16. Density of the Real Numbers
•The points on the real number line are
dense, meaning that the real line has
no gaps.
•Between any two distinct points on the
real number line is an infinite number
of other points.
17. Density of the Real Numbers
•The rational numbers are dense since
between any two rational numbers there is
always another rational number. You can
always add them and divide by 2. For
example 1/2 and 1/3. You can add them
and divide by 2. 3/6+ 2/6=5/6 and half of
that is 5/12 (5/12 is certainly between 4/12
and 6/12)
18. Density of the Real Numbers
•The whole numbers are not dense. Is
there a whole number between 1 and 2?
•Irrational numbers are dense as well, as
you can do the same thing you did with
the rational numbers. Just add them and
divide by 2, finding another number that is
halfway between the two numbers.
19. Density of the Real Numbers
Of course there are many other numbers
between each rational and each irrational
number. The idea of adding and dividing by
two just ensures the existence of at least
one such number. Now if the density
property applies to rational numbers and
irrational numbers, it must apply to real
numbers since they can be viewed as the
intersection of these two sets.
