Math: density property

Tutoring math, the density property is important. The tutor mentions it.

The following is by my understanding.

In math, the density property is applicable to some number sets. The integers aren’t dense on the number line, but the rationals are.

The density property applies to a set of numbers if, for any two members of it, arbitrarily many another members of it can be found between those two.

Integers (….-3,-2,-1,0,1,2,3….) aren’t dense on the number line; therefore, the density property doesn’t apply to the set of integers. Confirmation of this is that there isn’t an integer between 2 and 3.

The density property does apply to the set of rational numbers: between a/b and a/(b+1) is a/(b+0.5), where a and b are members of the rational numbers, a≠0, and b∉{0,-1,-0.5}. For example, 30/75 is between 3/7 and 3/8.

Source:

Travers,K.J., Dalton,L.C., Brunner,V.F., Taylor,A.R. (1977). Using Advanced Algebra. Doubleday Canada Limited.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.