11 Mar 2009, 12:41 am
lau wrote:
I don't know where people get "Hexidecimal" from. Base 16 is hexadecimal. Anyway, for practical purposes, I much prefer Sexagesimal.
Indeed, base 60 (and base 360) are two that I considered back when I was trying to decide what was the best for us as a species.
Binary is right out - 101011100100101001 is much harder to remember/say/write/grok then 178473.
Base 16 is much more appealing and a good alternative, but not perfect. Only 2 4 8 evenly go into 16 (~28% of numbers on 2..15).
It is also a bit tricky to count on your hands with this, though there are ways around this - ie, having hand up and hand down be distinct and counting the four non-thumb fingers twice per hand. If this seems like a silly consideration, remember that our numbering system should be universal.
Because of its applications in computers, base 16 should be taught more extensively to math/science-minded students. Specifically I believe we should teach not just translation to base 10, but each of the fundamental asthmatic operations in base 16 as well.
Base 60 requires that we teach kids 60 unique symbols. Basically anyone with Dyscalculia would cry at the thought of that. It is appealing because it is highly factorable and dense, so for me it might well have been superior (I excel in mental arithmetic), but that would not be universal. Instantaneous recognition of the full range of characters is mandatory, and this would complicate matters. Furthermore, multiplication would require memorization of 3600 operations, rather then our current 100.
Base 12 represents the smallest deviation from an accepted, standard system that returns the highest reward. Relative to base 10 it has the following benefits:
Density (distinct numbers expressible per length, n) is increased by 1.2^n at the cost of only 2 (20%) additional characters.
It requires only 144 operations to remember for multiplication (44% increase). Compare to base 60's 3500% increase, and base 16's 156% increase.
I am not counting addition and subtraction here since learning and utilizing them is actually a bit more tricky of a beast. For example, consider how I first memorized subtracting b from a, when b>a: 3-8 = 0-(7-2). More generally: a-b = 0-( A - B ), where A is a's compliment to 10, and ditto for B and b. My point is that there are too many different paths to first learning the system for it to be something you can honestly anticipate - contrary to multiplication, where pattern recognition pays off big time.
40% of numbers on 2..11 go into 12 evenly. Compare to 25% of base 10 on 2..9, ~28% of base 16 on 2..15, and ~17% of base 60 on 2..59. This makes mental shortcuts for addition, multiplication, and modulus functions much easier in base 12.
A refutation of an alternative numbering scheme for base 60, presented in advance: using number representation systems that require you to perform even the simplest of mathematical calculations to resolve a number (ie, the Babylonian and Roman systems, amongst others) is bad because it switches how your brain is thinking mid-problem. When I look at 632,887 I know how much it is because I have memorized what is basically a linked list of 10 elements. Each element is clearly distinct in my brain and has unique mathematical and representational properties.
Consider this simple case of Roman numerals: XIV. In order to know that number as 14 you have to perform both mental subtraction and addition. If you have something like IXL * IXL it is not immediately obvious that the answer is equivalent to XXXVIII * XXXVIII + II * XXXVIII + I. This is easy to see in our modern base ten representation: 39*39 = 38*38+2*38+1 (think about the expansion of a square by one unit of width, and then, one unit of height). This is because IXL and XXXVIII don't look the least bit alike, despite only being one apart.
Even the more simple Babylonian method, and indeed all schema that use simple pictographic patterns to generate the next digit, have a serious downfall: it is easy to mistake X and X+1 because they look similar. 5 and 6 are the closest we come to making this error, and its something that has irked me for a long while. Depending on the system it may be that X and X+Y appear very similar, but it is a problem that crops up.
Using unique digits is far superior for humans, mere mortals as we may be, even if it restricts us from using larger bases.
Actually, while Im thinking about it, Ive always been of the opinion that street lights should not use circles, but other shapes as well. This is for the benefit of synaesthetic and colorblind drivers, who may not be able to immediately intuit semi-rare traffic light configurations. I once saw a single traffic light that was configured as so:
Code:
XX
XX
X
X
XX
X
X
I have also seen horizontally mounted four wide lights before (and any other number of unconventional configurations), and was initially unsure what direction I should read it.
In general, I feel that too little thought goes into thinking about how we can change standards to fit real world limitations that seem uncommon, but are actually pervasive within a certain scope (ie, a certain individual who ubiquitously faces a unique challenge, or a common failing of a standard system that effects all people under certain situations). I believe that while widespread adoption of a superior solution is difficult, it can be incrementally accomplished with careful application in situational-specific cases.
Now, sorry for the hijack, and what was this thread about again?
11 Mar 2009, 1:14 am
11 Mar 2009, 1:46 am
11 Mar 2009, 8:39 am
11 Mar 2009, 11:34 am
11 Mar 2009, 12:09 pm
12 Mar 2009, 12:48 am
12 Mar 2009, 4:03 pm
CrimsonClown
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22 Mar 2009, 9:36 pm
22 Mar 2009, 11:56 pm
CrimsonClown
Tufted Titmouse
Joined: 21 Mar 2009
Age: 33
Gender: Male
Posts: 43
Location: somewhere in my mind
23 Mar 2009, 7:05 am
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