# Thread: can you have number bases that are non-integers?

1. ## can you have number bases that are non-integers?

if you were using base pi then pi would be 1.
but what would 1(in base 10) be?

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1/pi or ~0.318
10char

3. Actually, in any base, the number written as "1" isn't equal to the base; the number written as "10" is. And the numbers written as "100" and "1000" are the base squared and cubed (a hundred is ten squared, a thousand is ten cubed, four is two squared, eight is two cubed). And so on from there...

In other words, a 1 followed by any number of zeroes, in any base, equals the system's base number to the power of the number of zeroes. So if "10" in base-pi is pi, then "100" is pi squared and "100000" is pi to the fifth power.

That also means that in any system, "1" is the base number to the power of zero (or to the zero'th power, as some would say). And that's already universally defined: anything (or at least anything positive) to the zero'th power always equals 1. It might sound weird, but you can confirm this yourself in the decimal and binary and other systems; you know the value of a digit in any "column" is the value of that same digit in the one before it or after it, times or divided by the base number: the sixteens column in base 2 has half the value of the thirty-seconds column, the thousands column in base 10 has a tenth of the value of the ten-thousands column, the sevens column in base 7 has a seventh of the value of the forty-nines column, et cetera. So, to find the value of "1" based on the value of "10", you'd just have to divide it by the system's base number, which is always "10", whatever "10" is... and in any system, a number divided by itself can't help but be 1:
2/2=1
6/6=1
22.43568/22.43568=1
pi/pi=1

So, in base pi:
first digit is "ones"
second digit is "pis"
third digit is "pi squareds"
fourth digit is "pi cubeds"
seventeenth digit is "pi^17s"

The real problem is how many numerals the system would have aside from 1 and 0 (the equivalents of 2-7 in base 8, 2-9 in base 10, and 2-E in base 16), and what the value of each of those numerals would be...

4. so pi in base pi would be 10?

5. Yes, the base number in its own base system is always "10". 18 in base 18 is "10". 6 in base 6 is "10". 73 in base 73 is "10". Blienep in base blienep is "10".

6. FYI: semi-related discussion in General Science

7. All your base are belong to us

8. Can you have number bases that are non-integers?
In String Theory, the math-wizzards use number-bases that are not only non-integers, they are non-existent!

Oh wait. Never mind. I'm thinking dimensions...

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If you're going to use pi you might as well be extra difficult and add in some irrational imaginary constant as well. A complex base would be much more fun.

10. What about base 0.5? That's weird enough for me:
1 = 1
10 = 0.5
100 = 0.25
So the maximum value in base 0.5 tends to zero...

11. Originally Posted by Sp1ke
What about base 0.5? That's weird enough for me:
1 = 1
10 = 0.5
100 = 0.25
So the maximum value in base 0.5 tends to zero...
Even freakier:

110 = 0.75
1110 = 0.875
11110 = 0.9375

12. Originally Posted by Chuck
If you're going to use pi you might as well be extra difficult and add in some irrational imaginary constant as well. A complex base would be much more fun.
Going along that train of thought: is Base i possible?

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Consider the integers {...-3, -2, -1, 0, 1, 2, 3, ...}. We wish to have a system for representing these in convenient fashion, and a number base is an example of that. Only a number base which is itself a positive integer greater than 1 will do. How do you represent 4 (the integer) in base pi? 11 in base pi is not an integer. Now clearly we can represent 4.000 as a real number in base pi, it will be a non-terminating non-recurring "decimal", but this lacks convenience, and makes it very difficult to talk about integers. With a transcendental base, it also makes it impossble to talk about rational numbers, algebraic numbers etc.

Now a representation of a real number in a base is just a power series, which is bounded above but not below (since fractions need not terminate):
(Not having mathematical notation available on this keyboard you will have to imagine what this power series normally looks like

SUM [n varies from minus infinity to N] a_n base^n

where each a_n is a non-negative integer and a_n < base

Now provided that base is a positive real number greater than 1, then any non-negative real number can be represented as such a power series, and it will be unique (apart from the 0.999... = 1 identity).

