4 ms·

I don't think I've seen this notation before: (a,b) = 1 I guess it means gcd(a, b) = 1

• acchow 13y ago
  Yes. This is standard notation in many textbooks and university classes.
• vog 13y ago
  Yes, this is a short notation of gcd, and it is quite common among mathematicians.
  • Dewie 13y ago
    Is there any good reason for this short notation other than saving three letters for the well-initiated? Does it make sense in the context of other mathematical notation conventions (ie could you guess that it meant gdc(a,b) without a lot of incriminating context pointing towards it)? Mathematics is hard enough for laymen as it is.
    • wging 13y ago
      There is some precedent for the notation (a, b) or <a, b> or [a, b] meaning "some function applied to a and b". For example, inner products or commutators (for a group, the commutator [a, b] of a and b is equal to a^-1 b^-1 ab). The compactness really can matter sometimes... you might have a gcd appearing in the initializer of a summation symbol. http://en.wikipedia.org/wiki/Mobius_inversion http://en.wikipedia.org/wiki/Mobius_inversion gives you a good example of the "d divides n" notation appearing in such a position... pretty sure there exist similar constructions but with gcds.
      • vog 13y ago
        Also, when stretching analogies far enough, one could argue that the GCD serves a similar purpose in number theory like the inner product in linear algebra. :-) However, I think the real reason is the notation of GCD as an Ideal generated over two elements of a Ring, as described in the other reply: https://news.ycombinator.com/item?id=6090688 https://news.ycombinator.com/item?id=6090688
    • PieSquared 13y ago
      As I learned recently, there is actually a good reason why gcd(a, b) is often abbreviated as (a, b). Roughly speaking, in ring theory, an ideal I is some subset of a ring (which you can think of as a number system, sorta) for which ai is in I (if a is any element of the ring and i is in I). Given any element x, we can generate an ideal from x by just taking every other y in the ring and computing yx - if every yx is in the ideal, then multiplying by some other z to get zyx is the same as just having (zy)x ,which is just a different multiple of x. This ideal - the ideal generated by x - is written as (x); if it's generated by multiple elements, it's written as (x1, x2, ...). If you consider the ring of the integers, then the ideals are multiples of some integer. For instance, the multiples of three are an ideal, because multiplying a multiple of three by any integer yields a multiple of three (so if you multiply the set of multiples of three by any integer, you just get back something that's in the ideal). The final point is this: the ideal generated by two integers is actually just the set of multiples of their gcd. Therefore, if a and b are integers, (a, b) is the set of multiples of their gcd - just like (3) is the set of multiples of three. This is why the gcd is often written in this way. The cool thing is that this works in rings in general, not just integers. You can extend the concepts of gcd, primality, divisibility, etc to rings in general, and operate on things besides just integers, such as matrices, polynomials, or rotations of a cube. For more info: http://en.wikipedia.org/wiki/Ring_theory http://en.wikipedia.org/wiki/Ideal_(ring_theory)
      • tel 13y ago
        This is a great counterexample to those that decry mathematical notation. Ambiguity may not be introduced for raw efficiency, but instead to indicate powerf techniques through punning.
        • vog 13y ago
          The same holds not just for math symbols but also for math names. Many things have similar/same names because there is a deep mathematical connection that justifies the conflation.
          • ErsatzVerkehr 13y ago
    • deleted 13y ago
      [deleted]
• coherentpony 13y ago
  It's actually standard notation for the highest common factor.