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bertieboo
science forum beginner

Joined: 12 Jul 2006
Posts: 5

Posted: Wed Jul 12, 2006 11:50 pm    Post subject: Change of base

Change of base

This note deals with changes of base from any base to any other base
(positive numbers)
(Note: Numbers in a base other than ten required to be converted can
easily be dealt with by first converting them to base ten).

Note: Bases are represented by a suffix to a number in brackets, except
that base ten numbers are shown with no brackets or suffix. The base
itself is always depicted in base 10 (ten). The base, b, of any number
system is always written 10, ie (10)b = b

Rational bases, a general method

Consider conversion to a rational base.

A general method which may be applied to all cases is as follows:

Consider a number, n which is to be converted to base b.

First find the highest required power p of the base, thus:

bp < n whence p = [logn/logb] and [ ] is the integer part.

The required number is then a1bp + a2bp-1 + ...+ apb + ap+1 . + ap+2 +
ap+3 ...
!
Where a1 = [n/bp] and d1 = the decimal part of n/bp, !
a2 = [d1b] and d2 = the decimal part of d1b radix point
a3 = [d2b] and d3 = the decimal part of d2b
: etc.

Note: The radix point lies between ap+1 and ap+2.

Example 4: Convert 3178.22 to base 5.2.
Here, p = [log3178.22 ÷ log5.2] = 4
a1 = [3178.22÷5.24] = 4 and d1 = 0.346809...
a2 = [0.346809x5.2] = 1 and d2 = 0.803408...
a3 = [0.803408x5.2] = 4 and d3 = 0.177722...
a4 = [0.177722x5.2] = 0 and d4 = 0.924154...
a5 = [0.924154x5.2] = 4 and d5 = 0.805600...
a6 = [0.805600x5.2] = 4 and d6 = 0.189120...
a7 = [0.189120x5.2] = 0 and d7 = 0.983424...
a8 = [0.983424x5.2] = 5 etc
Note: Radix point is between a5 and a6
Therefore 3178.22 = (41404.405...)5.2

Generally, rationals converted to a rational base do not seem to
repeat.

Eg 1/5 = (0.10102101504...)5.2
1/2 = (0.2303112044311...)5.2

However, there are examples that do repeat with base 5.2
Thus, 120/217 = (0.24)5.2 and 1685/5817 = (0.123)5.2

So, is it true that, generally, rationals converted to a rational base
do not repeat?
Robert B. Israel
science forum Guru

Joined: 24 Mar 2005
Posts: 2151

Posted: Thu Jul 13, 2006 2:39 am    Post subject: Re: Change of base

bertieboo <bob@bertuello.com> wrote:
 Quote: Change of base This note deals with changes of base from any base to any other base (positive numbers) (Note: Numbers in a base other than ten required to be converted can easily be dealt with by first converting them to base ten). Note: Bases are represented by a suffix to a number in brackets, except that base ten numbers are shown with no brackets or suffix. The base itself is always depicted in base 10 (ten). The base, b, of any number system is always written 10, ie (10)b = b Rational bases, a general method Consider conversion to a rational base. A general method which may be applied to all cases is as follows: Consider a number, n which is to be converted to base b. First find the highest required power p of the base, thus: bp < n whence p = [logn/logb] and [ ] is the integer part. The required number is then a1bp + a2bp-1 + ...+ apb + ap+1 . + ap+2 + ap+3 ... Where a1 = [n/bp] and d1 = the decimal part of n/bp, ! a2 = [d1b] and d2 = the decimal part of d1b radix point a3 = [d2b] and d3 = the decimal part of d2b : etc. Note: The radix point lies between ap+1 and ap+2.

I think what you mean is this. You have (presumably) b > 1 and
n > 0, and want to write n = sum_{j=1}^infty a_j b^{P+1-j} where
a_j are integers, 0 <= a_j < b. If P = floor(log(n)/log(b)), so
b^(P+1) > n >= b^P, take r_1 = n/b^P, and then for each j,
a_j = floor(r_j) and r_{j+1} = b (r_j - a_j).

