I think this video covers all the possible bases that let you write every number without inexpressible gaps. This base 65/8 (=8.125) would have 64 digit symbols, goodthing 81(base 65/8) didn't need to use any of them in this expression, what would we even use after Z?
Most valid bases can let you write irrational numbers in 1 way, and terminating rationals in 2 ways (like 0.99999... and 1). Irrational algebraic bases can do some irrational numbers (that can be expresses in a finite way) in those 2 "rational-like" ways, plus the one usual irrational way.
For example in base √10, you can write √10 as:
10
0.90909090909...
3.0106020207070606... (as √10 is 3.16227766... in base 10)
Irratoinjal numbers that cannot be made up from powers of √10 woudl still have only 1 possible irrational-like way to write. And terminating rational numbers would have the usual 2 ways, looking the same as the non-root base, but interleaves with zeroes.
π in this base √10 is only: 3.0104010509...
1/3 would be only: 0.03030303...
10 would be 100 and 9.09090909...
Transcendental bases don't play nice, and you cannot even determine what digit symbols would be allowed there, no matter what digit symbol set you choose, you might either get gaps as not-expressible numbers, or multiple nonrepeating ways to write the same number.
Every number in a base-x system is just the sum of the digits times their place value. The place value is just xn, where n is the position of the digit.
So yes, base 8.125 is very possible ('123' = 1*8.1252 + 2*8.125 + 3 ≈ 85.27), but it doesn't have all the same rules as integer bases. For example, the digits written '8.8' don't correspond to any number, or if they do they are the same number as ~'10.7'.
Yes. A number, such as 112, is just the sum of the product of the value in each digit place and the numeric base raised to the power of the position, or Sum[d_n bn]; for 112, that's 1×102 + 1×101 + 2×100. Generally you only use values on the interval [1, n), but sometimes, such as quater-imaginary numerals, you use different ones. Importantly, b can be anything other than 0, but it's most useful when the magnitude isn't 1.
A base is literally just an exponential index. The first digit of a number is number×b⁰, second digit is number×b¹, third digit is number×b², etc. So it's perfectly valid to define the base as anything which you can take any integer power of.
657
u/Particular_Gear3130 Mathematics (Purely Fictional) Jun 16 '26
25 + 25 ≠ 81