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Number Theory Boring? Nein!

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Number Theory Boring? Nein! Running head: PATTERNS IN NINE 1 Number Theory Boring? Nein! Discerning Patterns in Multiples of Nine Gina Higgins Mathematical Evolutions Jenny McCarthy, Jonathan Phillips Summer Ventures in Science and Mathematics The University of North Carolina at Charlotte PATTERNS IN NINE 2 Abstract This is the process Gina Higgins went through in order to prove that the digital sum of a whole, natural number multiplied by nine would always equal nine. This paper gives a brief history on the number nine and gives a color coded, three-hundred row chart of multiplies of nine as an example and reference. Basic number theory principles were applied with Mathematical Implementation to successfully create a method that confirmed the hypothesis that the digital root of every product of nine equals nine. From this method could evolve into possible future attempts to prove why the digital root anomaly occurred. PATTERNS IN NINE 3 The number nine is the largest digit in a based ten number (decimal) system. In many ancient cultures such as China, India and Egypt, the number nine was a symbol of strength representing royalty, religion or powerful enemies (Mackenzie, 2005). The symbol for the number nine used most commonly today developed from the Hindu-Arabic System written in the third century. Those symbols developed from the Brahmi numerals found as far back as 300 BC. (Ifrah, 1981) Figure 1 Number theory is a branch of mathematics dealing specifically with patterns and sequences found in whole numbers. It was long considered the purest form of mathematics because there was not a practical application until the 20th century when the computer was invented. Euclid and Pythagoras were the greatest names in the field, each adding important concepts that were later built upon by others. Pythagoras discovered perfect numbers, those that equal the sum of their proper divisors like = 3 = 1 = 2 and 3 + 2 + 1 = 6. He put many 6 6 6 2 6 3 numerology values into these numbers; to him they represented beauty and perfection. Euclid took a much more practical position on the process and discovered prime numbers. (number theory, 2013) Nine is a very unique number in the fact that nine is the only number that, when multiplied by any natural number, the digits of the answer added together equal 9. Another way to express it is that the digital root of the product of a number and nine is always nine. 9 × 3 = 27 2 + 7 = 9 × 52 = 468→ 4 + 6 +ퟗ8 = 18 1 + 8 = 9 × 9806 = 88254→ 8 + 8 + 2 + 5→+ 4 = 27 ퟗ 2 + 7 = → → ퟗ PATTERNS IN NINE 4 9 × 95843 = 862587 8 + 6 + 2 + 5 + 8 + 7 = 36 3 + 6 = → → ퟗ This paper attempted to prove this occurrence holds true constantly, with any natural number multiplied by nine. Method I’ve known about the “nines trick” since I first learned my multiplication tables. How the tens digit of the product of nine goes up with each multiplication, and the ones digit goes down. I realized that the digits of the product added together to make 9 about half way through elementary school, it blew my mind, but I took the knowledge without question. I finally sought out to prove this “trick”, to do so I had to go back to basics. I wrote down every product of nine and a number between 1 and 100 and looked for patterns. I recognized that the digital sum of a product only equaled 9 half the time. (Table 2) Whenever the ones digit of the number (ex. 43) was less than or equal to the tens digit then its product’s digital sum would not equal nine. 43 × 9 = 387, 3 + 8 + 7 9 . ≠ At this point I marked down all numbers that worked and then proceeded to write out a basic method for finding the digital sum of products of nine. I stated the variables = a number being multiplied by nine, & = any number between 0 and 9. I added a specific푚 rule on when the method would work. When푚 푛 > × 10 + × 10 , then 9 = × 10 + × 10 + 1 0 2 1 1 2 + 10 and + + = 9푚. The푥 formula is 푥specifically for 푚two digit푛 numbers,푛 so I needed a 0 3 1 2 3 way푛 to multiply푛 nine푛 by any푛 digits, as long as they followed my rule on what m could be. To discover this, I rewrote the original formula so it would work for numbers with three and four digits but ran into a problem when I realized my