Irrational Numbers from Adjacent Natural Numbers (Part III)

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

This is a continuation of Part IIa and Part IIb which gave a proof by induction for the equation √n(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 √n(x + y). In addition, the equation √n(y² − x²) produced the same staircase sequence since one of the factors of y² − x² equals one. So, therefore, let's proceed as follows:

The sequence derived from the equation √n(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), is calculated to 6 decimal places, is irrational and each of these values indeed falls between their respective consecutive natural numbers. In addition, since √n(x + y) = √n(2x + 1), the two equations are interchangeable.

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.

It's coincidental that the staircase pops up as the Bohr model for the quantization of the electron in an atom, where one cannot go between steps but must instead step up from one energy level (rise) to the next. If no levels are present then it's like going continuously up a ramp, where the sequence from Part I would be the comparable analogue of the ramp.

Go to Part VIII for another new staircase sequences.
Go back to Part IIa and Part IIb for proof by induction for √n(x + y).

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