# Division using index notation

In summary, the use of index notation can be confusing when it comes to divisions. If an index is not repeated on the same side of an equation, there is no summation involved. However, if an index is repeated, the expression can involve summation. Divisions don't often come up with vector quantities. Additionally, a = 1/b_{ii} and a_i = b_i/c_{jj} are valid expressions.

Hello everyone,

Recently I started to use index notation, but still the division is not clear for me. I'll mention just some simple examples that I'm not sure about:

Does $a =\frac{1}{b_i}$ mean that $a = \sum_{i=1}^{3}\frac{1}{b_i}$ or $a = 1 / \sum_{i = 1}^{3}b_i$ ?

Similarly, does $a_i =\frac{b_i}{c_{jj}}$ mean that $a_i = \sum_{j=1}^{3}\frac{b_i}{c_{jj}}$ or $a = b_i / \sum_{j = 1}^{3}c_{jj}$ ?

thanks beforehand!

Generally speaking, there is no summation involved if an index is not repeated on the same side of an equation. An index that is "free" (not repeated) should be free on both sides of the equation. Hence, $a = 1/b_i$ is a nonsensical expression.

$a_i = b_i/c_{jj} = \sum_j b_i/c_{jj}$ is fine, however. Divisions don't come up very often with vector quantities, though.

Muphrid said:
Generally speaking, there is no summation involved if an index is not repeated on the same side of an equation. An index that is "free" (not repeated) should be free on both sides of the equation. Hence, $a = 1/b_i$ is a nonsensical expression.

Indeed, I'm sorry, what I wanted to write is $a = 1/b_{ii}$

Muphrid said:
$a_i = b_i/c_{jj} = \sum_j b_i/c_{jj}$ is fine

thanks! it is clear now.