Irrational Numbers from Adjacent Natural Numbers (Part XIb)

A Staircase Sequence of Irrational Numbers from √(x + 1)y² ∕ n

This is a coninuation of the induction proof Part XIa for √(x + 1)y² ∕ n.

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 is a common difference (Δ) between the irrational numbers generated from Part XIa for √(x + 1)y² ∕ n. So, therefore, let's proceed as follows:

The sequence derived from the equation √(x + 1)y² ∕ n is shown in Tables I and II where the values of n range from 1 to 20, while those of x and y range, respectively, from 1 to 18 and 2 to 19. The difference (Δ) between the irrational numbers are, respectively, about 1.0 and approaches the value of 1 as x and y increase without bound. The value for each √(x + 1)y² ∕ n, is calculated to 6 decimal places, is irrational and the value falls between two consecutive natural numbers. In addition, the sixth column heading is SQRT = √(x + 1)y² ∕ n. he column header ∑ 1 ∕ n sums up all 1 ∕ n up to that particular n value.

If the natural numbers are plotted on a grid vs 1 ∕ n (actually plotted in reverse, the zero value not shown). The following picture of a staircase, whose risers are composed of the harmonic sequence starting at one third (if zero were considered a natural number the first step level would be one half). The blue lines are the steps, the red stars the irrational values between the natural numbers and the orange lines are the risers of the staircase. In addition, the numbers on the vertical axis are from column three of Table I. The rational number adjacent to each step, 1 ∕ n, correponds to the level of each step in which n approaches zero as the sequence approaches infinity.

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

Go back to Part XIa for proof by induction of √(x + 1)y² ∕ n.

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