The Base of Money, and Other Adventures

Image: slgckgc, via Flickr.

Image: slgckgc, via Flickr.

The other day in class, there arose an interesting discussion about one of the most common topics in math classes. We were talking about bases. It is a very interesting topic, because there is so much room for creativity. One interesting thing about bases is that we don’t quite know why we are using a base ten number system. We have our theories, but nothing is quite concrete. (We just say that it’s the number of fingers we have, and um… yeah, we leave it there.) (I honestly think that’s what the ancients did.) In class it was mentioned that in the Bible the base ten number system is used, and it was theorized that maybe ten being the “holy base system” we cannot change it. Being in Utah, and with the predominant religion being Mormonism, Prof. Lamb asked what base they used in the Book of Mormon.

Bases Used in a Religious Text

In the Book of Mormon, while troop numbers are given as nice multiples of ten, indicating a base ten numbering system, in one part of the text (in Alma 11) the currency used by the culture is laid out, which has an interesting property of doubling from one unit to another.
Now, we know what doubling means, Binary! (only your computer is excited, sorry) In the text it lays out the following system:
Gold Coins
8= Limnah *(see bottom of entry)
4= Shum
2= Seon
1= Senine
Silver Coins
8= Onti *
4= Ezrom
2= Amnor
1= Senum
(and now for the interesting part where they go all fraction on us)
1/2= Shiblon
1/4= Shiblum
1/8= Leah
Now because we don’t know everything about this civilization, we don’t easily know how much any one of these coins actually bought, except for saying that either a “Senine” or “Senum” could be used to buy “a measure of barley” or any other kind of grain.
Let me be the first in saying that knowing that a “measure” of grain is equal to x is very little information, but if we decide that a “measure” is a useful quantity of grain that is enough to actually eat, we could say that 1 kilogram (or 2.2 lbs.) is a very useful amount of grain for a small family for a few days. To my knowledge the cost of that much wheat, or other grains is about $2. (I am assuming that there is a great deal of error involved in my guessing game.)

Now based on this, we might say that the system as it stands is a bit inflexible, unable to go to very high numbers, as it maxes out at roughly sixteen dollars, and hits a minimum at about 12.5 cents, but with a few tweaks on our part, applying the same pattern, we can achieve a wide array of numbers and a very intriguing property (at least to me). This most intriguing property, is that within this system any change given is rather simple to calculate, and give. This is based on the fact that if I used a theoretical 2^8 coin (256) for an item of value 102, I would get change in the form of I could get change as just a series of these coins (128), (16), (8), and (2) which is basically 10011010 (in binary)

Now I don’t know about you, but if I wanted to give you change in a way that I could just look at my coins and take out the biggest one that cuts down the difference with less of a need for calculations. This is called the greedy algorithm, where the largest coin possible is taken, and used until it can’t be used anymore, continuing until no more can be subtracted. In a base 2 coinage system, while performing this algorithm, or method for choosing, there is never a need for repeated coins.

The main drawback of this system is that it uses 14 different levels of currency to make it from 1/64 to 128 (roughly 1 cent to $100) while to go from $.01 to $100 we only need 12 (counting $.50, and $2, which are almost never used)

Basically the gist of the story is that within this religious text, The Book of Mormon, we find that they have a currency system with very interesting properties that come from being based on a binary system.

* In the text, it is stated that a Limnah, and an Onti are “as great as them all” which could mean that it is the sum of all the previous values, or in my interpretation, it is the greatest of them all, or worth more than all the others put together.

Sources

Original Text Where the Monetary system appears
https://www.lds.org/scriptures/bofm/alma/11.5-19

Why We use Base 10
http://ideonexus.com/2008/07/08/why-a-base-10-number-system/

Greedy Algorithm
http://math.stackexchange.com/questions/106317/for-what-coinage-systems-does-a-greedy-algorithm-not-work-in-providing-change
http://www.sciencedirect.com/science/article/pii/S0195669809001292

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s