[NineStarPoint]

Fractional numbers make sense to me as an extension, but it also requires an intuition of division on the same lines.

Take number n and multiply it by number x/y. To do this, you have to split number n into y parts and take x number of them. So to multiply 8 by 3/4, you split 8 into 4 parts (2 + 2 + 2 + 2) and then take 3 of those parts. This ends up being 2 + 2 + 2 = 6.

For multiplying two fractions, you have to extend it to n/m * x/y. Since you can multiply the top and bottom of a fraction by the same number, you can write n/m as ny/my. Then you can have n/my be your “equal part”, and take x of them. So 1/2 * 1/2, you take 2/4 and split it into 1/4 + 1/4 and 1 of them, so the answer is 1/4.

To me at least, this makes sense as an extension of multiplication is repeated addition. It’s when you get to irrationals that it starts to fall apart, and even then the intuitions the above way of thinking led me to have served me well.