Density of Irrational Numbers Theorem
Given any two real numbers α, β in R
, α<β, there is an irrational number γ such that α<γ<β.
We will first show that if r and s, s non-zero, are rational
, then r+s√2 is irrational
. Let r and s, s non-zero, be rational numbers. Suppose r+s√2 were rational. Then it could be expressed in some fashion as x/y, where x and y are integer
s. In addition, since r is rational
, it can be expressed as a/b, where a and b are integer
s, and s can be expressed as c/d, where c and d are integer
However, we know that √2 is irrational, so we have our contradiction. r + s √2 is irrational
Since α<β, β-α>0. √2>0 as well. We may use the Archimedean property
to conclude that there is some integer
m so √2<m(β-α), or equivalently,
Let n be the largest integer such that n≤mα. Adding √2 to both sides gives
But since n is the largest integer
less than or equal to mα, we know that mα<n+√2 and therefore that
We know from our above results that (n/m)+(1/m)√2 cannot be rational
, so we have shown that there is an irrational
number between α and β.
NOTE: Please see Euclid's Proof that 2^.5 is Irrational
for an explanation of the irrational nature of √2.
I like this proof
because it’s simplistic and low on vocabulary
; I suppose it’s more of a hoi polloi
-ish proof than the professors would prefer we use. The Archimedean property
is the most sophisticated tool you need to understand this, and there’s a good write-up on that. This proof is fantastic for someone being introduced to the study of analysis
or a non-major “stuck” taking a single semester of the stuff.
Taken from a homework assignment from a class titled "Fundamental Properties of Spaces and Functions: Part I" at the University of Iowa.