Math Is Fun Forum

Dries

Member

Registered: 2021-01-05

Posts: 7

Fractional base for numbers

So what happens if I use base 3/2 instead of base 10?

Now, this doesn't make any sense. But it is fun to do.
Any number in base 10, say 62, can be written in a form like this: 6 * 10^1 + 2 * 10^0
So I can start building numbers in base 3/2
It starts with 1, as 1 * (3/2)^0
There's no 2, since it would be larger than 10 in base 3/2, since 10 would be 1 * (3/2)^1
However, there's no 11, cause that would be 1 * (3/2)^1 + 1 * (3/2)^0 Which is 2.5 And that's larger than 100 in base 3/2, because 1*(3/2)^2 = 2.25
And so the number line seems to go as follows:
1, 10, 100, 101, 1000, 1001, 1010, 10000, 10001, 10010, 10100, 100000
And the corresponding decimal values are:
1, 1.5, 2.25, 3.25, 3.375, 4.375, 4.875, 5.0625, 6.0625, 6.5625, 7.3125, 7.59375

So we get "whole numbers" that aren't really whole numbers.
And those whole numbers are kind of peculiar. We might add two of these whole numbers, but the result might not be a whole number.
Which kind of makes sense, since 1 ( base 3/2) fits 1.5 times in 10 (base 3/2) and 10 fits 1.5 times in 100.

Now if there were some aliens somewhere in the universe that would only know base 3/2, they might start their number theory with these "whole numbers". But they would find no primes. So it seems. Since the only number that seems to divide 1, is 1 itself.

They would find composite numbers, however. Since, in base 3/2, 10 * 10 = 100 too.
And 10 *101 = 1010 Since 1.5 * 3.25 = 4.875
And 10 * 1010 = 10100 = 100 * 101

But not all numbers are composite.
The sequence of non-composite numbers goes as follows:
1, 101, 1001, 10001, ...

Interesting... An alien mathematician might want to prove a conjecture...

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,406

Re: Fractional base for numbers

Hi,

I came across a link here.

In the relevant topic.

Interesting!

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