151–160 of 198 posts
Hmm?
Ah.
> CDF files can be read using a proprietary CDF Player with a restrictive license, which can be downloaded free of charge from Wolfram Research. [1]
Fuhgeddaboudit.
[1] https://en.wikipedia.org/wiki/Computable_Document_Format
Earlier quoted context omitted.
Why? No matter what radix you pick, there's always going to be some rational-value you cannot accurately encode. Moving to binary just gives you a different distribution of "un-encodeable values". (I'd say a larger/worse set, but I'm not yet sure how to prove it.) In the long run, trinary gives you the best radix-economy, being closest to e: https://en.wikipedia.org/wiki/Radix_economy
Making a self-reply here, I think I know how to show binary is "worse" when it comes to un-representable rational numbers. I'm not a math major, so this is probably some incredibly obvious textbook stuff to somebody else, but... In base-N, you can accurately write any fraction of (1/y) provided that y can be expressed using the same prime-factors found in N. For example, 10 has the prime factors of 2 and 5, leading t…
But there are two approaches that we can use.
The first is to point out that in real life it is more common to encounter small numbers than big ones. So if you choose a probability distribution for what denominators you expect in practice, and sum up the values of all of the exactly represented numbers, you can get to a fixed probability of even division. And it won't be zero.
Unfortunately your choice of the relative probability of having a 2 in the denominator versus a 3 is extremely arbitrary. So while the approach makes sense, the numbers you get really will be made up. (Though no matter how you make them up, your general statement is guaranteed correct.)
The second approach is to just look at how the count of exactly representable numbers below N scales with both N and your set of primes. This has been studied. The exact counts are a mess, but if you have a set S of prime divisors of size k, then the count is (1+o(1)) * 1/k! * [product over p in S of (log(N)/log(p)].
The upshot is that for large N, more prime factors always wins. And for specific prime factors, you just look at the ratio of logs.
For instance with this approach we can say that log(3)/log(5) = 1.46497352071793... times as many denominators are exactly representable in base 12 as in base 10. (The log(2) factors cancel out.) Therefore base 12 is better.
Of course the second approach says that base 6 is as good as base 12, and it is smaller, so why not use it? Well, the reason why we keep on winding up with base 12 in practical situations is that divisibility by 4 comes up a lot. It isn't just exactly representable that matters, efficiency matters.
Plus inches in feet.
On a more serious note, the world does not end with 10.
Earlier quoted context omitted.
You have to be clear on the difference between the number and its representation. In English we represent numbers in base 10 unless otherwise specified. Hence you say things like, The base 12 number 10 is 12. I did not otherwise specify, so I represented my numbers in base 10 everywhere even though I was talking about base 12.
In math, we prefer to use either (101010)_2 or (101010)2 or 101010_2 – which represents (42)_10 (With TeX formatting implied)
In fact we mostly indicate the base when there is a good possibility that it isn't 10, and any stray numbers fail to have a base represented, they should be assumed to be base 10.
(Mathematical notation mostly uses sensible defaults except for differential geometry.)
Earlier quoted context omitted.
Making a self-reply here, I think I know how to show binary is "worse" when it comes to un-representable rational numbers. I'm not a math major, so this is probably some incredibly obvious textbook stuff to somebody else, but... In base-N, you can accurately write any fraction of (1/y) provided that y can be expressed using the same prime-factors found in N. For example, 10 has the prime factors of 2 and 5, leading t…
no. You can accurately represent anything which is k*2^n, where (k,n) are integers. Simple example: 3 is not 2^n but obviously expressing 3 as a floating point is not a problem. Similarly 1.5, etc. In this fashion, primes p seem "better" as they increase, because for a given size limits on k and n, you get more 'numbers' per unit space on the real line. Composite numbers are 'even better', but there are a lot of pain…
It's moved from "should" to must. "Every child in England will be expected to know their times tables before leaving primary school from next year. Pupils will be tested against the clock on their tables up to 12x12 in new computer-based exams that the Department of Education (DfE) said were part of the government’s “war on innumeracy and illiteracy”." http://www.theguardian.com/education/2016/jan/03/pupils-face... G…
I'm not sure people need to know the 12 x 12 table, so much as knowing how percentages and compound interest work.
How long will it take someone to pay off a credit card debt if the APR is 39.9% and they can't afford to clear the balance every month? If they don't understand why the answer is "Probably forever unless they get lucky with an inheritance" they shouldn't be using a credit card.
I suggest learning them up to 16, or at least learn the 16th column-row (some of the others not so much).
Really, if I could have come out of school with the 20x20 table memorized, I wouldn't mind now at all.
Earlier quoted context omitted.
But do we all learn tables up to 12? In the Netherlands it's only up to 10 for as far back as I know.
Same in New Zealand. The only times I ever saw anything with 12x tables it had escaped from the US education market and had exercises involving nickels, dimes and quarters etc.
Earlier quoted context omitted.
Making a self-reply here, I think I know how to show binary is "worse" when it comes to un-representable rational numbers. I'm not a math major, so this is probably some incredibly obvious textbook stuff to somebody else, but... In base-N, you can accurately write any fraction of (1/y) provided that y can be expressed using the same prime-factors found in N. For example, 10 has the prime factors of 2 and 5, leading t…
no. You can accurately represent anything which is k*2^n, where (k,n) are integers. Simple example: 3 is not 2^n but obviously expressing 3 as a floating point is not a problem. Similarly 1.5, etc. In this fashion, primes p seem "better" as they increase, because for a given size limits on k and n, you get more 'numbers' per unit space on the real line. Composite numbers are 'even better', but there are a lot of pain…
In fact, I think it exemplifies the upper bound for the number of digits you'd need for "related fractions". For example, we know "one eighth" is expressible in decimal as 0.125. Even if you pick a really big N value, N * (1/8) should never need more than three digits to the right of the decimal point.
I HATED memorization. I was able to do the math in my head up through 10, but 11x11, 11x12 and the rest of the 12s eluded me, and I just really, really was not able to sit still and just do flashcards.
I ended up only chancing back on math through programming and a course in logic in undergrad. Everything else was mostly self taught as needed.