Earth/matriX
Science in Ancient Artwork

Extract Nº 4

POWERS AND ROOTS IN THE
ANCIENT RECKONING SYSTEM

by
Charles William Johnson

Obtaining powers and roots in mathematics today are easily executed by pressing the correct keys on an electronic calculator. But, one can only wonder hoe the ancient peoples obtained such omputations. The ancient Babylonians and Egyptians seem to have had computational methods based on simple addition and division. One of the main methods employed was that of mediatio/duplatio, halving and doubling the numbers. We have used this same method throughout our research in an attempt to reconstruct some of the possible computations. In this Extract (N°. 4), we shall illustrate some of the possible methods that one could use to obtain certain products of powers and roots through simply adding and dividing numbers based on the concepts of doubling/halving numbers.

In earlier essays, we have seen how the maya long count numbers and the numbers of the ancient kemi system are apparently related. We shall use numbers that are readily accepted as being significant for those two ancient cultures. In reviewing the distinct examples of the Extract, one should keep in mind the concept of fractal numbers, whereby it is significant to visualize 4 x 4 = 16, as also being 4 x 40; 40 x 40; 40 x 400; etc.

    POWERS

The square of a particular number may be easily obtained throughout contemporary methods of multiplication. In fact, there are alternative ways of achieving this; consider the 52c numbers of the ancient Mesoamerican system:

208² = 43264
200 x 200 = 40000
400 x 8 = 3200
8 x 8 = 64

43264

416² = 173056
400 x 400 = 160000
800 x 16 = 12800
16 x 16 = 256

173056


In these cases, one simply takes the initial hundreds of the number under study (200, or 400 here) and multiply them as fractals and adding zeros. Then, the second step of computation doubles that particular number times the last two digits of the number studied; and, finally, the third step consists of multiplying the last two digits times themselves. Then, those products are added together for the final result. Consider another example:

273² = 74529
200 x 200 = 40000
400 x 73 = 29200
73 x 73 = 5329

74529


These steps in multiplication may be achieved through addition as well; we have made the presentation more brief.

Cube roots present another procedure which involves the maya long count numbers directly. Let us look at some examples of the same 52c series and the corresponding power of three.

208³ = 8998912 416³ = 71991296
200³ = 8000000 400³= 64000000
100³ = +1000000 200³ = +8000000


9000000 72000000
8³ = -512 16³ = -4096


8999488 71995904
-576 -4608


8998912 = 208³ 71991296 = 416³

In both of the cases above, the same integers of the numbers themselves (208, 416) are used for the computations, except for the fact, now, in powers of three for the 52c numbers, the maya long count numbers are used for obtaining the final result. Both 576 and 4608 are numbers within the maya long count and can be employed for obtaining the final results as shown above through subtraction. The rule appears to be that if the number subtracted initially (512) is used, then one would employ the maya long count number nearest it: i.e., 576 in this case; whereby 4608 is the nearest number for the 4096 constant number. One must also understand that the numbers 512 and 4096 lie on the series of constant numbers: 2,4,8,16,32,64,128,512,1024,2048,4096

In powers of three, then, significantly appear the three distinct series of numbers in relation to one another in simple computations of addition and subtraction:

13,26,52,104,208,416,832,1664...
18,36,72,144,288,576,1152,2304,4608,9216...
2,4,8,16,32,64,128,256,512,1024,2048,4096...

Once again, it becomes obvious that the selection of the 52c, the 360c, in relation to the constant numbers in relation to the power of three surely could not have been made through accident in ancient times. The ancient peoples just happenend to employ three series of numbers, which just happen to produce computations for the power of three. Obviously this was no accident. Let us observe the procedure for the 54c of the ancient kemi in Egypt:

432³ = 80621568
400³ = 64000000
200³ = +8000000

72000000
now +8640000 (432 x 2)

80640000
-18432 remember, 18432 is of the maya long count: 18432

9216
80621568 = 432³ 4608 2304 1152 .....
The 432 number lies on the 54c series: 54,108, 216, 432, 864, 1728...

In this manner, we also see how the 108c (54c) of the ancient kemi connects easily to the maya long count numbers for obtaining products to the power of three. Even complex looking numbers like
1664³ ( = 4607442944) are easily computed: 1600³ = 4096000000, plus 800³ = 512,000,000 yielding a maya number 4608000000, then has the 64³ = 262144 subtracted from it:
4608000000 - 262144 = 4607737856. Then, the maya number immediately greater than that 294912 is thereby subtracted from that: 4607737856 - 294912 = 4607442944. In other words: 1664³. This seven-step procedure makes computations for the power of three extremely easy for the 52c and the 54c of the ancient kemi-maya system.

