Why is this equal to 1? 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?
 A: 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 $1$.
A: 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.
A: 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.
A: Wolfram is probably contruing $\displaystyle\prod_{i=4}^0$ as the product over all values of $i$ that are $\ge4$ and simultaneously $\le0$.  There are none, so you are multiplying no numbers at all.
Multiplying by no numbers at all is the same as multiplying by $1$.
