Cyclic Addition Wheels Cyclic Addition, unlike Base 10 Number with just a range of names for each numerals position, has simply a ‘common multiple’, a tier and a Count to form both Base 10 Number whilst applying Cyclic Addition Step by Step Mathematics. Sure Base 10 Number in some form or rather has been with us for about 1400 years. And the invention of Place Value positions of a numeral forming a Number was put to good use in the Western world hundreds of years later. Cyclic Addition is as well as being a preserver of Base 10 Number, has many inherent features that preserve and protect the Mathematical nature of all integer Number. Like Base 10 Number Cyclic Addition Number is infinite. Its application to integers used in the World during a modern age is also like Base 10 Number applied to basically anything and everything. So all the more reason to guard its foundation of originally growing from Mathematics. A ‘Cyclic Addition Wheel’ is called hereupon a ‘Wheel’. A Wheel is 6 numbers in a circular form. Like those of the title page show 7 different Wheels forming a pattern of the relative positions of multiples of the ‘common multiple’ around the Wheel. The Wheel usually starts with 1×’common multiple’ at the top and rotating clockwise the 3, then the 2, then followed at the bottom of the Wheel the 6, then the 4 and then the 5×’common multiple’. Roughly forming a Numbers at hexagon points around the circular Wheel. The first tier sequences are from ‘1 3 2 6 4 5’ to ‘69 207 138 414 276 345’, counting the ‘common multiple’ 1 to 69 by 1’s. The ‘common multiple’ terminology can show either the first tier ‘common multiple’ and the tier or multiply both the ‘common multiple’ by 7n-1 where integer n= tier number. For example the third tier of ‘common multiple’ 3 is the same term as ‘common multiple’ 147=7(3-1) ×3=7×7×3. The first tier Wheels are all in the form ‘common multiple’ × ‘1 3 2 6 4 5’. The second tier Wheels, following the first tier, are a ‘common multiple’×7ב1 3 2 6 4 5’. The third tier Wheels, following the second tier, are a ‘common multiple’ × 7 × 7 × ‘1 3 2 6 4 5’. Each tier to follow the first tier is exactly 7 × Wheel of the previous tier. The number of possible Wheels for any given ‘common multiple’, following the tier structure and order, is potentially infinite. A Wheel is always declared, like an opening ceremony, written and / or oral, before counting with that sequence. The 1 × ‘common multiple’, usually the first number in a circular Wheel, is also formed from the Pure Circular Fraction 69 exponentials. The Pure Circular Fraction 69, its Exponentials of 7, and the Wheels for a ‘common multiple’ are presented together as an invaluable Reference Tool throughout a Cyclic Addition Count. This is largely the reason for presenting the 69 Reference Pages that
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follow. They are a whole and complete universe of Number (integer) for the first 7 tiers. These form a great chunk of this book ‘Laws within a Number Universe’. As they will remain constant no matter the changes made to the written text. All 69 Reference Pages are found in this Chapter. There is one page for each ‘common multiple’. Complements of 69 are on an open double page. For example 68+1=69. All members of the Pure Circular Fraction 69 sequence are together to aid navigation through the fraction. Cyclic Addition Mathematics is the complete way to navigate through the successive tiers of Wheels. A Cyclic Addition Count can be manifested in a variety of ways. The Guidebook ‘Mathematics with just Number’ has a plentiful array of Counts with multiple tiers. See the last pages of chapter 7 Hierarchy. Exponentials of 7n-1 × ‘common multiple’ × 1 derived from Pure Circular Fraction 69 are equal to the first Number in all Wheels, for all ‘common multiples’ 1 to 69 and all of the higher tiers. Integer n = tier number. The first number listed underneath the fraction sequence has a units matching the fraction and a transparent tens making the ‘common multiple’ of the page. This ‘common multiple’ forms the beginning of the Wheels that are found on the same page. In essence uniting a continuous circular sequence of numerals, forming the Pure Circular Fraction 69, with Numbers (integers) found as the ‘common multiple’ in every possible 6 Number Wheel. These Mathematical relationships of Fraction, Exponential and Integer or Natural Number are all found in unity with Cyclic Addition. This protects the formation of all three. Cyclic Addition