Algebraic Patterns – Monoid Morphism

The counting zeroes in &quot;100!&quot; problem is actually a common high school level math contest problem. You&#x27;re supposed to calculate it by hand, not with code. The trick was to notice<p>- number of trailing zeros is equal to times you can divide something by 10<p>- There are a lot more 2s than 5s as the divisor, so number of 5s is the limiting factor<p>- 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100 contribute (at least) a factor of 5 each<p>- 25, 50, 75, 100 contribute another factor of 5 each<p>So answer is 20 + 4 zeros.<p>I love monoids but I personally thought that was a bad example. Not only did it complicate the problem, it didn&#x27;t actually lead to any understanding that will let you write a generic and efficient solution for &quot;n!&quot;. Something like this O(log(n)):<p><pre><code> def countZeroes(n): count = 0 divisor = 5 while divisor &lt;= n: count += n &#x2F; divisor divisor *= 5 return count print countZeroes(100)</code></pre>

I use monoid homomorphisms extensively in xi-editor, and have a couple of slides on it at my RustConf talk earlier this month. I hope to be writing up the ideas in more detail soon (both what I&#x27;ve implemented, and speculative explorations such as doing incremental syntax highlighting).

I don&#x27;t fully understand how category theory terms are helpful in programming. It always feels backwards to me, that first you write the code, then you look through the code and pick out all the places where category-theory terminology might appear. Monoids are nice and all, but in the end it&#x27;s just a name given to a specific design pattern: you may as well reason about it in terms of code.

Ah... Monoids-shmonoids. While I appreciate abstract algebra, I feel that forcing it onto students of programming is a terrible idea. An entire generation of potential engineers and scientists (and even mathematicians) was lost when they started teaching children that geometric figures can only be &quot;congruent&quot; to each other, not &quot;equal&quot;. In the same way, I would have never been able to use, say, generic lists without giving this idea a second thought if they had told me first that those are an example of an important kind of &quot;endofunctor in the category of types&quot; (i.e. a monad).

Hi,<p>I&#x27;m new here :)<p>Each time I see something like that, I think it looks great, but I&#x27;m having a hard time understanding the meaning of the vocabulary used…<p>Is there any good (free?) resources out there to learn Monoids, Monads, traits, category theory, in short, anything used in functional languages?<p>This is something that clearly stops me from using functional languages more.