 128, so let’s put a 1 into that location and subtract it from 222.
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 
1 
1 
1 
1 
? 
? 
? 
? 
? 
? 
? 
The remainder is 94. We can add yet another 1 because clearly 64 can contribute to the sum.
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 
1 
1 
1 
1 
1 
? 
? 
? 
? 
? 
? 
94 – 64 = 30. Finally we get to write a 0 because 30 is smaller than 32, and, thus, we cannot include 32.
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 
1 
1 
1 
1 
1 
0 
? 
? 
? 
? 
? 
But 30 definitely leaves lots of room for 16, so we subtract 16 from 30 and have 14, with the 16 spot having earned its 1.
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 
1 
1 
1 
1 
1 
0 
1 
? 
? 
? 
? 
14 contains 8, leaving 6. 8 gets a 1.
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 
1 
1 
1 
1 
1 
0 
1 
1 
? 
? 
? 
6 is the sum of 4 and 2, so each of those two places gets a 1 because they are represented.
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 
1 
1 
1 
1 
1 
0 
1 
1 
1 
1 
? 
These figures add up to 2014. Since that's an even number we could have marked the 0 in the last spot right at the outset.
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 
1 
1 
1 
1 
1 
0 
1 
1 
1 
1 
0 
Now, I wouldn't have gone through all of this if I didn't want to take a further step and tell you about what I learned from Smullyan's book about dyadic numbers, but that will have to wait. Eventually I will also address the parable in Luke that has a few numbers.
IV. Introducing Dyadic Numbers.
Raymond Smullyan has invented an adaptation of the binary system, which he finds more useful for various mathematical reasons. He calls it dyadic notation. It works very much like the binary system, except that it uses 1s and 2s instead of 0s and 1s. The dyadic system, just like the binary system, can express any positive integer, but it does not allow us to write 0.
A. Basic Construction of Dyadic Numbers
Again, we have a series of boxes or placeholders, and again they represent the powers of 2, but each one may contain either the designated multiple of 2 or double that amount. There are no empty boxes.
PLACES 

2^{7} 
2^{6} 
2^{5} 
2^{4} 
2^{3} 
2^{2} 
2^{1} 
2^{0} 
2^{n} or 2^{n }x 2 
128 or 256 
64 or 128 
32 or 64 
16 or 32 
8 or 16 
4 or 8 
2 or 4 
1 or 2 
POSSIBLE VALUES 
The first two numerals are equivalent to their counterpart in decimals. In the far right position we can have either a 1 (2^{0}) or a 2 (2 x 2^{0}). [Remember that any number with the exponent 0 = 1.]
1 
2 
Counting upwards, you might be tempted to write a “2” as 10 (which would be correct for binary numbers), but, as we said, there is no 0 among the dyadic numbers (just as there is no crying in baseball). A 2 shows up as a “2” in the rightmost box this time (2 x 20^{ }aka 2 X 1).
Under this system, we need to shift boxes to the left whenever the number would be a 3 or higher.
So, when we want to write the decimal 3 in dyadic, we can’t put a 3 into the final position, and we must move forward.
1  1 
The dyadic version of 3 is 11. We see that there is a 1 in the 2^{0} (= 1) position, and 1 in the 2^{1} position (= 2). And, of course, 2 +1 = 3.
4 is 2^{1}+ 2 x 1, and we can write it as 12 (one 2 and two 1s). Here are the first twenty integers in Smullyan dyadic style:
Decimal 
Dyadic 

