I am learning about number systems in mathematics.
I am told that prime numbers are distributed in a specific pattern amongst the natural numbers, and that there is a notion of density amongst the primes, and that the primes are less dense, the farther you are along the natural numbers.
So if this is the case:
Let [x,y] be notation for an interval of natural numbers: the set whose elements are x, y, and every natural number between x and y, if the expression makes sense.
The length of an interval [x,y] is y - x.
If we have 2 intervals of natural numbers of the same length, say [a,b] and [c,d]. Then they may each contain a different number of prime numbers. If c is much larger than a, then its likely there are more primes in [a,b] than in [c,d].
Is this correct?
If we now let [a,b] and [c,d] be intervals of real numbers of the same length, then analogously, does the position of [a,b] and [c,d] in the number line affect the "density" of certain types of irrational numbers in the intervals?
For example are Quadratic Surds, or transcendental, or constructible numbers more "prevalent" earlier in the number-line (closer to 0) than farther out (as in the case of primes)?
I'm trying to see if intervals of real numbers are the "same" with regards to diversity of types of number. Or equivalently, if the density of the various types of special rational and irrational numbers is constant throughout the number-line (unlike the case of primes in the natural numbers).