What is a number base, and what’s special about them?
The base your number system is in doesn’t actually affect what numbers are, only how you write them, and until you get to more than one digit, all bases are the same. Base 10(numbers 1 0), would technically be an accurate way to describe any base, because the number 1 followed by 0 means that you’ve reached the number your base is in, and everything loops back around again. So 13 in base X(the one we’ve typically used lately in society), is 9+4, and in any base you use, it will always be 9+4, though you may need to convert 9 and/or 4 if you’re using a base lower than X. 1+1+1+1+1 will always equal 5 though, even if you have to write it in binary(1, 10, 11, 100, 101), or even unary 11111
The fibonacci sequence isn’t defined in terms of a base, but rather in terms of adding the two numbers immediately preceding it together. So the first number is 1, then 1 and 1, then 1+1 and 1, then 1+1+1 and 1+1, and so on. So no matter what base you use, you’ll still come up with the fibonacci sequence, and all the numbers will mean the same thing, it will just be written differently once it starts passing whatever 10 happens to be.
However, we might not even have a base, in which case, we’d just have to keep tacking numbers on endlessly, never repeating and never looping. Having a number base was actually dependent on coming up with the concept of 0, which came along with the arabic numeral system we still use today. Without it, we’d just been figuring out symbols for numbers, then tacking them together endlessly, larger numbers would just grow larger and larger until they became too cumbersome, and we’d give them another symbol.
I II III V VIII XIII XXI, XXXIV, the first 8 fibonacci numbers in baseless roman numerals 🙂
Here’s the first XIII numbers in several different bases.
| Roman | I | II | III | IV | V | VI | VII | VIII | IX | X | XI | XII | XIII |
| Binary | 1 | 10 | 11 | 100 | 101 | 110 | 111 | 1000 | 1001 | 1010 | 1011 | 1100 | 1101 |
| Base 3 | 1 | 2 | 10 | 11 | 12 | 20 | 21 | 22 | 30 | 31 | 32 | 40 | 41 |
| Base 5 | 1 | 2 | 3 | 4 | 10 | 11 | 12 | 13 | 14 | 20 | 21 | 22 | 23 |
| Base 6 | 1 | 2 | 3 | 4 | 5 | 10 | 11 | 12 | 13 | 14 | 15 | 20 | 21 |
| Base 8 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 10 | 11 | 12 | 13 | 14 | 15 |
| Base X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
| Base R | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | X | E | 10 | 11 |
| Base T | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | X | E | R | 11 |
The first few primes are easy enough to see in base X, but it’s actually nowhere near the best base for picking them out. In order for a number to be prime, it has to not be divisible by any other prime number that’s come before it. That’s why 2 is the only even prime, because anything else even is divisible by 2. 3 is also the only prime that’s divisible by 3, so every 3rd number will also never be prime.
Now, the pattern made by multiplying 2 and 3 repeats regularly every 6 numbers.
2, 3, 2, O, 2*3, O, 2, 3, 2, O, 2*3, O, 2 …. therefore, the only place a prime can ever exist is in the pattern left over by the Os.
That’s why in base 6, above, every prime that will EVER exist after 3, will be a number adjacent to a 10. In base 6, the first several primes after 3 are 5, 11, 15, 21, 25, 31, 41, 45… and you will never see a prime in base 6 that doesn’t end in 1 or 5.
You can also get a similar effect in any other base divisible by 6. In base R, primes will only ever end in 5, 7, E, or 1. Again due to being adjacent to 6s, which always end in 6 or 0. 🙂
Base R also has a number of other useful features, looking at the chart, its 10 lines up with 4 other bases above it, this means that those numbers all go evenly into it, and loop back around at the same numbers, every single time. So in base R, every number that’s divisible by 3 ends in 3, 6, 9, or 0. Every number ending in 4, 8 or 0 is divisible by 4, And every number ending in 6 or 0 by 6.
All just like any even number or 5 in base X is easily picked out at a glance for being divisible by 2 or 5.
Prime numbers such as 13 in X // 11 in R however would make terrible bases, as nothing would ever repeat. However, they WOULD be consistent. Even numbers would alternate every 10, Threes would cycle every 30, fours every 40 fives every 50 … and so on. In the meantime you’d come up with one ending in every number up to the base, but the order would be consistent with every loop.
However, this property also exists in every other base, in base X, the ending of numbers divisible by 3 cycles every 30, so 21, and 51, and 81 are all divisible by 3. And therefore 4, 44, 84, 124, and 164 are all divisible by 4. And actually, 4 is divisible by 2, which also goes into X, so 24, 64, 104 and 144 also count.
And since the order is consistent, if you remember the way it goes you can quickly count in multiples of that number endlessly on and on forever.