Decimal 
Dyadic 

Decimal 
Dyadic 

Decimal 
Dyadic 
1 = 
1 
6 = 
22 
11 = 
211 
16 = 
1112 

2 = 
2 
7 = 
111 
12 = 
212 
17 = 
1121 

3 = 
11 
8 = 
112 
13 = 
221 
18 = 
1122 

4 = 
12 
9 = 
121 
14= 
222 
19 = 
1211 

5 = 
21 
10 = 
122 
15= 
1111 
20 = 
1212 
Some of the dyadic numbers are the same as their binary counterparts, but constructing them calls for a very different kind of procedure. In binary code, if you want to express the decimal 16, you can just plunk a 1 into the 16spot and fill out the rest with 0s. But since the dyadic code does not have 0s, that’s not possible. No location can go entirely unfilled until the number is completed. Again, it must get either one or two instances of 2^{n}, as called for by the spot in question. Dyadic numbers can be read as easily as binary ones, but to construct them properly involves a little bit of trial and error, at least for me.
*****
B. My Age in Dyadic Form
So, if I want to write out my age (65) in dyadic, I can’t just put a 1 into the place for 64, bring in lots of 0s, and then finish with a 1 at the end. In fact, I won’t even need the slot for 64. For binary numbers it’s usually easiest to start from the left, but for dyadic numbers we need to do a little more manipulating to make sure that each place has either a 1 or a 2. There are no empty placeholders, and so we may need to do a little planning ahead. If someone reading this discussion has an idea on how to make this process automatic and mechanical, please share it with the rest of us.
Since the number 65 is odd, we know that it will need a 1 as the last digit, which stands for 2^{0}, which, as we said, equals 1.
32  16  8  4  2  1 
?  ?  ?  ?  ?  1 
That leaves 64 units to distribute over the row of boxes that we started with 32 on the left. We can do so by allocating them in the way I have depicted below. As mentioned, I arrived at this number by juggling them, so there is no methodology that I know of to walk us through.
32  16  8  4  2  1 
1  1  1  1  2  1 
111121
To write out the decimal 65 in dyadic, each slot wound up containing one representative, except for the one designated for 2s, where there are two 2s. You might note that, if we were to read the binary and dyadic numbers as decimals, I would be roughly 900,000 years younger in the dyadic format. But that's nonsense and unhelpful, though it's great fun to indulge in. Nevertheless, we must proceed.
The obvious question comes up: What can you do with this? It seems that there is not a whole lot of practical application for grocery shopping or playing music, though dyadic numbers figure prominently in contemporary presentations of arguments leading up to the Gödel's Incompleteness Theorem. There is also another mathematical operation tied to it, that I find intriguing.
*****
C. Chaining Dyadic Numbers
Please, if you've tried to follow the aboveor even if you haven'tand you're really quite tired of all this mathy stuff, please click down to the next section, which is about the Bible passage in Luke. It still has numbers in it, but that's not the entire focal point. I don't want anyone to feel like they're suffering when they're reading my blog. That is, of course, unless you have done something bad for which you need to do penance. In the latter case, if this is torture and you feel that you need a catharsis, by all means don the numerical hair shirt. On the other hand, I hope that at least some people will find this to be fun. 
I'm continuing to report about some things I learned about the notations of numbers from Smullyan's book, A Beginner's Guide to Mathematical Logic.
What you can do with dyadic numbers is add them up.  Well, it's not really addition, we're just pasting two dyadic numbers together in series. Smullyan calls it concatenation; we can also use the English equivalent, "chaining together."
Let us conjoin the dyadic versions of 6 and 7. The symbol Smullyan uses for concatenation is an asterisk: *. So, our problem is
6 * 7 (in dyadic) = ?
The dyadic equivalents are:
6 = 22 7 = 111.
Then
22 * 111 = 22111
That's right, we just pasted the two number together and got what appears to be nonsense.
When you translate 22111 back into a decimal number, you get two 16s, two 8s, and one each of 4, 2, and 1, all of which add up to 55. (32 + 16 + 4 + 2 + 1 = 55)
and consequently, to summarize the result of the concatenation back into decimals,
6 * 7 = 55
That is a bizarrelooking statement. How can one make sense of it? Is there a formula that actually provides a consistent basis for the result of chaining two dyadic numbers.
Let us call the first number x and the second number y, then
x * y = "x pasted to y" or x y,
and we still want to know what in the world one could possibly find of interest in this apparently meaningless application of mental super glue.  By the way, Smullyan cautions us to keep in mind that in this case x y does not stand for x multiplied by y (as in xy); that's why I put a space between the two variables.
Yes, there is something of interest, at least if you're fascinated by the way that numbers behave. It might help to know that Smullyan started out his career as a stage magician before turning to mathematics and Daoist philosophy. (Thanks to nephew Michael for pointing out the latter fact to me.) So, let's try some Smullyan magic.
