# Step By Step Calculus » 8.1 - Summation and Product

Synopsis
When we want to work with the sum or product of a long list of numbers, it is convenient to introduce efficient notation.
• An example of summation notation is: \sum\limits_{i=1}^{4}i\sum\limits_{i=1}^{4}i, which translates into english as “add up the natural number ii, starting with i=1i=1 and stopping with i=4i=4, in other words 1+2+3+41+2+3+4. The general form of summation notation is:
\displaystyle \sum\limits_{i=m}^{n}a_i = a_{m} + a_{m+1} + a_{m+2} + \dots + a_{n}, \quad \mathrm{where} \; i,\; m,\; n\in \mathbb{Z}, .
\displaystyle \sum\limits_{i=m}^{n}a_i = a_{m} + a_{m+1} + a_{m+2} + \dots + a_{n}, \quad \mathrm{where} \; i,\; m,\; n\in \mathbb{Z}, .
• Product notation is very similar, written as \prod\limits_{i=m}^{n}a_i=a_m \times a_{m+1}\times \cdots\times a_{n-1} \times a_n \, . \prod\limits_{i=m}^{n}a_i=a_m \times a_{m+1}\times \cdots\times a_{n-1} \times a_n \, .
• A frequently-used case of product notation is \prod\limits_{i=1}^{n}i\prod\limits_{i=1}^{n}i. It is written simply as n!n! and called \mathbf{n}\mathbf{n} factorial. Remember that 0!=10!=1.
• One of the basic summation formulae is \displaystyle\sum_{i=m}^ni=\frac{(m+n)(n+1-m)}{2}\displaystyle\sum_{i=m}^ni=\frac{(m+n)(n+1-m)}{2}.
The following are the basic properties of summations and products.