 Quote: Example 4: Convert 3178.22 to base 5.2. Here, p = [log3178.22 ÷ log5.2] = 4 a1 = [3178.22÷5.24] = 4 and d1 = 0.346809... a2 = [0.346809x5.2] = 1 and d2 = 0.803408... a3 = [0.803408x5.2] = 4 and d3 = 0.177722... a4 = [0.177722x5.2] = 0 and d4 = 0.924154... a5 = [0.924154x5.2] = 4 and d5 = 0.805600... a6 = [0.805600x5.2] = 4 and d6 = 0.189120... a7 = [0.189120x5.2] = 0 and d7 = 0.983424... a8 = [0.983424x5.2] = 5 etc Note: Radix point is between a5 and a6 Therefore 3178.22 = (41404.405...)5.2 Generally, rationals converted to a rational base do not seem to repeat.

Indeed: if p is a prime such that the p-adic order of b is -k < 0
(i.e. if b is expressed as a fraction in lowest terms, the denominator
of b is divisible by p^k but not p^(k+1)), and some r_n has p-adic
order -m < 0, then the p-adic order of r_j for j >= n is -m-(j-n)k.
So the sequence {r_j} never repeats, and since the sequence of digits
(a_j, a_{j+1}, ...) determines r_j, that sequence can't be
eventually periodic.

 Quote: Eg 1/5 = (0.10102101504...)5.2 1/2 = (0.2303112044311...)5.2

p = 5 in these cases, with b = 26/5 having 5-adic order -1,
r_1 = 26/25 and 13/5 respectively having 5-adic orders -2 and -1.

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
bertieboo
science forum beginner

Joined: 12 Jul 2006
Posts: 5

Posted: Fri Jul 14, 2006 10:43 pm    Post subject: Re: Change of base

Robert Israel wrote:
 Quote: In article <1152748256.731645.58760@s13g2000cwa.googlegroups.com>, bertieboo wrote: Change of base This note deals with changes of base from any base to any other base (positive numbers) (Note: Numbers in a base other than ten required to be converted can easily be dealt with by first converting them to base ten). Note: Bases are represented by a suffix to a number in brackets, except that base ten numbers are shown with no brackets or suffix. The base itself is always depicted in base 10 (ten). The base, b, of any number system is always written 10, ie (10)b = b Rational bases, a general method Consider conversion to a rational base. A general method which may be applied to all cases is as follows: Consider a number, n which is to be converted to base b. First find the highest required power p of the base, thus: bp < n whence p = [logn/logb] and [ ] is the integer part. The required number is then a1bp + a2bp-1 + ...+ apb + ap+1 . + ap+2 + ap+3 ... Where a1 = [n/bp] and d1 = the decimal part of n/bp, ! a2 = [d1b] and d2 = the decimal part of d1b radix point a3 = [d2b] and d3 = the decimal part of d2b : etc. Note: The radix point lies between ap+1 and ap+2. I think what you mean is this. You have (presumably) b > 1 and n > 0, and want to write n = sum_{j=1}^infty a_j b^{P+1-j} where a_j are integers, 0 <= a_j < b. If P = floor(log(n)/log(b)), so b^(P+1) > n >= b^P, take r_1 = n/b^P, and then for each j, a_j = floor(r_j) and r_{j+1} = b (r_j - a_j). Example 4: Convert 3178.22 to base 5.2. Here, p = [log3178.22 ÷ log5.2] = 4 a1 = [3178.22÷5.24] = 4 and d1 = 0.346809... a2 = [0.346809x5.2] = 1 and d2 = 0.803408... a3 = [0.803408x5.2] = 4 and d3 = 0.177722... a4 = [0.177722x5.2] = 0 and d4 = 0.924154... a5 = [0.924154x5.2] = 4 and d5 = 0.805600... a6 = [0.805600x5.2] = 4 and d6 = 0.189120... a7 = [0.189120x5.2] = 0 and d7 = 0.983424... a8 = [0.983424x5.2] = 5 etc Note: Radix point is between a5 and a6 Therefore 3178.22 = (41404.405...)5.2 Generally, rationals converted to a rational base do not seem to repeat. Indeed: if p is a prime such that the p-adic order of b is -k < 0 (i.e. if b is expressed as a fraction in lowest terms, the denominator of b is divisible by p^k but not p^(k+1)), and some r_n has p-adic order -m < 0, then the p-adic order of r_j for j >= n is -m-(j-n)k. So the sequence {r_j} never repeats, and since the sequence of digits (a_j, a_{j+1}, ...) determines r_j, that sequence can't be eventually periodic. Eg 1/5 = (0.10102101504...)5.2 1/2 = (0.2303112044311...)5.2 p = 5 in these cases, with b = 26/5 having 5-adic order -1, r_1 = 26/25 and 13/5 respectively having 5-adic orders -2 and -1. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
bertieboo
science forum beginner

