A. A General Proposition for the determination of the n th root (n = odd prime number) of a positive real number:
Here, a general discussion is presented for the determination of the nth root of positive real numbers using the interval-weighted denominator method where n is an odd prime number. For convenience, the discussion is restricted from 1n to 10n, which possesses nine intervals, namely 1n-2n, 2n-3n, 3n-4n, 4n-5n, 5n-6n, 6n-7n, 7n-8n, 8n-9n and 9n-10n. The following proposals are made:-
(i) The nth root of an positive real number x, lying between two numbers a and b that possess consecutive natural number nth roots k and m and denoted as NR(x) is determined according to :
$$NR \left(x\right)=NR \left(a\right) + \frac{D}{FWD} \dots \left(i\right)$$
Where NR (a) is the nth root of number ‘a’,
D = x-a and
FWD is the final weighted denominator.
In the scope of our discussion above, thus, for an interval such as 3n-4n, then a = 3n and NR (a) = 3.
(ii) “Weighting” is to be performed (n-1)/2 times to obtain FWD.
(iii)FWD can be obtained by weighting successive denominators, starting with the first base or denominator, B0, that may be described for a particular interval as follows:
$${B}_{0} = n*{a}^{n-1} \dots \left(ii\right)$$
Thus, for the determination of the 5th root, n = 5, the B0 in the interval 35-45 is
$${B}_{0 } = 5*{3}^{4}=405$$
(iv) First interval-weighted denominator, B1, may be obtained according to
$${B}_{1 }= {B}_{0} + \frac{{B}_{0}*D*WF}{I} \dots \left(iii\right)$$
Where WF = Weighting factor and may vary from interval to interval, and
I = b-a
(v) This process goes on till B(n-3)/2 as
$${B}_{2} = {B}_{1} + \frac{{B}_{1}*D*WF}{I} \dots . \left(iv\right)$$
…….. and all the way till\({B}_{(n-3)/2 }= {B}_{(n-5)/2} + \frac{{B}_{(n-5)/2}*D*WF}{I} \dots \left(v\right)\)
(vi) Finally, FWD may be obtained from the following last step:
$$FWD= {B}_{(n-1)/2} = {B}_{(n-3)/2 }+ \frac{D*(I-{B}_{\frac{n-3}{2}})}{I} \dots \left(vi\right)$$
(vii) The FWD obtained this way may be plugged into eq.(i) for obtaining the nth root of the given number ‘x’.
B. 5th root (FR):
- According to the above proposition, the number of weighting steps for the 5th root are (5-1)/2 = 2.
- There will then be two bases, B0 and B1 where FWD = B1.
- The interval 15-25 is ignored and it is assumed that any number in this interval can be converted into a number of higher intervals by multiplying and dividing with a number that has a perfect fifth root, for e.g. 32 for higher accuracy of determination.
- In this method, three different formulae are proposed for the following intervals:
- 25-35
- 35-45 and 45-55
- 55-65, 65-75, 75-85, 85-95 and 95-105.
B.1 5th root (FR) for the intervals 25-35:
- B0 can be determined from eq.(ii) above for all the intervals.
- B1 can be written as:
$${B}_{1} = {B}_{0} + \frac{{B}_{0}*D}{I}*\left(\frac{1}{FR\left(a\right)}\right) \dots \left(vii\right)$$
Where WF from eq.(iii) is defined as \(\left(\frac{1}{FR\left(a\right)}\right)\) and where FR(a) is the fifth root of ‘a’.
B.2 5th root (FR) for the intervals 35-45 and 45-55:
(i) B1 can be written as:
$${B}_{1} = {B}_{0} + \frac{{B}_{0}*D}{I}*\left(\frac{1}{FR\left(a\right)}- \frac{1}{{FR\left(a\right)}^{2}}\right) \dots \left(viii\right)$$
Where WF of eq.(iii) is defined as\(\left(\frac{1}{FR\left(a\right)}- \frac{1}{{FR\left(a\right)}^{2}}\right)\)
B.3 5th root (FR) for the intervals 5 5 -6 5 , 65-75, 75-85, 85-95 and 95-105.
(i) B1 can be written as:
$${B}_{1} = {B}_{0} + \frac{{B}_{0}*D}{I}*\left(\frac{1}{FR\left(a\right)}- \frac{FR\left(a\right)}{{2*FR\left(a\right)}^{2}}\right) \dots \left(ix\right)$$
For even values of FR(a) Where WF from eq.(iii) is defined as\(\left(\frac{1}{FR\left(a\right)}- \frac{FR\left(a\right)}{{2*FR\left(a\right)}^{2}}\right)\)
And\(\)\({B}_{1} = {B}_{0} + \frac{{B}_{0}*D}{I}*\left(\frac{1}{FR\left(a\right)}- \frac{(FR\left(a\right)+1)}{{2*FR\left(a\right)}^{2}}\right) \dots \left(x\right)\)
For odd values of FR(a) Where WF from eq.(iii) is defined as\(\left(\frac{1}{FR\left(a\right)}- \frac{(FR\left(a\right)+1)}{{2*FR\left(a\right)}^{2}}\right)\)
C. 7th root (SR):
- According to the above proposition, the number of weighting steps for the 7th root are (7-1)/2 = 3.
- There will then be three bases, B0, B1 and B2 where FWD = B2.
- The interval 17-27 is ignored and it is assumed that any number in this interval can be converted into a number of higher intervals by multiplying and dividing with a number that has a perfect seventh root, for e.g. 128 for higher accuracy of determination.
- In this method, two different formulae are proposed for the following intervals:
- First half of 27-37 interval
- Second half of 27-37 interval and all the rest of the intervals upto 97-107.
C.1 7th root (SR) for the first half of the intervals 27-37:
(i) This is the part of the interval from 27 to roughly midway of the interval, i.e. up to 1150.
(ii) B0 can determined from eq.(ii) for all intervals.
(iii) B1 can be written as:
$${B}_{1} = {B}_{0} + \frac{{B}_{0}*D}{I}$$
Where WF = 1 for determination of B1.
(iv) B2 can be written as:
$${B}_{2} = {B}_{1} + \frac{{B}_{1}*D}{I}*\left(\frac{1}{2*SR\left(a\right)}\right) \dots \left(xi\right)$$
Where \(WF= \left(\frac{1}{2*SR\left(a\right)}\right)\) for determination of B2 and where SR(a) is the seventh root of ‘a’.
C.2 7th root (SR) for the second half of the intervals 2 7 -3 7 and rest of the intervals:
(i) This is the part of the interval from 1150 to 37 and all subsequent intervals.
(ii) B0 can determined from eq.(ii) for all intervals.
(iii) B1 can be written as:
$${B}_{1} = {B}_{0} + \frac{{B}_{0}*D}{I}*\left(\frac{1}{2*SR\left(a\right)-2}\right) \dots \left(xii\right)$$
Where \(WF= \left(\frac{1}{2*SR\left(a\right)-2}\right)\) for determination of B1.
(iv) B2 can be written as:
$${B}_{2} = {B}_{1} + \frac{{B}_{1}*D}{I}*\left(\frac{1}{2*SR\left(a\right)-2}\right) \dots \left(xiii\right)$$
Where \(WF= \left(\frac{1}{2*SR\left(a\right)-2}\right)\) for determination of B2.