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Sep
26
comment Can a number have infinitely many digits before the decimal point?
Your second highlighted line should say that "0000.1111" is +1 rather than -1. Note that Boolean operators could be defined on real numbers using the above principles if one asserts that all negative numbers should be regarded as having an infinite string of "1"'s on both sides and all positive numbers with an infinite string of zeroes [under such rules, -1 would be regarded not as ...1111.0000... but ...1110.1111....].
Sep
26
comment Can a number have infinitely many digits before the decimal point?
@OscarCunningham: In binary, the value -1/3 would be "10...101010101.0"; the value 1/3 would be twice that, plus one, or "10...010101011.0". A fraction like 1/6 would be half of that, or "10...1010101.10" (which may be viewed as -1/3 +1/2). I think this approach will allow for all rational numbers, though those whose denominator is a multiple of 2 without being a power of 2 will have an infinite number of digits to the left as well as a non-zero finite number of digits to the right.
Sep
25
comment Can a number have infinitely many digits before the decimal point?
The normal formula for computing 1+n+n^2+n^3... is 1/(1-n). The fact that the formula yields -1 when n=2 might be viewed as nonsensical, but if one regards integers as "wrapping around" after value which, while infinite, is precisely one greater than the sum of that series, such a definition will yield an algebraic ring in which most of the normal properties of integer arithmetic hold, but some some others do as well.
Sep
25
comment Can a number have infinitely many digits before the decimal point?
That property is cool; I would guess it's a result of 10 being the product of two primes? Even $A^2=A$ will fail with a prime base, though I'm not exactly sure what's "going on" in base 10. What would happen in bases 6, 12 [has a duplicate factor], or 30 [smallest number with three factors]?
Sep
25
comment Can a number have infinitely many digits before the decimal point?
10-adic integers allow for the possibility that the leading infinite string is something other than all zeroes or all nines, and those possibilities get really weird, but the scenario of all nines (or, more generally, base-1) can be useful even if one doesn't allow for other leading infinite sequences. Since every number of that form has an additive inverse with only a finite number of non-zero leading digits, ordering relationships, as well as things like division, remain well defined.
Sep
25
comment Can a number have infinitely many digits before the decimal point?
A curious feature of the n-adics with prime bases is that they can represent any fraction except those whose denominator would be a multiple of the base. Thus, dynadic numbers can represent any fraction whose denominator is odd, but can't represent any fraction whose denominator is even.
Sep
25
comment Can a number have infinitely many digits before the decimal point?
A+B=1 and AB=0 implies that the two numbers end in 5 and 6. One can compute any number of digits of the one ending in 5 by beginning with 5 and squaring it until the digits stop changing. Is there any nice way to compute the one ending in 6 other than by adding two to the nines' complement of the one ending in 5?
Sep
25
comment Can a number have infinitely many digits before the decimal point?
If e.g. r is an integer greater than 1 and a=r-1, the series yields -1, a result consistent with the effect of subtracting one from zero in base r. If one has a number whose last k digits are zeroes, then after subtracting one the last k digits will be r-1. If there was an infinite string of zeroes before the number prior to the subtraction, there should be an infinite string of (r-1) after. What's weird is what happens if there's an infinite string of digits that aren't all zero or r-1.
Sep
25
comment Can a number have infinitely many digits before the decimal point?
If one thinks about how arithmetic is performed, it would make sense that (in decimal) subtracting 1 from 0 would yield an infinite string of nines. The meaning of an infinite string of digits other than zeroes or nines is a bit dodgy, but saying that an infinite string of nines is -1 can be quite useful (the binary equivalent is the way two's complement math works).
Sep
25
comment Can a number have infinitely many digits before the decimal point?
Two's complement arithmetic essentially says that numbers may have an infinite number of 0's or 1's to the left, but only a finite number of digits may follow the last 0-1 or 1-0 transition. Subtract 1 from 0, and one gets an infinite string of ones. Note that if one uses the power-series formula to compute the value of 1+2+4+8+16... one gets -1 (which makes sense, since it's the result of subtracting one from zero).l
Sep
22
comment Alternative ways to say “if and only if”?
