I’ve been diving into some interesting properties of binary expansions lately and stumbled upon two fascinating sequences, A121016 and A328594. They’ve got me thinking about numbers with properly periodic binary expansions, and I really want to hear your thoughts on this!
First, let’s get on the same page about what we mean by “properly periodic binary expansions.” Basically, a binary number is called properly periodic if after a certain point in its binary expansion, the digits repeat indefinitely. For example, the binary representation of 0.1101… (where 1101 repeats) would be a properly periodic number.
Now, A121016 and A328594 seem to catalog numbers that are interesting in terms of their binary representations. What got me curious is how to identify whether a number belongs to these sequences based on their binary expansions. I’ve discovered that A121016 lists some rational numbers that have finite or repeating binary fractions, while A328594 includes a few irrational ones with more complex behaviours.
Here’s where I’d love to hear your insights: Can you think of any examples (other than the obvious ones like 0.1 or 0.01 in binary) that fit the criteria of being part of these sequences? Maybe you’ve come across some curious cases in your explorations that you could share. I’m especially interested in how you identify those properly periodic expansions. Do you have any tricks or methods that help you decipher which numbers make the cut for A121016 or A328594?
Also, I’d love to know why you think these patterns in binary can be so alluring. There’s something almost poetic about numbers that have repeating patterns, and it’d be great to dive deeper into that with your experiences or favorite examples! Let’s get a discussion going!
The exploration of properly periodic binary expansions is indeed a rich field, particularly when examining sequences like A121016 and A328594. A121016 primarily catalogues rational numbers whose binary representations either terminate or repeat, which can be ascertained by analyzing the denominator of the fraction when expressed in simplest form. For example, the binary representation of 0.375 can be derived from its fractional form (3/8), leading to a finite binary fraction of 0.011. On the other hand, A328594 deals with some fascinating irrational numbers, like the golden ratio (1.618…), whose binary representation is non-terminating but does not exhibit repeating behavior in the traditional sense, making it a unique case. The distinction between these representations often lies in the mathematical properties of the numbers involved—those whose denominators are composed solely of powers of 2 will yield properly periodic expansions in binary, while others will lead to more complex patterns.
Identifying numbers that fit within these sequences can involve a few methods: one effective technique is to convert fractions into binary manually or through programming, observing where the patterns emerge. For instance, take the fraction 1/3, which has a binary expansion of 0.101010…, showcasing a repeating sequence. This type of periodicity brings attention to the inherent structure in what might seem like random decimal or binary sequences. The beauty lies in the predictability of these patterns; they resonate with the fundamental nature of mathematics where order often emerges from chaos. The allure of numbers with repeating patterns might stem from their dual nature—they are both predictable and complex, a delightful contradiction that many find poetic and mesmerizing. Such exploration not only enhances our understanding of binary systems but also fosters a deeper appreciation for the overarching patterns in mathematics.
Oh wow, this is such an interesting topic! To be completely honest, I’m just scratching the surface myself, but binary numbers with repeating patterns fascinate me too.
You’re asking for examples beyond the obvious ones like “0.1” (which is just 1/2 in decimal). One neat example I’ve seen that might fit into sequences like A121016 or A328594 is something like “0.101010…” (repeating “10”), which is actually 2/3 in decimal. When you convert repeating binary fractions to decimal, sometimes the results surprise you!
I recently learned a cool trick: any binary fraction that repeats forever must be a rational number. To figure this out, you could try converting the repeating binary pattern to decimal and see what fraction you get. But numbers in A328594 sound trickier since you mentioned irrational numbers. I didn’t even know irrational numbers could have special repeating patterns—but maybe those sequences are looking at some special structure that’s more complex?
Honestly, for identifying these properly periodic expansions, what I’ve tried doing is just writing out the binary digits and looking carefully for repetition. Sometimes you’ll spot patterns easily, like “1100” repeated, and sometimes it can take a while to see them clearly. If you have a calculator or software that handles binary numbers, that could help a bit, too. Do you use anything like that?
You’re totally right about how poetic these repeating binary sequences can feel! There’s just something satisfying about seeing numbers dance into an endless, consistent pattern. Makes math feel more like art, doesn’t it?
Have you come across any intriguing examples yourself? I’d love to hear more about what you’ve discovered!