Prouhet-Thue-Morse constant
Imagine you have a bunch of numbers that keep growing forever, and you want to find a pattern in them. That's what the Prouhet-Thue-Morse constant is all about.
You start with two numbers: 0 and 1. Then, you create a new set of numbers by repeating the pattern of the first two numbers. So, it goes like this: 0, 1, 1, 0.
Then, you take that new set of numbers and repeat that pattern again. So, it becomes: 0, 1, 1, 0, 1, 0, 0, 1.
You can keep doing this over and over again, creating longer and longer sequences. What you'll notice is that there's a pattern in how many times each digit appears.
For example, in the 8-digit sequence above, there are 4 zeros and 4 ones. In the 16-digit sequence, there are 8 zeros and 8 ones, and so on.
The Prouhet-Thue-Morse constant is a magical number that represents this pattern. It's a decimal number that goes on forever, just like the sequences of numbers we've been creating.
But what's really amazing is that this number has some special properties that scientists and mathematicians find fascinating. It's symmetrical, which means that the digits on one side of the decimal point are the same as the digits on the other side. It's also "normal," which means that every possible combination of digits appears in this number at some point.
So, even though it starts with just two simple digits, the Prouhet-Thue-Morse constant holds a treasure trove of information and patterns for those who study it.
You start with two numbers: 0 and 1. Then, you create a new set of numbers by repeating the pattern of the first two numbers. So, it goes like this: 0, 1, 1, 0.
Then, you take that new set of numbers and repeat that pattern again. So, it becomes: 0, 1, 1, 0, 1, 0, 0, 1.
You can keep doing this over and over again, creating longer and longer sequences. What you'll notice is that there's a pattern in how many times each digit appears.
For example, in the 8-digit sequence above, there are 4 zeros and 4 ones. In the 16-digit sequence, there are 8 zeros and 8 ones, and so on.
The Prouhet-Thue-Morse constant is a magical number that represents this pattern. It's a decimal number that goes on forever, just like the sequences of numbers we've been creating.
But what's really amazing is that this number has some special properties that scientists and mathematicians find fascinating. It's symmetrical, which means that the digits on one side of the decimal point are the same as the digits on the other side. It's also "normal," which means that every possible combination of digits appears in this number at some point.
So, even though it starts with just two simple digits, the Prouhet-Thue-Morse constant holds a treasure trove of information and patterns for those who study it.