What does ∏ mean and how can i use it?

[Vacrin]

ive only studied up to calculus 2, and have never encountered ∏, but from what i believe it means, it is a multiplier. much like summation but instead of adding things together they are times'd together. is this correct? and also can i see a example? thank you :)

 

[Borek]

http://en.wikipedia.org/wiki/Infinite_product

http://mathworld.wolfram.com/InfiniteProduct.html

 

[dkotschessaa]

Short answer - you are exactly correct. Read it exactly as you would a summation sign, but change the operation to multiplication.

So, for a really boring example, you had, for example, $\displaystyle\sum\limits_{i=0}^3 i^3$ it would be $0^2 + 1^2 +2^2 + 3^2= 0 + 1 + 9 = 10$

But if I changed that to $\prod_{i=0}^3i^2$ it would be $(0^2)(1^2)(2^2)(3^2)=0$

Probably less boring if I didn't make it start from zero, but you get he idea. It's a very different answer! (Also note that the formatting is usually the same, with the i=0 and the 3 above and below the sign respectively. For some reason it didn't format that way here. But generally the notation is very similar.

-Dave K

 

[D H]

For some reason it didn't format that way here.

That's because you used inline math mode for one, display mode for the other.

Display mode:
$$\prod_{i=0}^3 i^2$$
$$\sum_{i=0}^3 i^2$$

Inline: $\prod_{i=0}^3 i^2$ versus $\sum_{i=0}^3 i^2$

 

[dkotschessaa]

Thanks DH

 

[Vacrin]

Thanks guys :)

isnt there also a upside down one of ∏ these? is that the exact opposite? meaning that it is always division instead of multiplication?

 

[Mentallic]

I don't think I've ever seen an upside down Pi, but maybe you mean

$$\bigcup$$

which is from set theory and has nothing to do with the product symbol.

 

[micromass]

There is the symbol $\coprod$. But it is used in category theory and abstract algebra. It has nothing to do with the symbol $\prod$ as used in this thread.

 

[Curious3141]

There is the symbol $\coprod$.

I am familiar (intimately so) with upside-down cake. But this is the first time I'm hearing of upside-down pi.

 

[WannabeNewton]

I am familiar (intimately so) with upside-down cake. But this is the first time I'm hearing of upside-down pi.

http://instantrimshot.com/

 

[Vacrin]

There is the symbol $\coprod$.

How does that symbol work? I'm very curious to find out. It probably won't make any sense seeing as how I've never studied abstract algebra

 

[micromass]

How does that symbol work? I'm very curious to find out. It probably won't make any sense seeing as how I've never studied abstract algebra

In some sense, it is the "dual" of the product $\prod$. It is defined as such in category theory. The general definition probably won't make sense to you. But if you know sets, then we can define it as the disjoint union. That is, we define

$$\coprod_{i\in I} A_i = \bigcup_{i\in I}( A_i\times \{i\} )$$

Informally, we just take the union of the sets $A_i$, but we force them to be disjoint by taking the cartesian product with $\{i\}$.

 

[Vacrin]

I wish that made more sense, I recognise one symble, the U, but I'm not sure how to approach the A, and tridant thingy, as well as {i}, but the brackets, bring up a familiar idea such as {a,b,c}

 

[Ronnin]

What makes them disjoint? Is that because each A sub i set is mapped to a different i?