r/theydidthemath • u/zoroddesign • 16h ago
[Request] What number can be expressed as a 1 with some amount of zeroes behind it that is larger than just 1 in the most possible numerical bases?
1
u/Dependent_Order_7358 13h ago
The number you’re referring to is 10.
Here’s why:
In numeral systems, a number expressed as “1” followed by zeros is written as 10 in base b, which represents b1, or simply b.
For example:
In base 2, 10_2 is 21 = 2.
In base 3, 10_3 is 31 = 3.
In base 4, 10_4 is 41 = 4.
So, 10 is the number that can be expressed as a “1 with some zeros behind it” in a variety of numeral systems and gives different values in each base, always being larger than 1. The number 10, expressed as “1” followed by a zero, works in all bases greater than 1 and represents the base itself, meaning it’s expressed as “10” in the most number of numeral systems while being larger than 1.
Therefore, 10 can be expressed as “1 followed by zeros” in the most possible numerical bases, each time representing a larger value than 1.
2
u/FloralAlyssa 12h ago edited 12h ago
I think you have their question backwards. They are asking what base 10 number is 1....0 in the most number of bases.
10 is 1010 in base 2, 101 in base 3, 22 in base 4, 20 in base five, etc.
I believe the answer is that there is none, but any number X that has a prime factorization pN will have multiple bases, but the formula is eluding me right now.
For example, 16 is 24, and in base 16 it is 10, and in base 4 it is 100 and in base 2 it is 1000. 27 is 33 and it is 10 in base 27, and 100 in base 3.
1
u/Dependent_Order_7358 11h ago
Oh haha :) you are right! I’m a bit sleepy, thanks!
Your answer is cool :)
1
u/zoroddesign 8h ago
I did make this post right before falling asleep so I could have worded it better.
The example of what I am trying to get is
4096 in Binary is 1000000000000, Quaternary is 1000000, Octalnary is 10000, hexadecimal is 1000.
These were easy to find because they are multiples of each other.
But is there a point where they also include trinary, pentanary, decimal, etc?
1
u/gmalivuk 10h ago
It doesn't need to be a power of just one prime, it just needs to be a perfect power of something. And you'd get more possibilities based on the factors of the power.
For example, anything to the 6th power can be written as 1000000, 1000, 100, and 10, because n6 = (n2)3 = (n3)2 = (n6)1.
There's no limit but the best you can do up to a given maximum is with highly composite powers.
•
u/AutoModerator 16h ago
General Discussion Thread
This is a [Request] post. If you would like to submit a comment that does not either attempt to answer the question, ask for clarification, or explain why it would be infeasible to answer, you must post your comment as a reply to this one. Top level (directly replying to the OP) comments that do not do one of those things will be removed.
I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.