Steps all use the same Wheel. The same Wheel for Counting, for Place Value, for ‘Move tens to units’, and for Remainder. The 7×Multiple is confirmed by the next tier of this same Wheel. Thus a major reason why the Wheel is written first upon the Counting Page. As all Cyclic Addition Steps follow with Mathematical application of the Wheel. Some ‘common multiples’ share the same Wheels though they differ in tier number. ‘Common multiple’ 1 and 7 and 49 are unique ‘common multiples’ in their own right. However ‘common multiple’ 7 is considered the second tier of the ‘common multiple’ 1 and 49 the third tier of ‘common multiple’ 1. All ‘common multiples’ of 7 to 63 in multiples of 7 are also considered as the second tier of ‘common multiple’ 1 to 9 respectively. Some ‘common multiples’, with a common factor, share 1, 2 or 3 Numbers from applicable Wheels. Here’s a simple table of ‘common multiple’ 3, 9, 6, 18, 12 and 15. These 6 circular Wheels all share numbers with the first Wheel. 3 9 6 18 12 15 45 9 27 18 54 36 24 30 6 18 12 36 108 72 90 18 54 36 24 72 48 60 12 36 45 30 90 60 75 15
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The Practical ‘Wheels’ forms the largest Chapter. There are 69 pages of Wheels, one for each ‘common multiple’. There are 69 numbers from 1 to 69 by 1’s forming the pure circular fraction 69 sequences. The workings of the pure circular fraction 69 are discussed in Chapter 14 ‘Pure Circular Fractions’. Each page called a ‘Reference Page’ presents one ‘common multiple’. The pure circular fraction 69 sequence with exponentials of 7, together with the first 7 tiers of Wheels form a ‘common multiple’ Reference Page. A Reference Page has complements of 69 on opposite pages. Both pages have ‘common multiples’ adding to 69. The exponentials of seven, expressed as 7n-1 × ‘common multiple’, where integer n = tier number, are stepped down a place value to the left for each exponential, and add vertically to equal the numerals in the pure circular fraction 69. For example, consider ‘common multiple’ = 68, the first Reference Page, and its exponentials of seven. Upon the top line is a circular sequence from part of the pure circular fraction 69. The first exponential 68×70 = 68, the ‘units’ 8 is aligned to the corresponding number in the pure circular fraction. The second exponential, 68×71 = 476 is positioned one place value to the left of the above 68. The 6 ‘units’ and 6 ‘tens’ from 68 above add to 12. The 2 ‘units’ from 12 equal the numeral in the pure circular fraction, the 1 ten is added to the next place value left. The third exponential, 68×72 = 3332, is again positioned one place value left and below the 476. The 2 ‘units’ and 7 ‘tens’ from 476 and 1 ten from the previous column add to 2+7+1=10. The units 0 from 10 equals the numeral aligned above in the pure circular fraction. The fourth exponential, 68×73 = 23324, is positioned in the same stepped down one place value to the left. Add the 4 units to 3 tens to 4 hundreds and the 1 from the previous addition. 4+3+4+1=12. The 2 units from 12 equal the next numeral, left of the previous addition, in the pure circular fraction. The fifth exponential, 68×74 = 163268, is placed again in the same place value pattern to the left one numeral. Add the 8 units to 2 tens to 3 hundreds and 1 from the previous column 8+2+3+1 = 14. The 4 units equals the numeral, next left, in the pure circular fraction. The sixth exponential 68×75 =1142876, place left one numeral and just below the fifth exponential. Add the 6 units to 6 tens to 3 hundreds to 3 thousands from the exponentials above and 1 from the previous column. 6+6+3+3+1=19. The units 9 from 19 equals the next numeral left in the pure circular fraction. The seventh exponential, 68×76 = 8000132, place left one numeral and below the sixth exponential. Add 2 units to 7 tens to 2 hundreds to 3 thousands from the above exponentials and 1 from the previous column. 2+7+2+3+1=15. The 5 units from 15equals the next numeral left in the pure circular fraction. The Reference Page has the first 7 tiers of exponentials upon it. Similar Mathematics of further tiers of exponentials equal the circular sequence of numerals in the pure circular fraction 69 using the same method described in the previous paragraph. As there are a complete 69 numbers forming the pure circular fraction 69, just these and only these ‘common multiples’ are shown with exponentials of 7 and Cyclic Addition Hierarchy of the same ‘common multiple’. The number of ‘common multiple’ being 69 for the first tier has particular significance. The Pure Circular