@msouth: There is a continuum of possible pronunciations ranging from a dominant stress on the first syllable, to a second syllable which is stressed and elongated to the point of ridiculousness. I don't know exactly how far it would be necessary to push the second syllable to ensure that most listeners would notice the difference in pronunciation from a normal "if", but I would expect that with enough stress many people would notice it, and with insufficient stress people would not.
Sep
21
comment Alternative ways to say “if and only if”?
@msouth: Why can't "iff" be pronounced as two syllables "i-ff", with the second one stressed? Such an utterance would be concise but clearly distinct from an ordinary "if".
Sep
17
comment Why can't a set have two elements of the same value?
Clinton's answer doesn't specify how sets are defined. I would interpret this one as offering something of a definition. Although this definition is incomplete (it fails to mention the requirement that B must be a subset of all sets containing the proper elements), it would seem sufficient to answer the particular question asked. Would you prefer another definition?
Sep
13
comment Simplest examples of real world situations that can be elegantly represented with complex numbers
My point was that someone who isn't familiar with the use of complex math in describing circuits or similar phenomena is far less likely to understand talk of "phasors" than to understand "If inductors and capacitors are regarded as resistors with imaginary values, then techniques which would normally only be able to analyze combinations of resistors will be applicable to any combination of resistors, capacitors, and inductors." I don't know who first realized that Ohm's Law applies to RLC circuits, but I find it amazingly elegant.
Sep
13
comment Simplest examples of real world situations that can be elegantly represented with complex numbers
...one can solve for each frequency independently, process the complex values back into sine waves, and add them together to describe the circuit's behavior with the actual driving waveform. Trying to model the behavior of a network containing many resistors, capacitors, and inductors would be difficult without complex numbers, but with complex numbers the behavior can be computed using straightforward arithmetic.
Sep
13
comment Simplest examples of real world situations that can be elegantly represented with complex numbers
+1 -- The linked page for "phasor" is sufficiently complex that it might make many people's eyes glaze over. It might thus be nice to offer a simple real-world example: if a circuit consisting of resistors, capacitors, and inductors is driven with a signal composed of sine waves at one or more frequencies, one can determine the behavior of the circuit at each frequency by regarding inductors and capacitors as though they have "imaginary" resistance, and then applying Ohm's law the same way as one would if the network contained only resistors. If the input contains multiple frequencies...
Aug
19
comment Simulate a 7-sided die with a 6-sided die
@kasperd: Your last sentence certainly makes sense. Some sequences merge safely, and some don't; is there a general way to determine when merging will be safe or dangerous?
Aug
19
comment Simulate a 7-sided die with a 6-sided die
...reporting digits as soon as their values were known, rather than in sequence, would cause certain sequences of output values to be more probable than they should.
Aug
19
comment Simulate a 7-sided die with a 6-sided die
[correction: mod-7 difference between that digit and preceding digit]. For example, suppose that die rolls evaluated to 0.3s. One might need an unlimited number of rolls to find out the first digit of output should be 2 or 3, but any sequence of rolls which evaluates to at least 0.260 but less than 0.301 would yield a "4" as its second digit. Likewise any sequence that evaluates between 0.2660 and 0.3001 would have a zero as its third digit. Thinking about it, I guess there's an interesting problem because while one might be able to know the value of dozens of future digits...
Aug
19
comment Simulate a 7-sided die with a 6-sided die
@kasperd: Also, out of curiosity, if each roll was, rather than being a digit of the output number, a mod-7 checksum of that digit and all preceding digits, I wonder if that would help the "bunched-up" outputs problem? That would mean that an output sequence of n digits wouldn't be associated with a disjoint region of size 1/7^n in probability space, but a slightly larger region that would overlap its neighbors. This would increase the number of die rolls necessary to establish a sequence of output digits, but it would allow later digits to be established while earlier ones were ambiguous.