Thinking in different bases

Very few people ever stop to think about why numbers are the way they are. Have you ever stopped to consider why you live in a society that uses a ten based (or decimal) number system? This is a system where you start at 1 and go all the way up to 9 before you reset with one followed by a zero (or 10). Well if you haven’t taken the time to ponder this deeply complex situation that you find yourself in, fear not, for I will explain why. For the most part, human beings find themselves in the possession of 10 fingers. So the leading hypothesis for why we use a base 10 system is because it was convenient since we could count to 10 on our hands. For that single and simple reason, society decided that its number system would have a base of 10. There you have it, one of the great mysteries of life has been cleared up for you.

Now that you have an open space where a mystery used to rest, allow me to fill that spot with a new mystery. Why don’t we use a differently based number system? No really, think about it! Throughout history, multiple societies chose not to use a decimal system. For example, Babylonians used a base 60 number system and the Mayans used a base 20 system. Likewise, there are quite a few key things in our lives today that don’t really rely on the conventional base 10 way of doing things.

binaryFor example, we are surrounded by computers, which utilize Binary, a base 2 number system. At every position there is either a 1 or a 0, so the number ten in binary looks like 1010.

Another example of something that isn’t really 10 based is a clock. Take a second to look at an old school clock with an hour and minute hand (gasp! Not digital!) . You’ll quickly notice that it goes from 1 to 12 instead of 1 to 10. Weeks are broken down into 7 days and minutes as well as hours are in chunks of 60. So as you can see, the case could be made to switch to a different based number system. Let’s take a look at another option that society could use in place of the current base 10 system.

The system that we’ll look at is one of my personal favorites (thanks to my Computer Science bias), the hexadecimal system. As you can probably guess from the name, the hexadecimal system adds six to the base 10 system, leaving us with a grand total of base 16! The system uses 0 to 9 to represent numbers 0 to 9 and then uses A to F to represent the numbers 10 to 15. Hexadecimal is a positional numeral system just like the decimal system. Just to give you a better idea of what hexadecimal is, lets learn how we can represent a hexadecimal number in decimal.

Let’s take the random hexadecimal number 3FB1. Since it is a positional system, meaning the position of the symbol is part of its value, we can just take the symbol and multiply the symbol’s value by its base to the power of its position (position numbering goes right to left and starts with 0 rather than 1). So we would take (3 x 163) + (15 x 162) + (11 x 161) + (1 x 160). Simplifying this further we get (3 x 4096) + (15 x 256) + (11 x 16) + (1 x 1). At the end of this we are left with 16,305. So right off the bat we can begin to see the some of the potential benefits that would come with using a base 16 system. Firstly, we notice that it takes fewer symbols to represent numbers. Where in decimal we had to use 5 symbols (6 if you count the comma) to represent 16305, in hexadecimal we only had to use 4. We can also note that because of this space bonus, we could potentially represent higher numbers in the same number of hexadecimal characters. Even though this all sounds great, there are some disadvantages that would come with using the hexadecimal system. For one, performing mathematical operations on base 16 numbers can get complicated quickly (a base 16 multiplication table has 256 instead of 100 elements). Try performing long division on two hexadecimal numbers! Also, I personally believe that it would be trickier to set up equations with variables due to the fact that the characters “A”, “B” and “C” could no longer be used (there goes the quadratic formula song). On a similar note, there are many people who think that we would be better off on a duodecimal system (base 12), but that is a conversation for another time.

So next time when you are counting on your fingers, take some time to think about the effects of you simply having 10 fingers!

http://en.wikipedia.org/wiki/Binary_number

http://io9.com/5977095/why-we-should-switch-to-a-base-12-counting-system

http://en.wikipedia.org/wiki/Hexadecimal#Conversion

http://mathforum.org/library/drmath/view/63375.html

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