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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