[digama]

Oh but they do. When taking measurements in a scientific setting, it usually gives the most accurate and simple model to assume that your measurement is an approximation of a certain real number, and understand more accurate measurements as better estimates of this real number. Using rational numbers or other countable sets in this position usually leads to undesirable biases and or circuity in the model. When you need a "continuum", the real numbers have the best properties for the job.