*****
D. Example 1 of the "Dyadic Formula"
We'll stick with the above example.
x * y = x y
6 * 7 = 55
22 * 111 = 22111
1. We are going to start with a really weird move, namely to count the number of ciphers in the dyadic number that we are using for the variable y, which we shall designate as L for Length. In our example, the number 7 is translated as 111, giving us 3 ciphers. Thus, in this case
L of y = 3
or, to put it into even more formal terms,
L(y) = 3
2. Now we use L as an exponent for our steady companion, the number 2, and create 2^{L}. In this example, since
L = 3, then 2^{L} = 2^{3 }= 8
Then we multiply 8 by x, which was 6. So
8 x 6 = 48.
We have successfully applied an item from the multiplication table, but there still does not seem to be anything here that you'll want to share with your relatives over Thanksgiving. Can you see anything in that number 48 that could conceivably be of interest, if not over the Thanksgiving Turkey, then at least to someone who likes numbers?
Here's a clue. Let's go back to that very strange number 55, which was the result of chaining 6 and 7 dyadic form. We can now ask ourselves,
"Is there any significant relationship between 48 and 55?"
Suddenly something pops up. "What number do we get if we subtract 48 from 55?" The answer is, of course, 7.
Now, do you recall the importance of the number 7 in our example? Of course, you do. It's the number that we called y.
Then we get the following equation:
6 x 2^{3} + 7 = 55
6 x 8 + 7 = 55
As a matter of fact, what we have done is to display (though not prove) one instance of a relationship that is going to hold true for all results of chaining two dyadic numbers together is:
x multiplied by 2^{L} plus y = x and y chained together
x x 2^{L} + y = x y
*****
E. Example 2 of the "Dyadic Formula"
Do you want to try it out with a different set of numbers? Here we go:
3 * 16 = 3 chained to 16
11* 1112 = 111112
We translate the resulting dyadic number back into decimal form:
111112 = 32 + 16 + 8 + 4 + 2 + 2 = 64.
This time y (1112) is 4 digits in Length, so
L(y) = 4
Again here is the magic formula:
x x 2^{L} + y = x y
Then, in this case:
3 x 2^{4} + 16 =
3 x 16 + 16 =
48 + 16 =
64.
And, of course, 64 (111112) was our result of chaining 3 (11) and 16 (1112) together. Do you thing this is cool? I do.
Just for fun, try some numbers of your own.
Notsobytheway, I recommend Smullyan's book. As I intimated above, for a mathematical genius, such as Smullyan, the concept of a "beginner" is often slightly different than for the rest of us. Then again, Smullyan tells us that understanding the book will require multiple readings of various passages and chapters.
V. The Parable of the 122 Minas
Hover your mouse cursor over any dark blue number, and its decimal equivalent will pop up. Try it here: 111
It’s Reformation Day, the day on which we remember Martin Luther posting the 1100011 theses on the door of the castle church in Wittenberg. Here in Hoosierland we celebrate the occasion by having children go from house to house demanding candy from the residents. Actually, lest anyone misunderstand my attempt at being clever, there is no tiein between the Reformation and Halloween except that it happened to be on this day that Luther made his announcement. It was the day before All Saints Day, which in English would be called "AllHallows Eve," and somewhere the idea of ghosts and goblins and trickortreating arose, but it had nothing to do with Luther or the Pope. Actually, I don’t think that we’ll have much of a crowd on our front porch today because, as of this moment, it’s cold and rainy outside, and a drastic change in the weather for the better is not expected for the next few hourand, in retrospect, we didn't.
This is the last part of the Monster Entry, the one in which we go back to Luke. You have probably noticed already that messing around with those number systems has confused my mind more than normally, and I keep accidentally switching between binary, dyadic, and decimal notations. So, please forgive me, and use what you learned in the previous three posts to get through this. Keep in mind that there are some overlaps between the systems. For example 1 is 1 in all three systems, 2 is 2 in decimal and dyadic, 3 is 11 in dyadic and binary. If you see a 3 or higher cipher written out, it must be a decimal number (since I’m limiting myself to these three systems). A 2 immediately rules out binary, and a 0 tells us that the number cannot be a dyadic one.
If this is freaking you out, just make use of the hover feature, as mentioned above, that I have installed just for you.
This parable is clearly very similar to the one recorded in Matthew 11001:111011110. However, there are also significant difference between the two, so we cannot make the assumption that this is one parable recorded in two different ways. Instead, we have two similar parables here, incorporating some of the same elements. To be sure, both passages make the point of being faithful before Christ's return, but the parable here in Luke comes with a framework that includes a political reality check.
