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Back to the chapter on real numbers
The
false picture of the reals as a “row of points” suggests that to the right of
each real number is the “next” real number
That is
wrong: The points on the
real line are dense, meaning that the real line has no gaps. More precisely:
The proof is just like that for the density of the rationals: If r and s are two different real numbers with , then is between them:
(WA)
By
repeating this process you can produce as many points between r and s
as you want. More generally, if a
and b are any positive real numbers, then
which
gives an infinite
number of points
between r and s directly See detailed
analysis of this proof.
A subset S of the real numbers is dense in the reals if for every real number r you can find numbers that are as close as you want to r.
The subsets of rational numbers and of irrational numbers are both dense in the reals.
¨ If r is any real number you can find a rational number as close as you want to it by truncating the decimal representation of r. For example, , so is a rational number that is within a quadrillionth of .
¨ is irrational, so is also irrational for any integer n.
So if r is any rational number,
is irrational
and you make it as close as you want to r by making n big
enough. (You could use any irrational
number in place of .) If r
is an irrational number, might not be irrational, but that does not
matter because r itself is as close
to r as you want!
How do you go about understanding this inequality?
You can play with it and you can prove it. You really need to do both to see what is happening here.
For example,
you can calculate some values of for,
say, :
a |
b |
(ar+bs)/(a+b) |
1 |
1 |
2.45 |
2 |
3 |
2.48 |
5 |
2 |
2.38571 |
2 |
22 |
2.575 |
If you stare
at this formula and these calculations for awhile, you discover:
¨ You don’t have to worry about the denominator a + b being zero
because both a and b are positive.
(You should always
consider the possibility that a denominator might be 0.)
¨ When a = b =1, the inequality (WA) becomes
¨
which says: “The average of r and s is between r and s.”
¨ The calculations show that a and b are like
weights. When a is bigger the result is
closer to r and when b is bigger the
result is closer to s.
This
sort of random pondering and calculating instances is the sort of thing all
good mathematicans do to understand a situation.
You can also
prove (WA),
using these facts about the “<” ordering on the reals:
¨ (T) If x < y and y < z, then x < z.
¨ (A) If x < y and z is any real number, then z + x < z + y.
¨
(M) If x < y and
z is any positive real number, then zx
< zy.
Proof of (WA):
Note the use of pattern recognition in understanding this proof. For example, to prove the first “<”,
¨ We have br < bs by (M), with x = r, y = s, z = b.
¨ Then ar + br < ar + bs by (A) with x = br, y = bs, z = ar.
¨ Then by (M) with x = ar + br, y = ar + bs,
¨ Try proving the second “<” yourself. (WA) then follows from (T).