Wondering which teorem proves...

[sanctus]

...that all numbers can be created/represented in any base. For example why can we represent seventeen in any base, eg.

base_2: 10001

base_8: 23

base_10: 17 :-)

base_16: 11

 

In other words which is the theorem that states that any quantity can be obtained as a linear combination of the powers of any number?

I'm sure this must be a famous theorem I already heard much about but never looked at it this way...

[Donk]

It isn't a theorem, as far as I'm aware - just simple logic.

 

Think of a car's distance gauge. It rolls up to 9, then moves to 0 and the next digit goes up by one. It would work exactly the same if there were only six numbers on the wheel, or seven, or sixteen.

 

So you can represent any number in any base by simply adding ones, as above.

[CraigD]

...that all numbers can be created/represented in any base.

It’s worth noting that any finite nonnegative integer can be written as a numeral consisting only of digits in any base system, but that some finite nonnegative rational can’t be written as a numeral of only digits and “.” (the radix point), and some finite rational numbers – the transcendentals – can’t be written as a numeral of only digits and “.” in any base system.

 

Example of rational numbers that can be written in one base but not in another is [math]1/2[/math], which is [math]0.1_2[/math], [math]0.5_{10}[/math], but is [math]0.111 \dots_3[/math], which can’t be written using only “0”, “1”, “2”, and “.”, and [math]1/3[/math], which is [math]0.1_3[/math], but [math]0.010101 \dots_2[/math] and [math]0.333 \dots_{10}[/math]. In fact, there are an infinite number of bases in which any rational number that is not also an integer cannot be written.

In other words which is the theorem that states that any quantity can be obtained as a linear combination of the powers of any number?

It’s also worth noting that a numeral isn’t precisely the same thing as a linear combination of the powers of its base, because an interger can be uniquely written as numeral of any base, but there are an infinite number of linear combinations of the powers of the base for every integer, because the coefficients of a linear combination are not limited to the values of a base’s digits. For example,

[math]123 =1\cdot100+2\cdot10+3\cdot1 = 1\cdot100+1\cdot10+13\cdot1 = \frac12\cdot100+3\cdot10+43\cdot1[/math].

In other words, the powers of a number are not a basis.

 

Though some might consider it too obvious to mention, I think it’s also worth noting that numerals and numbers are not the same – that is, the map is not the territory.

[Turtle]

...that all numbers can be created/represented in any base. For example why can we represent seventeen in any base, eg.

base_2: 10001

base_8: 23

base_10: 17 :-)

base_16: 11

...

 

Here's some intersting reading on the topic, and some unusual results.

 

Base -- from Wolfram MathWorld

Some number systems use a mixture of bases for counting. Examples include the Mayan calendar and the old British monetary system (in which ha'pennies, pennies, threepence, sixpence, shillings, half crowns, pounds, and guineas corresponded to units of 1/2, 1, 3, 6, 12, 30, 240, and 252, respectively).

 

Bergman (1957/58) considered an irrational base, and Knuth (1998) considered transcendental bases. This leads to some rather unfamiliar results, such as equating [math]pi[/math] to 1 in "base [math]pi[/math] ,"[math]pi[/math]=10_{[math]pi[/math]} . Even more unexpectedly, the representation of a given integer in an irrational base may be nonunique, for example

 

10 = 10 100.010010101011_{[math]phi[/math]} (18)

10 100.0101_{[math]phi[/math]} (19)

 

where [math]phi[/math] is the golden ratio.

 

It is also possible to consider negative bases such as negabinary and negadecimal (e.g., Allouche and Shallit 2003). The digits in a negative base may be obtained with the Mathematica code

 

NegativeIntegerDigits[0, n_Integer?Negative] := {0}

NegativeIntegerDigits[i_, n_Integer?Negative] :=

Rest @ Reverse @ Mod[

NestWhileList[(# - Mod[#, -n])/n& ,

i, # != 0& ],

-n]

 

The base of a logarithm is a number used to define the number system in which the logarithm is computed. ...

[sanctus]

CRaigD, you said:

because an interger can be uniquely written as numeral of any base,

now let's be mathematicians, because I agree with Donk when you look at his example it is obvious/intuitive, but a "true mathematician" would like to prove it...or a real geek ;-) And don't get me wrong there is no critic there.

Completely agreed on the difference between numeral and linear combination.

 

Turtle, very intersting link indeed, base pi must be quite fun...

[CraigD]

now let's be mathematicians, because I agree with Donk when you look at his example it is obvious/intuitive, but a "true mathematician" would like to prove it...or a real geek ;-)

Though I’m not sure I fit this description, having little drive to prove the obvious proposition that any finite positive integer can be represented as a numeral, when I can’t imagine a useful consequence of failing to prove or disproving it, I’m a sucker for a direct challenge, so here goes…

 

Define:

1. A numeral base b is a string of zero or more digits base b, where b is a integer and b>1

2. A digit base b D is a glyph (character) mapping to an integer v_D, 0<=v_D<b

2a. For the sake of this proof, a numeral may begin with the digit mapping to 0 (the digit “0”), but may not begin with the digit mapping to b-1.

 

Assume the proposition to be disproven:

3. There exist one or more finite integers that can’t be represented as a base b numeral.

 

4. Because integers are ordered (consequence of the Peano postulates), there must then exist a smallest of these integer, m.

5. Because all integers have successors S(n)=n+1 (Peano postulates), an integer p that can be represented as a base b numeral exists such that m=S(p)=p+1.

 

6. Because p can be represented as a base b numeral, it must consist of

S_L A S_R

Where S_L is any string of zero or more digits base b, A a digit mapping to integer v_A<b-1, and S_R a string of zero or more of the digit B mapping to the integer v_B=b-1

 

7. The successor of p, m=p+1, then, must consist of

S_L B S_Z

Where B is a digit mapping to integer v_A+1, and S_Z is a string of zero or more of the digit “0” such that length(S_Z) = length(S_R)

 

8. For it to not be representable as a base B numeral, S_L, B, or S_Z must not exist. S_L exist by definition. S_Z exists if the digit “0” exists, so exists. Because every integer has a successor, if v_A exists, v_A+1 must exist.

9. So for B not to exist, there must be an v_A+1 < b that does not map to a digit, which contradicts [2].

10. Therefore B exists, and m=p+1 can be represented as a numeral base b, which contradicts [4], so [4] is false. If [4] is false, [3] is false, and the proposition to be disproven is disproven.