Joined: 12 Jul 2006
Posts: 5

Posted: Fri Jul 14, 2006 10:45 pm    Post subject: Re: Change of base

Robert Israel wrote:
 Quote: In article <1152748256.731645.58760@s13g2000cwa.googlegroups.com>, bertieboo wrote: Change of base This note deals with changes of base from any base to any other base (positive numbers) (Note: Numbers in a base other than ten required to be converted can easily be dealt with by first converting them to base ten). Note: Bases are represented by a suffix to a number in brackets, except that base ten numbers are shown with no brackets or suffix. The base itself is always depicted in base 10 (ten). The base, b, of any number system is always written 10, ie (10)b = b Rational bases, a general method Consider conversion to a rational base. A general method which may be applied to all cases is as follows: Consider a number, n which is to be converted to base b. First find the highest required power p of the base, thus: bp < n whence p = [logn/logb] and [ ] is the integer part. The required number is then a1bp + a2bp-1 + ...+ apb + ap+1 . + ap+2 + ap+3 ... Where a1 = [n/bp] and d1 = the decimal part of n/bp, ! a2 = [d1b] and d2 = the decimal part of d1b radix point a3 = [d2b] and d3 = the decimal part of d2b : etc. Note: The radix point lies between ap+1 and ap+2. I think what you mean is this. You have (presumably) b > 1 and n > 0, and want to write n = sum_{j=1}^infty a_j b^{P+1-j} where a_j are integers, 0 <= a_j < b. If P = floor(log(n)/log(b)), so b^(P+1) > n >= b^P, take r_1 = n/b^P, and then for each j, a_j = floor(r_j) and r_{j+1} = b (r_j - a_j). Example 4: Convert 3178.22 to base 5.2. Here, p = [log3178.22 ÷ log5.2] = 4 a1 = [3178.22÷5.24] = 4 and d1 = 0.346809... a2 = [0.346809x5.2] = 1 and d2 = 0.803408... a3 = [0.803408x5.2] = 4 and d3 = 0.177722... a4 = [0.177722x5.2] = 0 and d4 = 0.924154... a5 = [0.924154x5.2] = 4 and d5 = 0.805600... a6 = [0.805600x5.2] = 4 and d6 = 0.189120... a7 = [0.189120x5.2] = 0 and d7 = 0.983424... a8 = [0.983424x5.2] = 5 etc Note: Radix point is between a5 and a6 Therefore 3178.22 = (41404.405...)5.2 Generally, rationals converted to a rational base do not seem to repeat. Indeed: if p is a prime such that the p-adic order of b is -k < 0 (i.e. if b is expressed as a fraction in lowest terms, the denominator of b is divisible by p^k but not p^(k+1)), and some r_n has p-adic order -m < 0, then the p-adic order of r_j for j >= n is -m-(j-n)k. So the sequence {r_j} never repeats, and since the sequence of digits (a_j, a_{j+1}, ...) determines r_j, that sequence can't be eventually periodic. Eg 1/5 = (0.10102101504...)5.2 1/2 = (0.2303112044311...)5.2 p = 5 in these cases, with b = 26/5 having 5-adic order -1, r_1 = 26/25 and 13/5 respectively having 5-adic orders -2 and -1. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
bertieboo
science forum beginner