    ROOTS

A discussion of powers cannot be complete without examples for obtaining the roots of equally complex numbers. For our example, we shall use a number found in the ancient kemi system: 756, which is often cited as the base measurement of a side of the Great Pyramid. Let us say, we know the square of 756 to be 571536. Now, let us wish to computate the square root of a related number, for example, 571541. Without the assistance of the electronic calculator these computations become almost unmanageable. However, let us explore the idea of a constant number as of the 571536 value. Consider the following method.

756² = 571536 Need to know the square root of 571541
378 / 571536 = .000661375661375 (constant value)

we shall now take this constant value (at .000661375) and add it to 756 in an incremental manner in order to obtain the following roots and powers after 7562:

756² = 571536
Add .000661375 each time
756. 000661375² = 571537
756. 00132275² = 571538
756. 001984125² = 571539
756. 0026455² = 571540
756. 003306875² = 571541

Other results may be easily obtained by use of the constant value:

571914 - 571536 = 378
378 x .000661375 = .24999975
756.249999975² = 571914.062121

572292 - 571536 = 756
756 x .000661375 = .4999995
756.4999995² = 572292.249243

In order to make adjustments and eliminate the decimal fractions in these results, one would simply reduce the decimal places used in the computations; consider,

573048 - 571536 = 1512
1512 x .000661 = .999432
756.999432² = 573048.140048

The electronic calculator offers the number 756.999339497 for the rot of the number 573048; a difference of about .000093; something next to an incommensurable distinction.

Now, supposing one might wish to know the square root of 571541.5 of our example. Then, simply use 189 / 571536 = .0003306878307 as the constant value (189 being half of 378 which was employed above initially):

756.003306875 + .0003306878307 = 756.003637562
756.003637562² = 571541.500006

which is an extremely close approximation. The electronic calculator offers the value 756.003637557; a difference of .000000005; almost nothing to speak about.

Understandably, the electronic calculator must be computing numbers on a similar basis of constant values. However, what we find to be of interest is the manner in which the series of historically significant numbers relate to one another in terms of powers and roots. For example, consider some historically significant numbers of the maya long count: 1872000 and 151840. Other numbers may also be considered, but for now we shall exemplify these:

The number 1872000 (days) constitutes the cut-off period for the maya long count and reflects a significance for powers and roots:

1368² = 1871424
1368.21051² = 1872000

-576 a difference of a maya long count fractal

If we take .2105 to be the constant value for the series as it is, then:

1368² = 1871424 that is: 1871424 + 576
1368.21051² = 1872000 1872000 + 576
1368.42102² = 1872576.088 1872576 + 576
1369.63153² = 1873152.265 1873152 + 576
1368.84204² = 1873728.53 1873728 + 576
1369.05255² = 1874304.885 1874304 + 576

The terminations of the numbers in the series reflect the maya long count fractals: 576, 1152, 2304..., whereby the termination 728 (in 1873728) reflects the 1728 integers (that is 1152 + 576 = 1728).

The number 151840, another historically significant maya number often cited in the literature, reflects another series of powers/roots as of the number series 390c.

390²
= 152100
389.666524094²
- 151840

260
a difference of the ancient 260c calendar (Olmec)

The constant value in this case would be .666524094; producing a constant series of numbers differentiated by the value of the 260c.

153400 + 260 153140 + 260 152880 + 260 152620 + 260 152360 + 260 152100 + 260 151840 ......

Other number series exist: for example, on the same maya series of 1872000, if one employs the kemi number 1296 in subtracting it from that maya period, the companion number 1366560 obtains:

1872000 -1296 1870704 -1296 1869408 -1296
...
1369152 -1296 1367856 -1296 1366560 ....

There would be a corresponding constant value for each computational procedure for the powers of these numbers.

The numbers of the maya system and the kemi system are computationally related as we have illustrated herein. Every time we review a distinct aspect of these different systems, one obtains the impression that both systems represent a single conceptual idea for computational math.

***************************************

©1996-2012 Copyrighted by Charles William Johnson. All Rights Reserved


Your comments and suggestions are greatly appreciated:
e-mail: johnson@earthmatrix.com


Earth/matriX
Science in Ancient Artwork
Extract Nº.4

POWERS AND ROOTS IN THE ANCIENT RECKONING SYSTEM
30 November 1996
©1996-2012 Copyrighted by Charles William Johnson. All Rights Reserved
Earth/matriX

Reproduction prohibited without written consent of the author.


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