Tag Archives: Number bases

Drop the Base

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A demonstration of exactly how a shift of base can change our perception of time. Image: Jeremy Keith via flickr.

As a kid, when we were first introduced to numbers, they were just something we memorized, learned to accept, and started using on a regular basis. While this seems almost second nature to most of us, there was a time where the idea of a number system was a new thing and, like all new things, it was discovered multiple times by different people who had different setups. One of the more interesting areas of variation between different number systems would be the base that different number systems used.

Before going into detail about what a base is, it is important to understand that base systems are primarily used by number systems which also use position to determine how large a number is. For example, the Arabic numeral system is positional because I can use the same symbol in a different position to change the value of the number. While 01 is only the value one in this system, by switching these numbers around to 10, I have changed the value now to ten. This is different from something like the Roman numeral system which, for the most part, wouldn’t be considered a positional system because in two different numbers, like X and XIII, the value of the symbol X doesn’t change.

Now, what does this have to do with the base of a number system? The thing is, the base of a positional number system is the number of different symbols you can have in any single position. For example, the Arabic numeral system is base 10 because we can have ten different symbols in a single position (1, 2, 3, 4, 5, 6, 7, 8, 9, 0). In addition to defining how many different symbols you can have in any one position, the value of the base will also affect how much of a change in value a symbol will have based on its position. As I had mentioned earlier, different number system have different bases. The primary reason why would most likely be just because they may have had a different system for counting which lead to that decision. Having a base 10 system is the more common one and a lot of people give credit to that due to the fact that the average number of fingers we have on our hands combined is ten and people like to count using their fingers. On the other hand, the Mayans had a numeral system which consisted of base 20. Unlike most people from Europe, the Mayans wouldn’t wear shoes which meant they could count using ten fingers and ten toes. Even the Babylonian’s had a numeral system with base 60. I honestly couldn’t say why but I am sure they had a good reason for doing so.

Even current day computers use a different base than 10. Instead, computers count using base 2 which means they can only have a 0 or 1 in any position. How can something like this work? The reason why different numeral systems can have different bases is because all positional systems use mathematics in combination with the base size to determine how important a certain symbol is based on its position. This means that it is easy to convert from any base system into a different one. For example, if I want to convert the binary number (100010) into a base 10 number, all I need to do is figure out the base 10 value at each position and add them together. Since this is base 2, every position will be multiplied by 2i with i being the current position. This means:

100010 -> 1*25 + 0*24 + 0*23 + 0*22 + 1*21 + 0*20 -> 1*32 + 0*16 + 0*8 + 0*4 + 1*2 + 0*1 -> 32 + 2 = 34

To make things even more bizarre, a base can be found in more than just numeral systems. Another great example of a system which has a base is the alphabet. In our case, the Roman alphabet has a 26 base system or 52 (if you include capitalization). In addition, a lot of the different measuring units we have also have their own setups for bases. There are 12 inches in a foot, 16 ounces in a pound, 60 seconds in a minuet, and even 12 months in a year. And yet, for all of these we use a base 10 counting system instead of creating our own symbols for each measuring units. Then again, imagine how confusing that would be. In most places, people realize how difficult it can be constantly converting from one base system to another which is why certain measuring systems like the metric system uses a constant base of 10 between unit sizes to make things easier.

In the end, the point is that different bases are used everywhere. Whether you are dealing with numbers or some other system entirely, you will usually be able to find a base of some kind connected to the system. While it may be difficult to have to constantly deal with different kinds of bases, bases are necessary for people to be able to have such a large variety with such a limited number of symbols. Bases are here and they are here to stay.

Sources:

http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_numerals.html http://mathcentral.uregina.ca/RR/database/RR.09.00/hubbard1/MayanNumerals.html

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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

We Need more Fingers

2010-07-20 Black windup alarm clock face

Image: Sun Ladder, via Wikimedia Commons.

Our base ten number system is so ingrained in us that it is difficult to imagine using anything else.  With our ten fingers to count on, it makes sense that we have ten symbols to represent numbers.  Despite this, other cultures have used extremely different number systems.  The Ancient Mesopotamians used a sexigesimal, or base 60, number system.  The Mayans used a vigesimal, or base 20 system.  Roman numerals are used to number the Rocky movies despite them being almost completely useless.  Most computer techies are familiar with binary and hexadecimal.  Many early peoples even used a system with only 5 symbols (Boyer 3).  Our current number system may be intuitive but it may not be the best one around.  What if we had an extra finger on each hand?  We would be using a much more useful number system.  We should move away from the decimal numbers we currently use, and switch to a base twelve, or dozenal, number system.

