Linear Independence of Irrational Numbers
Lesson · Intermediate
Algebra
Instead of √5, we can consider ∛5 or any other irrational number.
Assume b ≠ 0.
This is a contradiction because √5 is irrational.
By the first property,
Suppose
where x, y ∈ ℚ. Find x and y.
Compare the rational parts:
so
Compare the coefficients of √7:
Thus
so
Suppose
where a, b ∈ ℚ. Find a and b.
Take reciprocals:
Rationalize:
By uniqueness,
Suppose a, b ∈ ℚ and
Find all possible values of a + b.
Expand:
Compare rational and irrational parts:
and
so
Since a ∈ ℚ,
So
The statement
requires
It is false if arbitrary real values of a and b are allowed. For example, taking
gives
even though a, b ≠ 0.