Why is
$$ \prod_{i=4}^0 (4i -1) = 1 $$
At least according to: http://www.wolframalpha.com/input/?=prod_{i%3D4}^0+%284*i+-+1%29 It is rather unintuitive, why would the product even be defined? One could however interpret it as letting i decrease from 4 to 0. But that clearly is not the case. So what's going on?
On
WA probably considers this product (where the upper bound is less than the lower bound) to be empty. And the empty product is 1 by definition (just like the empty sum is zero). So it's not really a math question, more like a Mathematica question.
On
What you have here is a product in terms of $i$ from $4$ to $0$. Since $4\leq0$ is false, the set of $i$ such that $4\leq i\leq0$ is empty, and so this is an empty product. The empty product is defined to be $1$.
What you likely wanted was the following $$\prod_{i=0}^{4}\left(4i-1\right).$$ Putting this into Wolfram Alpha gives -3465.
As far as I am aware, the standard way to interpret your product notation is that it would be the empty product, as Wolfram Alpha seems to interpret it.
An empty product, by definition, is equal to $1$. An empty sum is defined as $0$.
You can think of it as a generalization of the idea that $a^0=1$ and $0!=1$.
More generally, if you have a finite set, $X$, and a function $f:X\to \mathbb R$, then for any subset $A\subset X$, you can define:
$$\prod_{x\in A} f(x)$$
this product has the property that if $A,B\subset X$ and $A\cap B=\emptyset$ then $$\prod_{x\in A\cup B} f(x) = \left(\prod_{x\in A} f(x)\right)\left(\prod_{x\in B} f(x)\right)$$
For this to be defined for all subsets of $A$, we have to define it for $A=\emptyset$. The only value for $\prod_{x\in \emptyset} f(x)$ that keeps the above property is $1$.