Do humans mentally discretize numbers?
One common way of framing numerical cognition is in terms of a "mental number line". This mental number line is thought to have a logarithmic scale, so perceived differences are inversely proportional to their magnitude. For example, the difference between 6 and 7 is perceived as bigger than the difference between 76 and 77. This is just a variant of the Weber-Fechner Law, which applies in many perceptual domains (sound loudness, object weight, light brightness, etc.). So people don't exactly put numbers into discrete bins, but small differences begin to blur together for larger numbers. 

In typical, educated adults there is also thought to be an explicit symbolic or linguistic representation of numbers that captures their exact value and allows us to do exact calculations.

For theories of numerical cognition, one good place to start is:

Feigenson, L., Dehaene, S., &amp; Spelke, E. (2004). Core systems of number. Trends in cognitive sciences, 8(7), 307-314.