Irrational Numbers from Adjacent Natural Numbers (Part VIIIb)

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

This is a coninuation of of the induction proofs Part VIIc and Part VIId 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).

Table I Irrational (Sequence)
n	x	y	SQRT	Δ₁/Δ₂
1	1	2	1.581139
1	2	3	2.081666	0.500527

2	3	4	3.535534	1.453868
2	4	5	4.049691	0.514157

3	5	6	5.522680	1.472989
3	6	7	6.035609	0.512928

4	7	8	7.516648	1.481040
4	8	9	8.027730	0.511082

5	9	10	9.513149	1.485419
5	10	11	10.022701	0.509553

6	11	12	11.510864	1.488163
6	12	13	12.019215	0.508351

7	13	14	13.509256	1.490041
7	14	15	14.016657	0.507401

Table II Irrational (Sequence)
n	x	y	SQRT	Δ₁/Δ₂
8	15	16	15.508062	1.491406
8	16	17	16.014699	0.506637

9	17	18	17.507141	1.492442
9	18	19	18.013153	0.506012

10	19	20	19.506409	1.493256
10	20	21	20.011901	0.505492

11	21	22	21.505813	1.493912
11	22	23	22.010867	0.505054

12	23	24	23.505319	1.494452
12	24	25	24.009998	0.504679

13	25	26	25.504902	1.494904
13	26	27	26.009258	0.504356

14	27	28	27.504545	1.495288
14	28	29	28.008619	0.504074

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):

1, 1.581139, 2, 2.081666, 3, 3.535534, 4, 4.049691, 5, 5.522681, 6, 6.035609, 7, 7.516648, 8, 8.027730, 9, 9.513149, 10, 10.022702

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.