The Life Well Lived: The Importance of ‘Attitude’

Let ‘R’ denote the set of Real Numbers. Also, let ‘N’ denote the set of Integers. The naturalness of R is evident in the realization that two numbers, x and y, both in N can span some number, v=x/y which itself is not in N, but is in R. To sort of belabor the point, with x=1 and y=3, whereas v=(x/y)=(1/3) is in R, it is not an element of N. The set of Real Numbers makes sense, ensures we do not entertain any consternations whenever we engage in mathematical exercises that, initially consist solely of the set of Integers.

Relative to N, an important property of R is the reality that there does not exist any empty space between any two adjacent numbers. For illustration, let x=1 and y=2 be domiciled in N. Is there any integer in-between? Well, the answer is a resounding No. We say then that the space that resides between any two adjacent integers is an Empty or Null Set.

Contrarily, suppose we domicile x=1 and y=2 in R. Now we have that each of 1.25, 1.50, or 1.75 reside in-between 1 and 2. If we reset x=1.50 and y=1.75, we have that 1.625 resides in-between the two numbers; if, otherwise we reset x=1.25 and y=1.50, yet again, we have that 1.375 resides in-between the two numbers. By now, I guess you get the drift, namely that on the Real Line, always there exists some number between any two adjacent numbers. It is not ever the case…