# Multiplication as a convolution

As a brief aside, we will touch on a rather interesting side topic: the relation between integer multiplication and convolutions As an example, let us consider the following multiplication: .

In this case, we might line up the numbers, like so:

Here, each column represents another power of 10, such that in the number 123, there is 1 100, 2 10s, and 3 1s. So let us use a similar notation to perform the convolution, by reversing the second set of numbers and moving it to the right, performing an element-wise multiplication at each step:

For these operations, any blank space should be considered a . In the end, we will have a new set of numbers:

Now all that is left is to perform the carrying operation by moving any number in the 10s digit to its left-bound neighbor. For example, the numbers or 58. For these numbers,

Which give us , the correct answer for integer multiplication. I am not suggesting that we teach elementary school students to learn convolutions, but I do feel this is an interesting fact that most people do not know: integer multiplication can be performed with a convolution.

This will be discussed in further detail when we talk about the Schonhage-Strassen algorithm, which uses this fact to perform multiplications for incredibly large integers.

## License

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The text of this chapter was written by James Schloss and is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

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