THE FINITE PART OF NUMBERS WITH PERIODIC DECIMALS

THE FINITE PART OF NUMBERS WITH PERIODIC DECIMALS By Matheus Muniz∗ The issue is inaugurated from the analysis of non-zero integers divided by 11, which are not divisible by it, when generating repeating decimals. It was noticed that the repeating decimals generated by the division, where the prime number 11 is the denominator, create a period always of 2 digits, here represented by AB, which I also call the reflector term (Tr), where A is the first digit and B is the succeeding one. That is, the decimals generated by the division by 11 always have in their structure a pair of digits that repeat into the infinity of the decimals, which emerge from the reflected term (To). When looking at the division, for example, of the number 985, by dividing it by 11, the result is the number 89.545454. In this result, there is a whole part, which is the number 89, a reflector term, which is 0.54, followed by the reflected term (To), which repeats throughout the entire periodic decimal expansion that is 0.0054. The assertion can be proven as follows: The number 985, used in the example, has in its composition a finite part and an infinite part, i.e., one part will generate a finite decimal and the other will generate the decimal. The finite part of the number 985 is the number 984.94, and the infinite part, or that generates the part of the infinite periodic decimal, is 0.06. Thus, dividing the number 984.94 by 11 we get the finite decimal 89.54. That is, here we have the whole number and right after the comma we have the reflector term which is not a decimal, since it is just a finite decimal. By dividing the remaining part last mentioned (0.06) by 11 we get the number 0.00545454, which is from there that one can consider the decimal starts. ∗ http://www.matheusmuniz.com 1 Therefore, this study proposes a new thought regarding what has been understood by decimal until today. Thus, moving to the first conjecture. CONJECTURE 1: There is no pure periodic decimal in the division of any non-zero integer, different from the denominator, by the number 11, when generating decimals. Every decimal starts after its first reflector term occurs, commonly known as the period. In every number with a decimal composition, there is always a whole part, null or non-null, a non-periodic decimal part, here called the reflector term, and, right after, the start of the periodic expansion of the decimal, whose first term is called the reflected term (To). It is to assert that the infinite parts of the numbers for, until then, the division by 11, follow a successive and limited order of 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, and 0.1. Despite the regular arithmetic progression, the regularity does not follow the number on which the division will be made, that is, although until a certain point the infinite parts accompany their numerators 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, the regularity ends when the divisor finds its equal, i.e., the number 11, which does not enter the division as a numerator, derailing the path and losing the regular reference with its similar digits. One might believe that, from the number 12, the infinite part would be 0.02, by logical-intuitive deduction, which would be corroborated by the jump from the preceding number 11. However, the infinite part of the number 12 is 0.01 and not 0.02. Such irregularity gains strength over the course of hundreds, thousands, and millions of subsequent numbers. Since throughout the interval there will always be numbers divisible by 11 that will continue changing the order of the part that will be infinite from the numerator when divided by 11. For example, we can cite the number 315 which has its infinite part at 0.07, or we can cite the number 999972, whose infinite part is 0.06. There is more utility in identifying the infinite part of the number about to be divided by 11. One of them is the simplification through the decomposition of the number into another fraction. For example, we can cite the number 315 itself, whose infinite part is 0.07. If we put the division 315 , and need to facilitate 11 the division, we could decompose such fraction in a way that would facilitate its manual division if we knew which infinite part corresponds to the number in question, we would multiply this part by 100, and it would serve as a numerator in the decomposition of the originating fraction. 2 By doing so, the infinite part that would be 0.07 would become just the integer 7, where it could be used to expand the original fraction to 315 = 11 7 308 + . In this way, one can see that 308/11 = 28, leaving the rest of the 11 11 7 decomposition thus 28 + 11 The numbers, when divided by 11, have a strict relationship with the 7 number 9. For example, in the situation where one reached with 28 + 11 , it would be possible to use the same idea to verify what the infinite part of the number 7 when divided by 11 is, which is 0.07, which by subtraction by its reference integer would give 7 − 0.07 = 6.93 and which, when divided by 11, would give us 6.93 = 0.63. 11 In the case of this last decomposition, the removal of the infinite part from the finite demonstrated the reflector term before the periodic decimal 7 . So the value of 0.63 is precisely the reflector term (Tr), whose decimal of 11 will follow it. Therefore, the final value would give 28.636363. Knowing these values would greatly facilitate the calculation done manually. Therefore, the formulas follow. Given a number N , here called the numerator prior to division, not divisible by 11 and non-null, and that generates decimals, it is understood that N = W + D, where W is the finite part of the number and D is the infinite part, as well as Z is the whole part of the result of the division that has a decimal, which can be null, i.e., zero, or non-null; AB is the reflector term (Tr) or period; A, the first digit, and B, the second digit of it and r, as being the number of repetitions for the notation of W , whose minimum repetition is equal to 1, we have: ***Formula for the composition of the number with the finite and infinite parts:*** N =W +D ***Formula to find the finite part of the number*** W = (A + 9 r∗2−1 X 10−K + B · 10−r∗2 ) + (Z · 11) K=1 ***Formula to find the infinite part of the number*** 3 AB + 10−r∗2 9 ***Formula to decompose a number not divisible (ND) by the multiples of 11 (MP) into two fractions*** D= DC = ( AB + 10−r∗2 ) · 100 9 Example: N D − DC DC ND = + MP MP MP ***Formula to find the reflector term (Tr)*** (W − (Z · 11)) 11 I have also created a formula that, for now, reconstructs the number to its value before division by 11 starting from the number with the decimal composition. I named it R to signify the return of the number to its state before the division. M= AB ) + (Z · 11) 99 I need to mention that the particularity of r being 1 by default is that I sought with this to verify the infinite part from the Tr without duplication. This occurs because there is an AP in the ratio of 2 when we take the part of the reflector term (Tr), for example in the division of 2 by 11 would be 0.18, and we multiply this number by 11, which would give 1.98. If we duplicate the reflector term in 0.1818 and multiply by 11 we will have 1.9998 (which is the finite part of the number). That is, the amount of 9’s would increase in an arithmetic progression of 2 for the total given to r, which would result in the finite part of the number with three 9’s, being the infinite part 0.0002 (in the case of the division of 2 by 11) with 2 decimal places below what it would be if r were 1. That is, if r is 1 we will have the value of 0.02. When r is equal to 1 we obtain the largest finite part of the number that will be subjected to division by 11, when we progressively increase the value of r, we start to have infinitely smaller finite parts. R = 11 · ( 4 As an example, we can cite the largest finite part of the number 2 when r is equal to one: 1.98, because if we divide this part by 11 we will have only one decimal and not a periodic decimal. If r is 2, the infinite part of the number 2 would be 1.9998, which divided by 11 would not change the whole part of the number, just add one more reflector term after the decimal point. Solving it would be 1.9998 = 0.1818. 11 So it can be affirmed that there are infinite finite parts in a number, for the division by 11, that generate, instead of decimals, finite decimal numbers. Therefore, these formulas offer a structured approach to analyze numbers in relation to their division by 11, focusing on their decomposition into finite and infinite parts, demonstrating that the understanding about decimals starting right after the comma is mistaken, defining that, although the number seems to be a perfect decimal, in fact, the first term is just a decimal. The concept of the defined reflector term and the reflected term in the scenario of decimals was also explored. 5