How can we define the notion of multiplication without numbers?

Here is an Euclidean construction for the product. It does need a unit segment.

(Picture from this answer)

Each line is constructed from individual segments of unit length (represented by underscores). For convenience, lets say that we want to multiply two shorter lines:

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First, we insert spaces into the first line so that we can see the individual underscores.

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Then we replace each underscore in the first line by the second line.

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Finally, we remove the spaces to obtain the result.

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This process corresponds to rewriting multiplication in terms of addition.

The product of two positive real numbers $a$ and $b$ can be thought of as the area of the rectangle with side lengths $a$ and $b$.