Collatz Conjecture Plot Explanation
The Collatz Conjecture is a conjecture that has not been proven yet. It is where you take any number, and if the number is even, you divide it by two and that is the next number in its sequence. If your number is odd, then you multiply it by 3, add one, and divide by two. You continue this process until your process begins repeating. Once your sequence begins to repeat, you count all of the numbers before the repetition (including the seed/number you started with), count one repetition, and that number is the size or length of that number’s sequence. The conjecture is that no matter what number you have, the sequence will end up reaching one. On the table I made of seed numbers and the sequences they made, I did not see many patterns. Once the table was coded, patterns began to emerge. The table was coded with odd numbers being replaced by ones, and evens being replaced by zeros. I began to see that when you looked at the rows, the first number in each sequence began to make a repetitions in relation to evens vs. odds. The second row made a longer one, the third did the same, and so on. The first repetition was 0110. The second was 10010110. The third was even larger. The size of the repetition could be described by the equation n=2^(n+1). Also, if you split the repetition in half, you can notice that the second half is the opposite number of the first half (i.e. ones would be replaced by zeros, and zeros replaced by ones). My graph had the seed value as the x-axis, and the sum of all the numbers in the seed’s sequence as the y-axis. Patterns I noticed with this graph was that every even number had a significantly lower sum than the odd number before and after it. I also noticed that the trend line showed a continuous increase.
I have learned different tactics that are used in mathematical research. Certain tactics I have learned are coding data in a binary manner, graphing data to view trends and patterns, and making modifications to find similar data. All of these tactics help find patterns. The usefulness of patterns is that once you find a pattern, you can figure out why that pattern is there, and this occasionally helps you find out why the data is how it is. This is similar to research in Chemistry. In Chemistry, you keep completing labs noting patterns you notice. You can use these patterns to help discover why the results of your labs are occurring. The importance of patterns are to help explain what results you are getting. For example, if you notice that a line shown on a graph is increasing at a constant rate. If you note this pattern, and draw an equation to describe the rate with an equation, you can use this to describe your line, and solve for any value on the graph.
The Collatz Conjecture is a conjecture that has not been proven yet. It is where you take any number, and if the number is even, you divide it by two and that is the next number in its sequence. If your number is odd, then you multiply it by 3, add one, and divide by two. You continue this process until your process begins repeating. Once your sequence begins to repeat, you count all of the numbers before the repetition (including the seed/number you started with), count one repetition, and that number is the size or length of that number’s sequence. The conjecture is that no matter what number you have, the sequence will end up reaching one. On the table I made of seed numbers and the sequences they made, I did not see many patterns. Once the table was coded, patterns began to emerge. The table was coded with odd numbers being replaced by ones, and evens being replaced by zeros. I began to see that when you looked at the rows, the first number in each sequence began to make a repetitions in relation to evens vs. odds. The second row made a longer one, the third did the same, and so on. The first repetition was 0110. The second was 10010110. The third was even larger. The size of the repetition could be described by the equation n=2^(n+1). Also, if you split the repetition in half, you can notice that the second half is the opposite number of the first half (i.e. ones would be replaced by zeros, and zeros replaced by ones). My graph had the seed value as the x-axis, and the sum of all the numbers in the seed’s sequence as the y-axis. Patterns I noticed with this graph was that every even number had a significantly lower sum than the odd number before and after it. I also noticed that the trend line showed a continuous increase.
I have learned different tactics that are used in mathematical research. Certain tactics I have learned are coding data in a binary manner, graphing data to view trends and patterns, and making modifications to find similar data. All of these tactics help find patterns. The usefulness of patterns is that once you find a pattern, you can figure out why that pattern is there, and this occasionally helps you find out why the data is how it is. This is similar to research in Chemistry. In Chemistry, you keep completing labs noting patterns you notice. You can use these patterns to help discover why the results of your labs are occurring. The importance of patterns are to help explain what results you are getting. For example, if you notice that a line shown on a graph is increasing at a constant rate. If you note this pattern, and draw an equation to describe the rate with an equation, you can use this to describe your line, and solve for any value on the graph.
I Am Math Project Reflection
Challenges that I faced along the way were mostly while I was creating my table. When I was filling out the table I made for my altercation on the equation, I made the mistake of not dividing by two for every odd number. This messed up my whole table, and I realized it when I was on the seed number 22... I had time though, so with no worries, I just erased it all, and started over. I knew that my final draft would be my final draft when I had other people look at it, and explain what was on it as if they were the ones who made it. I had troubles coming up with the titles mostly, so once I figured out how to word my titles properly, and create a presentable looking graph, I knew my final was made. The numeracy in this project mostly lied within the tables we made. We had to follow equations to make them, and when we found patterns, we had to write equations to represent them. Also, we had to use different techniques to find patterns in our table, which required the use of numbers. When we made altercations to the equation, it was as if we had to find all new patterns, and write all new equations for that. The same goes for when we used binary coding for our tables.
Challenges that I faced along the way were mostly while I was creating my table. When I was filling out the table I made for my altercation on the equation, I made the mistake of not dividing by two for every odd number. This messed up my whole table, and I realized it when I was on the seed number 22... I had time though, so with no worries, I just erased it all, and started over. I knew that my final draft would be my final draft when I had other people look at it, and explain what was on it as if they were the ones who made it. I had troubles coming up with the titles mostly, so once I figured out how to word my titles properly, and create a presentable looking graph, I knew my final was made. The numeracy in this project mostly lied within the tables we made. We had to follow equations to make them, and when we found patterns, we had to write equations to represent them. Also, we had to use different techniques to find patterns in our table, which required the use of numbers. When we made altercations to the equation, it was as if we had to find all new patterns, and write all new equations for that. The same goes for when we used binary coding for our tables.