Joined: 12 Jul 2006
Posts: 5

Posted: Fri Jul 14, 2006 10:47 pm    Post subject: Re: Change of base

bertieboo wrote:
 Quote: Robert Israel wrote: In article <1152748256.731645.58760@s13g2000cwa.googlegroups.com>, bertieboo wrote: Change of base This note deals with changes of base from any base to any other base (positive numbers) (Note: Numbers in a base other than ten required to be converted can easily be dealt with by first converting them to base ten). Note: Bases are represented by a suffix to a number in brackets, except that base ten numbers are shown with no brackets or suffix. The base itself is always depicted in base 10 (ten). The base, b, of any number system is always written 10, ie (10)b = b Rational bases, a general method Consider conversion to a rational base. A general method which may be applied to all cases is as follows: Consider a number, n which is to be converted to base b. First find the highest required power p of the base, thus: bp < n whence p = [logn/logb] and [ ] is the integer part. The required number is then a1bp + a2bp-1 + ...+ apb + ap+1 . + ap+2 + ap+3 ... Where a1 = [n/bp] and d1 = the decimal part of n/bp, ! a2 = [d1b] and d2 = the decimal part of d1b radix point a3 = [d2b] and d3 = the decimal part of d2b : etc. Note: The radix point lies between ap+1 and ap+2. I think what you mean is this. You have (presumably) b > 1 and n > 0, and want to write n = sum_{j=1}^infty a_j b^{P+1-j} where a_j are integers, 0 <= a_j < b. If P = floor(log(n)/log(b)), so b^(P+1) > n >= b^P, take r_1 = n/b^P, and then for each j, a_j = floor(r_j) and r_{j+1} = b (r_j - a_j). Example 4: Convert 3178.22 to base 5.2. Here, p = [log3178.22 ÷ log5.2] = 4 a1 = [3178.22÷5.24] = 4 and d1 = 0.346809... a2 = [0.346809x5.2] = 1 and d2 = 0.803408... a3 = [0.803408x5.2] = 4 and d3 = 0.177722... a4 = [0.177722x5.2] = 0 and d4 = 0.924154... a5 = [0.924154x5.2] = 4 and d5 = 0.805600... a6 = [0.805600x5.2] = 4 and d6 = 0.189120... a7 = [0.189120x5.2] = 0 and d7 = 0.983424... a8 = [0.983424x5.2] = 5 etc Note: Radix point is between a5 and a6 Therefore 3178.22 = (41404.405...)5.2 Generally, rationals converted to a rational base do not seem to repeat. Indeed: if p is a prime such that the p-adic order of b is -k < 0 (i.e. if b is expressed as a fraction in lowest terms, the denominator of b is divisible by p^k but not p^(k+1)), and some r_n has p-adic order -m < 0, then the p-adic order of r_j for j >= n is -m-(j-n)k. So the sequence {r_j} never repeats, and since the sequence of digits (a_j, a_{j+1}, ...) determines r_j, that sequence can't be eventually periodic. Eg 1/5 = (0.10102101504...)5.2 1/2 = (0.2303112044311...)5.2 p = 5 in these cases, with b = 26/5 having 5-adic order -1, r_1 = 26/25 and 13/5 respectively having 5-adic orders -2 and -1. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
bertieboo
science forum beginner

