(lcm)

[ocyzl]

https://www.scheme.com/tspl4/objects.html#./objects:s110 says:

Although lcm should probably return infinity when called with no arguments, it is defined to return 1.

1 makes sense to me. What's the argument for infinity?

[soegaard]

We can compare with multiplication. Here we have a neutral element 1. ( 3 4 1) => 12 ( 3 4) => 12 The general rule is that we can remove the neutral element from the arguments without changing the results. Therefore: ( 1) => 1 () => 1 In other words we are forced to define the empty product as 1.

We attempt to transfer this principle to lcm, and thus needs to find a value neutral which satisfies:

(lcm a neutral) = (lcm a)    for all a

In particular (lcm 2 neutral) = (lcm 2) = 2 (lcm 3 neutral) = (lcm 3) = 3 etc So all integers a needs to be a factor of our neutral element.

The value we need is the "product of all integers" - which strictly speaking isn't a number - so we compromise and choose +inf.0 instead.

Another way of thinking: compare lcm and gcd to min and max The neutral elements for min and max respectively are +inf.0 and -inf.0.

[ocyzl]

We attempt to transfer this principle to lcm, and thus needs to find a value neutral which satisfies:

(lcm a neutral) = (lcm a)    for all a

I agree, but 1 satisfies that equation, not infinity. "lcm" stands for "least common multiple", i.e., the smallest number that is a multiple of all the arguments. Since every number is a multiple of 1, including 1 as an additional argument doesn't change the result.

[soegaard]

You are right!

søn. 17. mar. 2019 kl. 09.10 skrev ocyzl notifications@github.com:

We attempt to transfer this principle to lcm, and thus needs to find a value neutral which satisfies:

(lcm a neutral) = (lcm a) for all a

I agree, but 1 satisfies that equation, not infinity. "lcm" stands for "least common multiple", i.e., the smallest number that is a multiple of all the arguments. Since every number is a multiple of 1, including 1 as an additional argument doesn't change the result.

— You are receiving this because you commented.

Reply to this email directly, view it on GitHub https://github.com/cisco/ChezScheme/issues/407#issuecomment-473627878, or mute the thread https://github.com/notifications/unsubscribe-auth/AAcLxQSJcLdfyqUpf0cPdpF5ndB84V9Bks5vXfiNgaJpZM4b4Ogk .

-- -- Jens Axel Søgaard