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September 23, 2003
For Muriel, About Math
For Muriel, About Math.
In a note responding to the previous diary entry, my dear friend Muriel asked me why there is a “ones” place on the left side of the decimal and no corresponding “oneths” place on the right side. This is an interesting question that I myself have often pondered.
The idea of “place value,” as far as we know, was first used by the Babylonians many, many moons ago. Previous cultures used notation more like Roman numerals with different symbols standing for 100’s, 10’s, etc. The place value system, though, allowed people to calculate rapidly without nearly as much training as, say, a Roman scribe (if you don’t believe me try calculating MCMVII * MLCXXVI rapidly, chop chop!).
All place value systems of representing quantities operate basically the same way. There is a small set of symbols which represent a certain number quantities—for us, there are ten {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, for the Babylonians there were sixty, for computers there are two {0,1}. The number of symbols used is called the “base” of the number system. Humans now typically use “base 10,” or decimal numbers and computers use “base 2,” or binary numbers.
“Place value” means that the position of these symbols relative to the other ones and the “radix point” (it is only a “decimal point” if you’re using the “decimal system”) indicates what actual quantity is represented. The number 13, for example, means 1 ten plus 3 ones. Similarly, the number 1024 means 1 thousand + 0 hundreds +2 tens + 4 ones. The way mathematicians think of this is that each place in the number has a corresponding multiplier, which is a power of 10.
Take the number 2135, for example. We can think of this as 2 thousand + 1 hundred + 3 tens plus 5 ones. “Two thousands” is the same as 2*10^3 (10*10*10=1000); 1 hundred is the same as 1*10^2; 3 tens is the same as 3*10^1 and 5 ones is the same as 5*10^0 (any number raised to the zero power equals one). Notice that our decimal system, that is our “base 10” system, has everything multiplying by a power of 10. This patter continues onto the right side of the decimal point as well. For example,
237.9 = 2*10^2 + 3*10^1 + 7*10^0 + 9*10^(-1)
You may be asking yourself what the hell ten to the negative one means. Negative exponent usually means one divided by whatever the same number would be with a positive exponent. For example
2^3 = 2*2*2 = 8
2^(-3)= 1/ (2*2*2) = 1/8, or one eighth.
Similarly, 10^(-1) = 1/10 – one tenth and 10^(-3) = 1/(10^3) = 1/1000 or one “thousandth.”
So where is the “oneths” place? Well, if 10^(-3) is thousandths, 10^(-2) is hundredths, 10^(-1) is tenths, then “oneths,” if they existed, would have to be 10^(-0), right? What is negative zero, then? Negative zero is –1*0, which is zero. We already have a name for the 10^0 place, we call that the “ones” place.
I guess, then Muriel, that the reason that there isn’t a “oneths” place is because, unlike every other number, negative zero is the same as zero
I hope this is helpful without being too geeky,
-James
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P
Posted by james at 10:43 PM | Comments (0)
September 11, 2003
Plato
Plato was wrong. Don't ask him about it, though. He'll just tell you that he's merely a corporeal avatar reflecting universal wrongness and is therefore really not wrong, per se. He's just a shitty incantation of wrong--a copy of wrongness that is flawed and therefore must possess some non-wrongness (rightness?). Poor fellow, he can't even get being wrong right. I wonder if he inherited Socrates' lackeys, Glaucon and the rest, after the hemlock incident. I wonder if he made Aristotle do his photocopies and fetch his coffee when he was a TA. I wonder if togas are as comfortable as they look.
Posted by james at 08:57 PM | Comments (0)