
Glossary: Prime Pages: Top 5000: 
Often mathematicians use expressions such as "almost
every positive integer is composite" or "almost all real
numbers are irrational." In each case almost every
or almost all means all but a "negligible" fraction,
but how we define that fraction (and negligible) depends on
the underlying set.
In the positive integers (the usual case in this glossary): Let P(n) be a predicate (a statement about the integer n such as "n is prime"). Let #P(N) be the number of positive integers n less than N which satisfy P(n). For example, if P(n) = "n is even," then #P(N) is floor(N/2).In other sets: In other sets the concept "almost every" is defined in different ways. For example, in the real numbers a common way to define "almost every" is to specify all but a set of measure zero (usually using the Lesbegue measure). These ideas are beyond the scope of this glossary.
Chris K. Caldwell © 19992018 (all rights reserved)