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Fraction has both a fraction line forming ‘units’ of a ‘common multiple’ number and a remainder line underneath forming ‘tens’ of the same ‘common multiple’. The 69 numbers represent a finality and completion of all remainders from ‘ 1 3 2 6 4 5’ being presented to every numeral. The remainders form the ‘tens’ and the numerals form the units making every number from 1 to 69. The Wheel 1ב1 3 2 6 4 5’ = ‘1 3 2 6 4 5’ is actually the Remainder Sequence for number 7. Shown in Chapter 3 ‘Attributes of the Original Sequence’ as an infinite remainder from dividing a number in the sequence by 7. Counting with ‘1 3 2 6 4 5’ using Cyclic Addition presents all 7 × Multiples (multiples of 7×1=7). Preparing the Mathematician to Count with 71 × ‘1 3 2 6 4 5’ = ‘7 21 14 42 28 35’. Likewise Counting with the second tier of 1 presents all 7 × 7 × Multiples. Again preparing to count with 72 × ‘1 3 2 6 4 5’ = the third tier. Counting with the third tier presents all 73 × Multiple. Introducing the fourth tier 73 × ‘1 3 2 6 4 5’. Counting with the fourth tier presents a beginning of 74 × Multiples, from the fifth tier, 74 × ‘1 3 2 6 4 5’. Counting with Cyclic Addition with the fifth tier presents the start of 75 × Multiples from the sixth tier 75 × ‘1 3 2 6 4 5’. Counting with the sixth tier, the Remainder joins the sixth tier with the seventh tier 76 × Multiples. Thus Wheels of a ‘common multiple’, derived from exponentials of pure circular fraction 69, have a hierarchy with an order of 7n-1 × ‘common multiple’ × ‘1 3 2 6 4 5’. The next higher tier is always 7 × Wheel of the previous tier. This meshes the exponentials of 7n-1 × ‘common multiple’ with the Wheel ‘7n × Multiple’ perfectly, where n = positive integer. The Reference Page has 7 tiers of Wheels upon it. Each tier following the first is 7 × ‘Wheel’ of the previous tier. Though this is a practical presentation limit the actual number of possible tiers for any common multiple is infinite. It’s good practice with Cyclic Addition to display the Reference Page of Wheels when counting with a particular ‘common multiple’. This aids cyclic addition mathematics between the tiers. When counting with Cyclic Addition and applying a Remainder there exists no evidence to hide higher tiers. The pages that follow show the first 7 tiers of ‘Wheels’. The counting Wheel and higher tiers, to 7 tiers, provide for Mathematical navigation through a Wheel and when numbers are submitted to higher tiers, the Mathematics between tiers. The next tier Wheel is Mathematically required when a Count presents a Remainder to submit the Count to a higher order. In some cases a Count presents a Remainder to submit itself to an order two or more tier higher than the counting Wheel.
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When writing a count with a Wheel upon the paper, and a count has no remainder, the Mathematician can at that point present upon paper the next tier Wheel and continue counting with the higher tier. No matter which tier is being counted with Cyclic Addition, all lower tiers are written upon the counting page and higher tiers, to 7 tiers, are presented next to the counting page to find the 7 multiple and aid Cyclic Addition inter-tier Mathematics. The exponentials, shown on each Reference Page, above the 7 tiers of Wheels vertically add to the above pure circular fraction 69. Starting from the ‘units’ in the lowest exponential to the ‘units’ in the highest exponential. Use the display of Wheels of a ‘common multiple’ as a Reference Page and tool for cyclic addition with any tier and subsequent tiers always. A keen student can create their own Wheels by working through each number of the pure fraction 69. Show at least 7 tiers of exponentials with the pure fraction. Present the 7 tiers of Wheels of each ‘common multiple’ beginning with the pure circular fraction exponential. These Reference Pages that follow go hand in hand to present a map book of ‘Cyclic Addition’.
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. . 4782608695652173913043 63 441 3087 21609 151263 1058841 7411887 63 189 126 378 252 315 441 1323 882 2646 1764 2205 3087 9261 6174 18522 12348 15435 21609 64827 43218 129654 86436 108045 151263 453789 302526 907578 605052 756315 1058841 3176523 2117682 6353046 4235364 5294205 7411887 22235661 14823774 44471322 29647548 37059435 56
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