Write Multiplication Using Sigma?

Question

Let's suppose I have the following two numbers:

$$ a= a_{N/2} a_{N/2-1} \cdots a_1 $$

and

$$ b = b_N b_{N-1} b_{N-2} \cdots b_1 $$

Where a_1 is the first digit of a, a_2 the second digit and so on.

Then how can I write their multiplication using sigmas?

For example something like:

Answer 1

Hint: Multiply both numbers by corresponding powers of 10 to make them integers. Then, any $n$ digit integer can be written as a sum: $$a=\sum_{i=0}^{n-1} a_i\cdot 10^i$$ To multiply two such sums together, use the distributive property of real numbers.