Textbook
For a finite collection of numbers, $\prod_{i=m}^na_i$ means $a_ma_{m+1}\cdots a_n$, with integer bounds $m\le n$ included. The symbol $\prod$ is a capital Greek pi. The index selects factors rather than being added to them.
A finite index set may also specify the factors: $\prod_{i\in I}a_i$. For an empty set $I$, the result is 1, the multiplicative identity. A zero factor makes the entire product zero. Infinitely many factors require additional definitions and convergence conditions; the finite rule does not extend automatically.
Multiplication of numbers is associative, so splitting a finite collection into parts and multiplying the partial products preserves the result. An empty part must contribute 1 for that rule to continue working. Inclusive bounds $m\le n$ specify $n-m+1$ factors rather than $n-m$. Index conditions select the values themselves: omitting an index removes a factor rather than replacing it with zero.
Definitions and further reading: Mathematics for Computer Science.
In plain language
A finite product resembles repeated scaling: factors 2, 4, and 6 act on the size already accumulated. Adding those factors performs a different operation and gives a different result. An empty chain leaves one because multiplication by one changes nothing. A zero factor collapses the size, and later scaling cannot restore it; starting the accumulator at zero sabotages the operation before it begins.
Example
The example uses the original values 2, 4, and 6, with an accumulator initialized to 1. Selecting only indices $\{1,3\}$ would skip 4 and give 12.
Finite Products: Reading the Product Sign
First factor
Multiplication of the neutral starting value 1 by 2 gives 2.
Step 1 of 3: First factor
Read all steps
First factor
Multiplication of the neutral starting value 1 by 2 gives 2.
- State
- 1currentΓ2current2current
Second factor
The accumulated product becomes 8.
- State
- 2currentΓ4current8current
Third factor
The last factor gives 48.
- State
- 8currentΓ6current48current
After the second factor the accumulator holds 8, so the last step computes $8\cdot6$, rather than $4\cdot6$ after forgetting the first value. The compact notation expands as $1\cdot2\cdot4\cdot6$ to check the sequence. The initial one records the starting state but is not another observed measurement. The result 48 belongs to all three selected indices; omitting either 2 or 4 changes the calculation.
Related concepts: Finite sums.