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 Jun 9 comment What is the difference between natural numbers and positive integers? Monoids are often more useful than semigroups without additive identities. Counting numbers (natural numbers without "zero") are a semigroup that can be used to define 1x=x and (N+1)x = Nx+x for any semigroup element x. Natural numbers including zero can be used to extend that definition so that applying 0x to any monoid element x will yield its additive identity. Being able to apply Nx to semigroups (which won't work if N can be zero) can be helpful, but so too can be the ability to use 0x to get the additive identity for x's monoid. May 4 revised Are there rings whose multiplicative identity is not the number 1 or number 1-based? added 161 characters in body May 4 answered Are there rings whose multiplicative identity is not the number 1 or number 1-based? Feb 19 comment Is π equal to 180$^\circ$? Is radians a "dimension" or a "direction"? If something is rotating once per second, a point one inch from the center will be moving at 2π inches per second in a direction perpendicular to the radius; likewise a point one cubit from the center would be moving at 2π cubit per second in a direction perpendicular to the radius, etc. The difference between torque and energy is that in the former force and displacement are perpendicular, while in the latter they are parallel; the magnitudes of the units, however, are the same. Feb 17 comment Is π equal to 180$^\circ$? Even within the realm of mathematics, I would posit that there are many cases where angles would be most usefully measured in units of whole circles than in units of radians, but multiplying by 2π can be done concisely enough that it's no big deal. It's a shame that programming languages never standardized on separate sets of trig functions for radian-based and whole-circle-based angles; even if hardware only supports radian-based angles, a function to compute sin2pi(x) could be faster and more accurate than code which multiplies x by 2π and then computes the sine. Feb 14 comment Is there a size of rectangle that retains its ratio when it's folded in half? @DanielAndersson: A spec could be written such that a request that a piece of paper be A0 size with any particular dimensional tolerance would be satisfied if its dimension were within that tolerance of $1:\sqrt{\sqrt{2}}$m by $1:\sqrt{\sqrt{8}}$m, even if it wasn't possible to construct a paper with perfect dimensions. Dec 26 comment Do we really need reals? Would the set of compass-and-straightedge constructable points if one was given (0,0), (0,1), and (pi,0) as a basis, be sufficient to cover the irrationals one would encounter in Physics if every measurement apparatus always yielded rational results? If not, how many additional points would be needed? Dec 8 awarded Caucus Dec 8 comment Why are integers subset of reals? @David: The equality as written doesn't work when b is negative either. Otherwise, it's worth noting that while there's one "normal" definition of division for real numbers, and one for whole numbers, integers are another story. Real division upholds both (n+d)/d=(n/d)+1 and (-n)/d=-(n/d); whole number upholds the first, but the second is meaningless. Integer division operations can be defined to uphold the second equality, but only by breaking the first. 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.