There are several reasons to seek more mainstream use of base 12.  The factors of a number, or the numbers that divide into it evenly, determine a lot about the number.  Twelve can be broken down into more factors than ten can be.  Ten is divisible by only 2 and 5, whereas 12 is divisible by 2, 3, 4, and 6.  This gives 12 an advantage over 10.  The additional factors make it easier to think of many fractions, such as fourths and sixths, since they will now have only have a single significant digit after the point.  This is particularly effective when dividing 1 into thirds because it will not leave us with an infinite series of 3s like it does in decimal.  The simple tricks that help us do arithmetic, such as the fact that in base ten if a number ends in an even number the whole number is even and thus is divisible by 2, depend on the factors that make up our number base.  Since 12 has more factors, similar tricks can be used for more numbers.  In base twelve, if a number ends in 0, 4, or 8 the entire number is divisible by 4, if it ends in a 0, 3, 6, or 9 then it is divisible by 3.  We will not even miss out on the trick for evens that base ten has since twelve is also divisible by 2.  The trick we now currently use for 11, where you alternate adding and subtracting the digits of a number and see if the resulting number is divisible by 11, will work for 13 when we make the switch, because now 13 will be the number that is one larger than our base.  Don’t be concerned about 11, because we will have a new trick for 11 in dozenal.  The trick we currently use for 9, where we just add up the digits of a number and then check if the sum is divisible by 9, will work with 11 once we change to dozenal.  It is easy to check the divisibility of far more numbers in dozenal than it is to check in decimal.

For a dozenal system, we would have to make some changes to the actual symbols we use.  We have 10 symbols to use for the numbers 0-9 and a base twelve number system would need 2 more symbols to represent ten and eleven.  There are many different sets of symbols we can use to fill the two new places.  Some number sets use *, and # to represent ten and eleven in order to correspond to the symbols on most phone number pads.  Others use X, and a backwards 3, and some use a backwards and upside down 2 and 3.  Some sets completely replace all the symbols we use for 0-9, along with adding two new symbols.  We could use any group of numbers that would help us acclimate to a base twelve system.

Some things that we already do everyday would assist in our transfer to a base twelve system.  We already have specific terminology for 12 and several of its powers.  We use the word dozen to refer to twelve, a gross for a dozen dozens, or twelve sets of twelve, and a great gross for a dozen gross, or twelve gross.  When looking at a clock, we already deal with twelves to determine the time.  Figuring out what the time is 5 hours after 10 pm is basically the same thing as adding 5 to 10 in base twelve.  As Professor James Monroe notes, thinking of egg cartons makes thinking of dozenal numbers easy.  If 1 egg carton holds 12 eggs, and 1 case holds twelve cartons, a number like 426 in base 12 can simply be thought of as 4 cases, 2 cartons, and 6 loose eggs.  Twelves seem almost as prominent in daily life as the number ten.

Despite the familiarity we already have with base twelve, switching to dozenal will still be incredibly difficult.  Because we would have more digits, kids would have to memorize larger multiplication tables.  Luckily, the tables will not be anywhere near as large as they would be if we still used Cuneiform.  However, the real difficulty in switching has little to do with what number base we want to use.  The trouble will be in converting all the numbers on everything that we use.  Every road sign, price tag, page number, and countless other places have numbers that will need to be converted to base twelve.  This will be more difficult than changing from using imperial units to metric units, and America still has not completely converted to metric.  Here in the U.S., some things are measured with metric units, but we still measure distances in miles and a sack of potatoes at the grocery store is measured in pounds.  Also, to add more difficulty, in metric we would have to come up with new prefixes that are based on powers of 12 instead of powers of 10.  Changing to dozenal numbers is such a monumental task we may not be able to accomplish it.

In spite of the difficulties, I believe we should do it.  The conversion will not happen overnight.  We must look further down the road.  Perhaps, start by teaching people how to do arithmetic in dozenal in addition to teaching them the usual decimal system that we use.  Then when people are comfortable with it, we could move on to using base twelve alongside decimal.  Eventually, our ancestors will be able to move on to a better number system than what we currently have.  Like the Kwisatz Haderach from Frank Herbert’s Dune, we must endure temporary struggles in order to achieve the Golden Path.

Source

Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. Hoboken, NJ: Jon Wiley and Sons, 2010. Print.

Additional Material

http://www.dozenal.org/drupal/

http://www.dozenalsociety.org.uk/

http://youtu.be/U6xJfP7-HCc