Irrational Numbers from Adjacent Natural Numbers (Part VIb)

A Sequence of Irrational Numbers Derived from √2xy² ∕ (x + y)

It has been shown (Between_two_Rational_Numbers on Wiki) that between two real numbers (ℝ) there exists an irrational number. Accordingly, it will be shown here that between two adjacent natural numbers (ℕ) there exists an irrational number (ℝ\ℚ). In addition, an arithmetic progression or sequence of irrational numbers will be generated in this section where the common difference (Δ) between the irrational numbers generated from √2xy² ∕ (x + y). The inductive proof for this irrational equation is found in the previous section Part Vb. So, therefore, let's proceed as follows:

The sequence derived from the equation √2xy² ∕ (x + y) is shown in Table I where the values of x and y range, respectively, from 1 to 12 and 2 to 13. The difference (Δ) between the irrational numbers is approximately 1.0 in the sequence, and approaches the value of 1 as x and y increase without bound. The value for each √2xy² ∕ (x + y), is calculated to 6 decimal places, is irrational and each of these values indeed falls between their respective consecutive natural numbers. Table I shows the results where The third column heading SQRT is equal to √2xy² ∕ (x + y). The last two rows shows data for large values of n, giving a Δ value close to one.

The sequence can be written as one line sequence (for the first ten natural numbers, in red):

Go back to Part Vb for proof by induction of √2xy² ∕ (x + y).

Go to Part I or Part IV for non-staircase (ramp) methods. Go back to homepage.