is pi capital similar to exponents (in some way)?

[mohammad muazzam]

does
0
Π 4 = 1?
1
and does
-1
Π 4 = [tex]\frac{1}{4}[/tex] ?
1

[HallsofIvy]

No, [tex]\Pi[/tex] has nothing directly to do with "exponents"- but in these particular examples an exponent "sneaks in"! [tex]\Pi[/tex] is "similar" to [tex]\Sigma[/tex]. While "[tex]\Sigma x_i[/tex] means to add the various [tex]x_i[/tex] values, [tex]\Pi x_i[/tex] means to multiply them.

In your examples, there are two peculiar things. First you have the upper limit less than the lower limit. That is unusual but doesn't really change anything. Second the items to be multiplied don't depend on any index.

In the first one, when the index is 1, the quantity to be multiplied is "4". When the index is 0, the quantity to be multiplied is also "4". Since those are the only two items their product is [tex]4(4)= 4^2= 16[/tex], NOT "1".

In the second example the three values from -1 to 1 are -1, 0,and 1 so there are three numbers to be multiplied. It happens that those three values are all "4" so the product is [tex]4(4)(4)= 4^3= 64[/tex], NOT "1/4",

But the fact that these are exponents is peculiar and not a general property of [tex]\Pi[/tex]. More common would be something like [tex]\Pi_{i=1}^3 i= (1)(2)(3)= 6[/tex]