Question

@Limitless can you tell me what that capital pi meant?

\(n!=\prod_{1 \leq f \leq n}f=1\cdot 2\cdot 3\cdots n\)

Can I have more examples please?

Answer 1

multiply

Answer 2

just like \(\sum\) means add

Answer 3

I don't get how that represents factorial. what's up with the f?

Answer 4

\(f\) is the index of the productand. You multiply through all \(f\) in the set \(\{1,\dots,n\}\).

Answer 5

the \(1\leq f\leq n\) tells you to multiply all the numbers together between 1 and n

Answer 6

Hmmm I think I'm getting it.... more examples please?

Answer 7

just like you could write
\[\sum_{1\leq k\leq n}k\] to mean add all the numbers between one and n

Answer 8

\[\prod_{1 \leq q \leq n}(q+1)=(1+1)(2+1)(3+1)\cdots(n+1)=(q+1)!\]

Answer 9

\(k\) is constant, if I'm not wrong

Answer 10

No, \(k\) is being used as an index.

Answer 11

try here
http://www.saddleback.edu/faculty/aorrison/mathhelp/sumprod1.htm

Answer 12

What if I want to multiply all numbers between 1 and 10?

Answer 13

Let \(n=10\) :P

Answer 14

\( \color{Black}{\Rightarrow \prod_{1 \le f\le 10} f }\)

Something like this?

Answer 15

i.e.
\[\prod_{1 \leq q \leq 10}q=1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10=. . .\]

Answer 16

Yes. I lagged.

Answer 17

yeah, but you can also write it as you do with a summation
lower limit at the bottom, upper on top

Answer 18

i prefer \[n!=\prod_{\begin{matrix} 1 \leq f \leq n \\f\in\mathbb{Z} \end{matrix} }f=1\cdot 2\cdot 3\cdots n\]

Answer 19

\[ \prod_{n=1}^5\frac{n}{n-1}\]

Answer 20

Why do we have that \(\le\)?

Answer 21

Oh ok that lower and upper thing is a lot more easier.

Answer 22

Zarkon, that's a weebit pedantic. \(f \in \mathbb{Z}\) is implied.

Answer 23

That's how I used to do in summation.

Answer 24

yeah in some circumstances

Answer 25

not always

Answer 26

exactly, it is the same only with products

Answer 27

Not always, no. But in most cases, yes. Particularly elementary ones such as factorials. But, I have to say, there is a weebit of elegance to your definition.

Answer 28

\[n!=\prod_{k=1}^nk\]

Answer 29

I think I get it......

\( \color{Black}{\Rightarrow \Large \prod_{n = 1} ^{10} k}\)

Is this to add all numbers between one and ten?

Answer 30

Or maybe you'll put a \(n \epsilon \mathbb{Z} \) too.

Answer 31

Parth, I am using what's basically a notation employed by computer scientist Donald E. Knuth. It is a much superior notation to the delimited form (what Satellite is showing you). It has much easier manipulations. It's a weebit confusing, at first, however. But the two notations are equivalent. The one I'm using, though, is more easily manipulated.

Answer 32

I'm in trouble because I'll have to choose someone to give a medal to :P

Answer 33

If you pick up a copy of Concrete Mathematics, he has an entire chapter dedicated to this. It's an elegant notation.

Answer 34

i have enough, thank you. if i get any more they will take my bike away

Answer 35

ima help u @ParthKohli :u give someone i give the other ;)

Answer 36

Hmm so maybe Limitless is fine. Would you please not mind showing me how to write the product of numbers 1 to 10?

Answer 37

I have already.

Answer 38

Haha I want sat's notation.

Answer 39

limitless is certainly correct about usage and indexing. later you will see infinite products, such as
\[\prod_{p\in P} \frac{1}{1-p^{-s}} = \prod_{p\in P} \sum_{n=0}^{\infty} p^{-ns} = \sum_{n=1}^{\infty} \frac{1}{n^s} = \zeta(s) \]

Answer 40

\[\prod_{k=1}^{10}k\]

Answer 41

@Limitless where will I write k \(\epsilon \) \(\mathbb{Z}\)?

Answer 42

you might wan to index over numbers other than integers. the one i wrote above is the riemann zeta function, and the index is over prime numbers

Answer 43

I'll see the page that you linked me to. Thank you @satellite73 @Limitless !!!

Answer 44

Parth, you're getting into another question now.

Answer 45

you dont need \(k\in \mathbb{Z}\) with this notation \[\prod_{k=1}^{10}k\]

Answer 46

Also @Zarkon