I must tell you that in this long time of working through Luke, I have been surprised by the number of parables whose fundamental interpretation is already given in the text, either by Jesus in reciting it or by Luke in his introductions. In this case, Luke helps us understand the meaning of the parable by noting that Jesus told it in response to his disciples’ expectation that the time had come for him to set up the kingdom of God. We’ll come back to that point presently.
First, though, we need to recognize the fact that Jesus alluded to some real historical events in the story. This observation does not mean that the entire parable is supposed to be a historical narrative. Just because it contains some historical allusions does not mean that the parable isn’t still a parable. Nevertheless, the way in which Jesus framed it would have struck a chord of recognition in his audience.When King Herod the Great (the baby killer) died in 12 BC, it had only been a few weeks after he had changed his last will, in which he named his son Archelaus as his successor. Archelaus was known as a cruel and harsh man, and the populace was afraid of having him as their king, preferring his younger brother Antipas, who had been Herod's designee until then. Archelaus had to travel to Rome in order to have his kingship confirmed by the emperor; he was followed by Antipas, their halfbrother Philip, and other messengers opposing his rule. Nonetheless, the emperor Augustus confirmed Archelaus’ position over Judea, Samaria, and Perea, though he did not permit him to carry the title “king.” Instead, he was only allowed to be called “tetrarch,” which means ruler of a quarter of a kingdom, which put him on a par with Antipas, who governed Galilee, and Philip, who received territory further north. Apparently the title “tetrarch” was appropriate even when the land was divided among 11 rulers rather than 100. When Archelaus returned home, he vented his anger on those who had opposed him within his allocated land by staging a blood bath.
It is important to realize that the closeout of the parable in v. 11011, “But bring here these enemies of mine, who did not want me to rule over them, and slaughter them in my presence,” should not be attributed to God in some sort of allegorical fashion, but that they are something that the evil ruler in the parable says. In Jesus’ parable, the man who returned had become king, though, as we said, Archelaus did not have that title.
There were diverse opinions among first century Jews as to the attributes of the coming messiah, and not everyone even expected one. But in general, the anticipations included a political mission for the messiah. He would reestablish the Kingdom of David. Thus, as Jesus was heading up to Jerusalem, accompanied by a large parade of people, many of them thought that they were about to witness how Jesus would expel the Romans and take over as the true king.
The details of the parable are fairly straightforward. Before the wouldbe king left, he gave each of his servants 1010 minas, a unit of weight and currency (weight of the precious metal), whose actual value changed from time to time. It seems to have hovered around 60 shekels, which does not tell you much unless you know that a shekel weighed 211 grams or a little more. The wouldbe king expected his servants to use that money wisely on his behalf and hopefully make a profit in the process.
When the ruler returned as king, the first servant announced that he had made another 122 minas, and the ruler gave him authority over 1010 towns as a reward. Clearly the realistic basis of the parable ends here. The second servant earned 21 minas, and the king, displaying a fondness for symmetry, put him in charge of 101 towns. A third servant did nothing with the money entrusted to him. In fact, he made use of the occasion to let the ruler know what a bully he was. This man apparently was not very smart. First of all, he should have tried to put the money to work in some way, even if it was nothing more than to put it into a savings account and accumulate interest. Second, given the fact that the ruler was unjust and violent, it would have been best if he had not thrown this accusation at him. The king consigned him among those people who would be executed.
Jesus spoke this parable in light of the fact that a number of his followers expected him to reestablish the kingdom in just a short while. So, how does the parable respond to those hopes? I see 10 points here.
1. Those who belonged to Jesus should be prepared for a lengthy interval before Jesus would actually return as king. It is simply not true that everyone in the early church was living in the anticipation that Jesus would return in a very short time. He could, but he also might not. Please remember that Jesus spoke these words, but that early Christians recorded them and would have had this message right in front of them.
10. During the time of waiting, we should remain faithful to whatever God has called us to be and do. I need to make something clear here. There is no threat to Christians in this parable. The allegiance of the unfaithful servant was against the king, and a Christian’s allegiance is not against Christ. It may take a while yet, but when Christ returns, our standing in him is going to be an occasion for celebration, thanks to his grace.