Irrational Numbers from Adjacent Natural Numbers (Part VIIIa)

A Staircase Sequence of Irrational Numbers Derived from √(n((x ∕ y) + x + y)

This is a continuation of the induction proofs Part VIIa and Part VIIb for √n((x ∕ y) + 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 there are two common differences (Δ₁) and (Δ₂) between the irrational numbers generated from Part VIIb for √n((x ∕ y) + x + y). So, therefore, let's proceed as follows:

The sequence derived from the equation √n((x ∕ y) + x + y) is shown in Tables I and II where the values of n range from 1 to 14, while those of x and y range, respectively, from 1 to 28 and 2 to 29. The differences (Δ₁) and (Δ₂) between the irrational numbers are, respectively, about 0.5 and 1.5 where Δ₂ corresponds to the end of one Δ₁ to the start of the next Δ₁ in the sequence, and approach these values as x and y increase without bound. The value for each √n((x ∕ y) + x + y), is calculated to 6 decimal places, is irrational and each of these values indeed falls between their respective consecutive natural numbers. In addition, the fourth column heading is SQRT = √n((x ∕ y) + x + y).

If the natural numbers are plotted on a grid vs n the following picture of a staircase is obtained. The blue lines are the Δ₁ differences of 0.5 within the run of the staircase, while the black diagonal, is the difference Δ₂ of 1.5; the red, in addition, is a jump from one level to the next (the rise of the staircase).

Is still possible to write out the as one line sequence (for the first ten natural numbers, in red):

However, while when we look at the sequence in this way (as a ramp), there's no way of knowing if the sequence is a staircase or a ramp like the sequence derived from √xy in Part I which has no n levels. The Δ₁ and Δ₂ can still be read off as well as the differences between a natural number and the Δ₁ to its right which approaches 0.25 and 0.75 for a pair of natural numbers. In the staircase sequence the n is required in order that the irrational numbers fall between their two adjacent natural numbers. It wasn't expected that the irrational numbers fell right smack-dab in the same general areas as shown in the figure. In addition, connecting all the 0.25 terms together and all the 0.75 terms together gives "rise" to two parallel ramps where the sequence may have both ramp and staircase properties depending on the view.

Go back to Part VIIa and Part VIIb for proof by induction for and Part VIIb for √n((x ∕ y) + x + y).

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