What does the $\prod$ symbol mean?

Question

This is one of those cases where I would google if I could, but I don't know what to search for.

I've come across this symbol a few times, but I have no clue what it means or what it is called.

$$\prod$$

Furthermore, what are the $\coprod$ and $\amalg$ symbols for?

I could not list all the places I found it, but the one that sparked it was a discussion on solving the Diophantine equation, $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$.

Can you give the context in which you've found this symbol? $\Pi$ is frequently used for products, and $\coprod$ is frequently used for disjoint unions or for coproducts. — , Dec 28 '13 at 05:30
Excellent Question. Please note, in future, incase you ever need to know what a certain symbol means, refer to Wikipedia's list of Math symbols. I've asked the MSE community several times to make a list of their own but nothing's been done yet. But until it is, that little old wikipedia page is the best resource. — Nick, Dec 28 '13 at 09:43
I did go there, In fact I looked through quite closely.
I didn't see this symbol there because it looks more like a staple or upside down "U" on that page. See ->Π, not even close to $\prod$. (Edit: Okay, in the Math.SE font it does. Go look and see!) (Edit_2: Ahh, I was only looking at the HTML style symbols. Didn't even notice the TeX style ones.) — NictraSavios, Dec 28 '13 at 10:01
@Nictra: XD Yes, there are variants to the font. Mathematicians have never agreed on which is the standard font. But incase you ever have trouble searching for a notation in the list, go to the page and press Ctrl+F and in the search bar that pops up on the top right corner (if you have Chrome), paste the symbol you want to find. The list doesn't have all the symbols but I'm sure someone (someone from MSE) will fix it. — Nick, Dec 28 '13 at 10:05
I had no idea the find command was able to handle unicode characters. Thank you very much! — NictraSavios, Dec 28 '13 at 10:18
I did some research to come up with a good explanation, and then I came across this $\downarrow$ $$\mathop{\LARGE\mathrm K}_{n = 1}^{\infty}$$ uhhh... any ideas on what this means? I saw it used in a formula to express the value of $\pi$ — Mr Pie, Oct 27 '17 at 10:31
@user477343 https://en.wikipedia.org/wiki/Generalized_continued_fraction#Notation — Bruno Le Floch, Nov 04 '17 at 18:24

score 18 · Accepted Answer · answered Dec 28 '13 at 05:33

18

The symbol $\Pi$ is the pi-product. It is like the summation symbol $\sum$ but rather than addition its operation is multiplication. For example, $$ \prod_{i=1}^5i=1\cdot2\cdot3\cdot4\cdot5=120 $$ The other symbol is the coproduct.

answered Dec 28 '13 at 05:33

mathematics2x2life

13,322

Is there a general equation for it interms of factorials. I'm sure the OP would love to see that. – Nick Dec 28 '13 at 09:34
3

I can figure that part out for myself. $\prod\limits_{i=1}^{n} i = n! $ – NictraSavios Dec 28 '13 at 10:05
2

@Nick My example was a demonstration of $\Pi$. Indeed, there is a formula for $n!$ in terms of $\Pi$ but not the other way around, as you suggest. – mathematics2x2life Dec 28 '13 at 19:27
@mathematics2x2life why not? is there something wrong with the suggestion (genuinely curious)? – User Mar 17 '23 at 06:16

score 3 · Answer 2 · answered Mar 16 '19 at 15:12

The uppercase Pi $\prod$ symbol stands for the $product$ operator throughout mathematics, just as the uppercase Sigma $\sum$ symbol would describe the sum operator. Think of the following analogy alliteration:

Pi is to a Product ... as Sigma is to a Sum.

mathlove · Answer 3 · 2013-12-28T10:20:27.197

1

For example, $$\prod _{i=0}^{3}a_i=a_0\cdot a_1\cdot a_2\cdot a_3$$

This is a symbol for product similarly as $\sum$ for sum.

edited Dec 28 '13 at 10:20

answered Dec 28 '13 at 05:34

mathlove

139,939

Your use of the $\times$ symbol begs the question, if the products are vectors is the pi-product scalar or vector? (cross or dot product) If it is not vector, then the $\cdot$ symbol would be more appropriate. – NictraSavios Dec 28 '13 at 10:14
I think you are right. Thanks. – mathlove Dec 28 '13 at 10:20

score 0 · Answer 4 · edited Dec 09 '17 at 16:06

0

$$\prod_{i=1}^n=1\cdot2\cdot3...\cdot (n-1)\cdot n$$

edited Dec 09 '17 at 16:06

Parcly Taxel

103,344

answered Dec 09 '17 at 15:59

fed

19

1

Welcome to stackexchange. It's a good thing that you want to help people by answering questions - but why post a duplicate answer to a very old question that already has a good accepted answer? Pay attention to the new, unanswered questions and contribute when you can. – Ethan Bolker Dec 09 '17 at 16:04
1

His answer is good enough for its own reply – Mr-Programs Dec 18 '19 at 10:57
@EthanBolker Why not? It's a different way of showing the same thing. The cool thing about question answers is that no one is forcing people to read them. You also cannot display this math in a comment. – BigMistake Sep 29 '23 at 06:06
@BigMistake "Why not" because answers like this to old questions puts the questions at the top of the active queue. True, no one is forced to read them - but they are distracting to folks like me who watch that queue. To not read them I have to parse the dates carefully. – Ethan Bolker Sep 29 '23 at 12:33
@EthanBolker This is a fair point. However I think that is the fault of SE's interface instead of the answer's author – BigMistake Sep 29 '23 at 14:21

What does the $\prod$ symbol mean?

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