rule for didn’t work with numbers 푚 PATTERNS IN NINE 5 that had more than two digits. So I carried on with rewriting the formula for higher numbers, writing down each possible digital sum as I went, hoping to find a pattern. (Table 1) Table 1 Amount of Digits in the Number Possible Digital Sums being Multiplied by Nine 1 9 2 9, 18 3 9, 18,27 4 9, 18, 27, 36 5 9, 18, 27, 36, 45 I realized how blind I had been around writing the third case, I had missed an obvious pattern. Whenever I had gotten a digital root other than 9 I ignored it, because it didn’t work with what I was doing. Table 1 made it more obvious that the digital roots that weren’t nine simply weren’t finished being added. I could take any digital root on the table, add their digits and it equaled nine. At this point I could get rid of my m rule, and just add a step staying to repeat the expansion and addition process over until nine is summed. At this point I had all the necessary information and a basic method, I just had to make sure it was general and could apply to all natural numbers. I wrote out the steps to finding the digital sum below. Multiply nine by a number (x). PATTERNS IN NINE 6 9 × = 9 푥 푥 Expand the product. 9 = × 10 + × 10 + × 10 2 1 0 푥 푛 푛1 푛2 Add the digits (n). + + = 9 푛 푛1 푛2 푥 Repeat expansion and summation steps until a one digit number is found. Set equal to nine to check Here I used Mathematical Induction, a way to mathematically prove a statement for all natural numbers in two steps. (Franklin & Daoud, 1996) The first step is to prove the formula is true for the base or lowest number, generally 1 or 0. I used 11 as my base because it has already been proven that the digital sum of the product of nine and a number for the numbers 1-9 simply by logic. I used 10 simply because it was more comprehensible when used with two nonzero digits. Nine times eleven equals ninety-nine. 9 × 11 = 99 Expand ninety-nine. 99 = 9 × 10 + 9 × 10 1 0 Add the expanded digits; repeat expansion and addition until a one digit number is created. 9 + 9 = 18 PATTERNS IN NINE 7 18 = 1 × 10 + 8 × 10 1 0 1 + 8 = 9 Set the sum equal to nine to check. 9 = 9 So I proved that the digital root of ninety-nine, the product of nine and eleven, equals nine. Next I proved that this method would work with every number still using mathematical induction. The second step of mathematical induction is to prove the formula using + 1, assuming that holds true in the formula. 푘 푘 Multiply nine by k plus one. 9 × ( + 1) = 9 + 9 푘 푘 Expand nine k plus nine into its digits. 9 + 9 = (9 × 10 + × 10 ) + 9 × 10 1 0 0 푘 푘 Add the digits. 9 + + 9 +푘18 푘 As in the first step, I repeated the expansion and summation until all non-variables equaled nine. + 18 = ( × 10 ) + 1 × 10 + 8 × 10 0 1 0 푘 푘 + 1 + 8 푘 + 9 푘 PATTERNS IN NINE 8 Set the sum equal to nine to check it. + 9 = 9 푘 This obviously did not work, so confused I went on with the algebra, hoping it would work out. Solve for k. Subtract nine from both sides. + 9 9 = 9 9 푘 − = 0 − 푘 I was stuck at this point; I tried rewriting the formula and trying again but to no avail. I knew the only way my method would work was if k was cancelled out. After adding variables and expanding all the numbers more to try and have the k cancel and having no success, I decided to experiment. I substituted the = 0 back into 9 + 9 = (9 × 10 + × 10 ) + 9 × 1 0 10 푘 푘 푘 0 9(0) + 9 = (9 × 10 + (0) × 10 ) + 9 × 10 1 0 0 9 + 9 = 18 This proceeded to take the formula back in the direction I needed it to be in. 18 = 1 × 10 + 8 × 10 1 0 1 + 8 = 9 Set the sum to 9 and check. 9 = 9 PATTERNS IN NINE 9 Results and Disscussion I set out to prove that every whole, positive number multiplied by nine would produce a number with a digital sum that equaled nine and by using my basic understanding of multiplication and my algebra skills I created a method to mathematically show this. I then used Mathematical Induction to prove that it would work every time. Proving that every product of nine has a digital sum of nine could be useful when dividing large numbers by hand; any number that digital sum is nine must be divisible by nine. An interesting way the information in this paper could be expanded upon is solving why this pattern occurs only in multiples of nine and no other numbers.
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