But notice that I needed the integers to write the definition. So a non-integral base that doesn't have the integers is not very useful even in this case.

If 0<base<=1, then there are no longer any integral a_n < base other than 0, and if we release the requirement that a_n<base then our series cease to be unique. And we require some limit on the a_n, so what should it be? Also unless there become numbers we cannot represent. So it don't work.
The problems are even worse if base is negative or imaginary.

Now we can have logarithms to base pi, like you find logarithms to base e on your calculator. But that is a different use of the word base.

14. Hmmmmmmmm.
Logs to base e.
Logs to base ten I can manage, even use and value.
Logs to base other I can grasp.

But why logs to base e?

John
(PS I know what 'e' is - the exponential number that describes growth and decay. I also know that 'e' logs are called 'natural' logs, and preceded base 10 logs in use, but not why!)

15. Originally Posted by JohnD
But why logs to base e?
Well, it's possible to use any positive real as your logarithm base, so a choice is made for convention and convenience.

A factor in favor of e is that the derivative of ex is ex and the derivative of ln(x) is 1/x.

Compare: derivative of 10x = 10x*ln(10) and derivative of log10(x) = 1/(x*ln(10))

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Originally Posted by Frog march
if you were using base pi then pi would be 1.
No it wouldn't. If you were using base pi, then the number expressed as "10" would be pi.

but what would 1(in base 10) be?
The same thing it's always been in base ten.

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Originally Posted by JohnD
Hmmmmmmmm.
Logs to base e.
Logs to base ten I can manage, even use and value.
Logs to base other I can grasp.

But why logs to base e?

John
(PS I know what 'e' is - the exponential number that describes growth and decay. I also know that 'e' logs are called 'natural' logs, and preceded base 10 logs in use, but not why!)
I am not a mathematician, but I believe that natural logs are useful in calculus.

(They're also a nifty feature to have on your calculator when doing growth and decay problems.)

18. Originally Posted by PaleBlueDot
(They're also a nifty feature to have on your calculator when doing growth and decay problems.)
When working with half-lifes, I'd think base two logarithms would be more handy

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e, e^x and log_e (x) gets to a lot of places in maths, and by extension natural processes:

Calculus:
Indefinite integral of 1/x is log_e (x)

Trignometry:
e^(i * x) = cos (x) + i * sin (x)
for example:
e^(i * pi) = -1

Number Theory:
Legendre's Prime Number Theorem of 1808
If we define P(n) as the quantity of prime numbers less than n, then
P(n) tends to n / (log_e (n) - 1.0836)

(Some slightly more accurate asymptotes have been devised since, but they all involve natural logs, or a variation called the logarithmic integral function.)

Base 2 logs are useful in computing theory.

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Originally Posted by hhEb09'1
When working with half-lifes, I'd think base two logarithms would be more handy
With half-lifes, they are. With continuous growth or decay, e and natural logs are more useful.

21. Originally Posted by PaleBlueDot
With half-lifes, they are. With continuous growth or decay, e and natural logs are more useful.
By half-lives, I meant continuous decay

22. Originally Posted by EricDerKonig
Going along that train of thought: is Base i possible?
Possible but rather useless, since you have
i0=1
i1=i
i2=-1
i3=-i
i4=1
you can't really represent a lot of numbers.

23. Originally Posted by HenrikOlsen
Possible but rather useless,
Similar to base 1, where 17 is written as 1111111111111111

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Originally Posted by hhEb09'1
Similar to base 1, where 17 is written as 1111111111111111
Actually, base 1 is very useful and commonly occurs in scoring card games and such. While it may not be easy to read, it is easy to write.

25. What do you think the bridge world would do with imaginary numbers?

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Originally Posted by HenrikOlsen
Possible but rather useless, since you have
i0=1
i1=i
i2=-1
i3=-i
i4=1
you can't really represent a lot of numbers.
You could represent any integer. Five would be 10001000100010001000. I guess you'd be stuck with fractions for non-integers.

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