Count Those Numbers | The Probabilist . com

Count Those Numbers October 24, 2006

Posted by The Probabilist in : [Articles], Calculus, Creativity, Society, History , trackback

Some of us hate them and some of us are fascinated by them. Whatever your thoughts about numbers are we all rely on them daily. This article will specifically focus on number bases. Why do we use ten as our numeral base in today’s society and is it the best one? We do use other bases directly and indirectly - maybe you’ve consciously noticed this or maybe you haven’t. But have you ever questioned their use?

Initially I will start by describing our currently used decimal system and moving on to other bases that could be more efficient and easier for us to use. The ancient or quite recent origins of each of them will be shortly explained and in what areas of development you can stumble upon them today. Following are some examples of what kind of peculiar challenges a change of base system would face and in which situations you might apply them. Lastly, why can pondering about issues like this be useful to you?

The decimal base (10) is widely believed to have been created due to the human being counting with the fingers on both hands. The first people estimated to have used the decimal base are either the Elamites (3500 - 2500 BC) in today’s Iran or the Egyptians (c. 3000 BC). The way we write the decimal base today is naturally called the Arabic numerals, but without getting too boring on the origins it’s time to focus on its limits. First off, ten (10) is divisible by four numbers, 1, 2, 5 and 10. And of these four divisors, none actually work well when dividing or multiplying consistently. Let me explain this better after introducing the next base system.

The octal base (8) doesn’t contain the figures 8 or 9, instead after 7 comes 10. This system has the same amount of divisors, which are 1, 2, 4 and 8, but they make more sense in calculus. Here’s the awaited example, divide the number 10 consecutively with 2. This results in the following sequence: 10, 4, 2, 1, 0.4, 0.2, 0.1 and so on, always cutting the previous amount in half. Now do the same on a decimal base: 10, 5, 2.5, 1.25, 0.625, 0.3125… and you get the picture. Dividing and multiplying with 4 using the octal base works naturally equally easy. So, it has the same amount of divisors with a lower base and is easier to calculate with. What else is there?

The duodecimal/dozenal base (12) is quite intriguing. It has six divisors (1, 2, 3, 4, 6 and 12). Just by adding two numbers to the base you lose the 5, but gain the 3, 4 and 6. You can find dozenal societies both in the US and UK who wish to change the world to this more applicable numeric system. However, deciding on what the two extra numbers should look like is undecided, A and B, X and E, * and # (like on a phone) or upside down 2 and 3.

The hexadecimal base (16) is most used in today’s world of computers. It signifies the scale of colors and is used in different forms of computer programming. Even music software like trackers use hexadecimal systems and the extra numbers are usually displayed as A to F. Naturally, the binary base is even more familiar to computer techies, and musicians grasp different numeral bases with less effort as well. Ok, the next base system is the last one I’ll present.

The sexagesimal base (60) comes to the point that you can argue if it’s a base or simply a multiple of another base. What’s especially notable about the sexagesimal system is how old it is and how it’s still used today. Originally the ancient Sumerians, which can be considered as the first people developing a civilization used base 60 and the Babylonians adopted it later. The base 60 system has a whopping twelve divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60), which leads to an adequate explanation why it’s still used today to measure time. We’re able to split the seconds in a minute and the minutes in an hour in so many ways that it explains why the way we measure time has withstood the test of time. When taken further, this base was multiplied by 6 by the Persians to create a calendar for the days in a year. The applicability of the number 360 is still used today to measure the degrees of a circle.

Without question, there are better bases than the decimal base. The real resistance lies of course in adaptation. There was already more than enough media coverage on the supposedly difficult transfer from national currencies to the euro. Then imagine the time and effort required to change mankind into a octal or dozenal system. The decimal base is so deeply rooted in today’s technology and people’s thinking it could take a generation - to get those really stubborn and/or old ones along as well. However, it’s not necessary to go that far when we still have our imagination and creativity to use all this information if only to consider some amusing thoughts.

Imagine reporting a financial statement with an octal system. Now, I’m in no way suggesting that one should break the law, but consider this. How long would it take for someone to notice that there aren’t any eights and nines in it? How easily would you notice such a thing? Ok, if you feel legitimately offended by that example consider pulling a prank on somebody using this approach. In what different ways can you pull somebody’s leg and still mathematically prove that what you did is totally acceptable? You’re just calculating differently than others.

Here’s another example. I’ll let you decide if this is more or less practical than the previous one. Imagine that we get contacted by another intelligent life form. Suppose also that we can communicate with them and they tell us their population is 100 million. Sure, it sounds like a small amount compared to us, but what if they use a quadrovigesimal (base 24) numerical system? Well then their population in our decimal terms is just above 110 billion. So much for that ego boost.

Consider that we change into an octal base and also redefine the seconds and minutes to 100 instead of 60. Decimally this would mean 64 seconds and 64 minutes so the chronological impact wouldn’t be that notable. But then think about how hard it is for 100 meter dashers to once again beat the 10 second mark. Would they even still run the same distances? Now you might really grasp what a numeral base change would bring forth. All of the results and data that humanity has stored for so long would then have to be redefined. How’s that for a day job?

Changing to an octal or a dozenal base wouldn’t have to change the way we measure time and degrees or anything else though. As pointed earlier we can still count to 60 minutes even if the base is different. This just leads to the decimal base number 60 to look in a different way, still meaning the same amount while still maintaining it’s excellent dividing capability.

Conclusively, it looks like we’re just too far on the wrong road to turn back and make a difference. But, you as an individual can find many creative and intuitive methods for yourself to effectively use different numeral bases to whatever you want to accomplish - intentionally creative and productive ones of course. I hope to hear from your endeavors. At least we can agree on one certain notion that would have made life easier today if understood in ancient times. The thumb is not a finger.

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Comments»

1. Tamer Hosny - November 11, 2006

arabic numerals…

Interesting post. I came across this blog by accident, but it was a good accident. I have now bookmarked your blog for future use. Best wishes. Tamer Hosny….

2. The Probabilist - November 12, 2006

Thank you for your interest, Tamer. Accident or not, I’d rather use the word synchronicity or chain of events. I got inspired about that subject so I’ll write an article about it someday during next week. I hope you’ll like it.

Edit: Direct link

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