# Dealing with discrete variables as continuous for K-means clustering (or not)

It is well established that k-means works best, and is designed for, continuous variables. I am considering a clustering problem where I have data like this:

total spend / \$ number of items in basket hour of purchase (out of 24 hours: 8 = 8am; 20 = 8pm)
20.00 6 13
23.34 7 7

This data is all mathematically discrete and I am considering two approaches to enable me to include this in my data along with the rest of my continuous data :

(a) Just pretend the data is continuous (is it actually a problem if the centroid of a cluster sits at 6.23 items in a basket and the hour of purchase is 13.6?)

(b) Categorise them (like the example below) and use k-modes clustering:

number of items in basket - normal number of items in basket - categorised
2 small
9 medium
19 large

My concern with approach (b) is that I loose the ordinality of the data - k-modes won't differentiate the greater size difference between small and large vs small and medium vs approach (a), where the difference between 3 and 20 vs 7 and 9 is clear for k-means.

Based on this data (total spend, hour of purchase etc.) is one of these approaches more likely to work better for k-means than the other? I know there is never one right answer here and it's very situational however any guidance and thoughts about the approach to take here would be appreciated.