Joined: 12 Jul 2006
Posts: 5

Posted: Fri Jul 14, 2006 10:52 pm    Post subject: Re: Change of base

bertieboo wrote:
 Quote: Change of base This note deals with changes of base from any base to any other base (positive numbers) (Note: Numbers in a base other than ten required to be converted can easily be dealt with by first converting them to base ten). Note: Bases are represented by a suffix to a number in brackets, except that base ten numbers are shown with no brackets or suffix. The base itself is always depicted in base 10 (ten). The base, b, of any number system is always written 10, ie (10)b = b Rational bases, a general method Consider conversion to a rational base. A general method which may be applied to all cases is as follows: Consider a number, n which is to be converted to base b. First find the highest required power p of the base, thus: bp < n whence p = [logn/logb] and [ ] is the integer part. The required number is then a1bp + a2bp-1 + ...+ apb + ap+1 . + ap+2 + ap+3 ... ! Where a1 = [n/bp] and d1 = the decimal part of n/bp, ! a2 = [d1b] and d2 = the decimal part of d1b radix point a3 = [d2b] and d3 = the decimal part of d2b : etc. Note: The radix point lies between ap+1 and ap+2. Example 4: Convert 3178.22 to base 5.2. Here, p = [log3178.22 ÷ log5.2] = 4 a1 = [3178.22÷5.24] = 4 and d1 = 0.346809... a2 = [0.346809x5.2] = 1 and d2 = 0.803408... a3 = [0.803408x5.2] = 4 and d3 = 0.177722... a4 = [0.177722x5.2] = 0 and d4 = 0.924154... a5 = [0.924154x5.2] = 4 and d5 = 0.805600... a6 = [0.805600x5.2] = 4 and d6 = 0.189120... a7 = [0.189120x5.2] = 0 and d7 = 0.983424... a8 = [0.983424x5.2] = 5 etc Note: Radix point is between a5 and a6 Therefore 3178.22 = (41404.405...)5.2 Generally, rationals converted to a rational base do not seem to repeat. Eg 1/5 = (0.10102101504...)5.2 1/2 = (0.2303112044311...)5.2 However, there are examples that do repeat with base 5.2 Thus, 120/217 = (0.24)5.2 and 1685/5817 = (0.123)5.2 So, is it true that, generally, rationals converted to a rational base do not repeat? ***********************************************

NB: I don't think I have put this reply in the right place.
Sorry!
Thank you Robert. I now deal with fractions to a fractional base
Change of base (I trust that I am not too longwinded)
In the following general method, the base, b may be a positive integer,
a mixed number, an irrational number or <1. (Note: I later dropped the
underscore for the suffix)

Consider a number, n which is to be converted to base b.
First find the highest required power p of the base, thus:
b^p <= n whence p = [logn/logb] and [ ] is the integer part.

The required number is then: a1bp + a2bp-1 + ...+ apb + ap+1 . + ap+2
+ ap+3 ...
a_1b^p + a_2b^p-1 + ...+ a_pb^1 + a_p+1b^0 . + a_p+2/b + a_p+3/b^2 ...
Where a_1 = [n/b^p] and d_1 = the decimal part of n/b^p,
a_2 = [d_1b] and d_2 = the decimal part of d_1b
a_3 = [d_2b] and d_3 = the decimal part of d_2b
: etc.
Note: The radix point lies between a_p+1 and a_p+2.

Example 4: Convert 3178.22 to base 5.2.
Here, p = [log3178.22 ÷ log5.2] = 4
a_1 = [3178.22÷5.2^4] = 4 and d_1 = 0.346809...
a_2 = [0.346809x5.2] = 1 and d_2 = 0.803408...
a_3 = [0.803408x5.2] = 4 and d_3 = 0.177722...
a_4 = [0.177722x5.2] = 0 and d_4 = 0.924154...
a_5 = [0.924154x5.2] = 4 and d_5 = 0.805600...
a_6 = [0.805600x5.2] = 4 and d_6 = 0.189120...
a_7 = [0.189120x5.2] = 0 and d_7 = 0.983424...
a_8 = [0.983424x5.2] = 5 etc
Note: Radix point is between a_5 and a_6
Therefore 3178.22 = (41404.405...)5.2

Fractional bases
In the above general method, the base, b, need not be restricted to >1.
Therefore, it is required to express any number to a base b < 1.
It is clear that some proper fractions can be expressed to the base of
another fraction.
Examples are as follows:
Take base, b = 1/3
Then (1)b = b^0 = 1, (10)b = b^1 = 1/3, (100)b = 1/9, (110)b 1/3 + 1/9 = 4/9
(1000)b = 1/27, (1010)b = 1/27 + 1/3 = 10/27, (1110)b 1/27+1/9+1/3 = 13/27

Now, if we want to express, say, 2/5 to base 1/3, the answer will lie
between (1010)b and (1110)b, since 10/27 < 2/5 < 13/27

Note: If we use digits to the right of the (radix) point, we will never
get proper fractions.
Thus: (0.1)b = 3, (0.01)b = 9, (0.11)b = 3+9 = 12, etc.

NB Some cases will not be possible.
Eg. Consider base 1/3. To obtain a fraction to this base, we are
limited to digits on the left of the radix point except 1 in the first
place ( b^0 = 1). Thus, we can only write (....1110.0)1/3 for any
fraction. In effect, this forms the series:
1/3 + (1/3)^2 + (1/3)^3 + ... where the sum to infinity 1/2
It is therefore impossible to represent say, 3/5 as a fraction to
base 1/3, and, to express 1/2 to base 1/3, we will need an infinity of
1's.

However, if we make the base big enough, any fraction can be
represented to that base.
Eg. If we take 3/4 as base, then,
(...11110.0)3/4 will represent 3/4 + (3/4)^2 + (3/4)^3 + ...
whose sum to infinity = 3.

So we require to find the base that will allow any fraction to be
expressed in that base.
Now, from knowledge of series, we know that 1/2 + (1/2)^2 + (1/2)^3 +
.... = 1.
Therefore, any base greater than 1/2 can be used to express any
fraction in that base.

To convert a number to a base, b < 1
Method: First convert the number to the base 1/b (see above), move the
radix point one place to the left, and then reverse the digits.
Proof:
If (...a1 a2 a3 . a4 a5...)b = (...a5 a4 a3 . a2 a1...)1/b
then, using place values, we see that:
LHS = ...+ a1b^2 + a2b + a3b^0 + a4/b + a5/b^2...

RHS = ...+ a5/b^2 + a4/b^1 + a3b^0 + a2b^1 + a1b^2... = LHS QED

Example of a mixed number converted to a fractional base: Convert 5.6
to base 2/3.
First convert to base 3/2 = b
p = [log5.6 ÷ log1.5] = 4
a1 = [ 5.6 / b^4 ] = 1 & d1 = 0.106...
a2 = [ d1 x b ] = 0 & d2 = 0.159...
:
a7 = [ d6 x b ] = 1 & d7 = 0.209...
:
a11 = [ d10 x b ] = 1 & d11 = 0.0599...

Therefore 5.6 = (10000.010001...)3/2 = (...1000100.0001)2/3 5.59473594...

Example of a fraction converted to a fractional base:

Example : Convert 2/3 to base 3/4. First convert to base 4/3.
p = [log 2/3 ÷ log 4/3] = -1; therefore radix point lies between a0
and a1
a1 = [2/3 ÷ (4/3)^-1] = 0 and d1 = 0.888...
a2 = [d1 x b] = 1 and d2 = 0.185...
:
a8 = [d7 x b] = 1 and d8 = 0.040...
and a20 = [d19 x b] = 1 and d20 = 0.287...
a25 = [d24 x b] = 1 and d25 = 0.172...
a32 = [d31 x b] = 1 and d32 = 0.294...
etc
Therefore 2/3 = (0.01000001000000000001000010000001...)4/3
= (...100000010000100000000000100000100.0)3/4
= 0.666637...

It should be noted that, in trying to convert to a base < 1/2, on
converting to the inverted base, the answer may contain digits other
than 0 and 1, and these digits will then appear in the final answer.
Example : Convert 100 to base 0.4. First convert to base 1/0.4 2.5.
p = [log 100 ÷ log 2.5] = 5
a1 = [100 ÷ 2.5^5] = 1 and d1 = 0.024
a2 = [d1 x 2.5] = 0 and d2 = 0.06
and a6 = [d5 x 2.5] = 2 and d6 = 0.343...
a8 = [d7 x 2.5] = 2 and d8 = 0.148...
a11 = [d10 x 2.5] = 2 and d11 = 0.319...
a13 = [d12 x 2.5] = 1 etc
Therefore 100 = (100002.0200201...)2.5
= (...10200202.00001)0.4
= 99.9983684...

Looking at the earlier example, we can easily derive the value of
3178.22 to the base 1/5.2. Thus:
3178.22 = (41404.405...)5.2
= (...5044.0414)1/5.2

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