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Converting Base Numbers Directly Without Using Base 10

Date: 01/26/2004 at 01:53:32
From: John
Subject: Bases changes

How do you convert from one base to another without going through base
10?  I understand conversions between binary based bases, but how do I
go from base 3 to base 7, for instance?  Would the same algorithm work
to go from base 5 to base 12?



Date: 01/26/2004 at 13:14:00
From: Doctor Greenie
Subject: Re: Bases changes

Hi, John --

The process is straightforward and can be used to convert between any
two bases.  But to use it you need to be proficient in the arithmetic
of both the bases you are working with.  The key to the process is the
method you use to evaluate a number in a particular base.

Given the base-10 number "1234", there are two basic ways to evaluate it:

(1)  (using place values)

  1*1000 + 2*100 + 3*10 + 4 = 1000 + 200 + 30 + 4 = 1234

(2)  ("building" the number from left to right using only the base)

With this method, you start from the left in the number; for each 
digit, you add that digit and then multiply by the base:

  1                ["add" the next (first) digit]
  1 * 10 = 10      [multiply by the base]
  10 + 2 = 12      [add the next digit]
  12 * 10 = 120    [multiply by the base]
  120 + 3 = 123    [add the next digit]
  123 * 10 = 1230  [multiply by the base]
  1230 + 4 = 1234  [add the next digit]

I used a base-10 example first so you could see the process; let's use
the two methods to evaluate 1234 (base 5):

(1)  (using place values)

  1*125 + 2*25 + 3*5 + 4 = 125 + 50 + 15 + 4 = 194

(2)  ("building" the number from left to right using only the base)

  1
  1 * 5 = 5
  5 + 2 = 7
  7 * 5 = 35
  35 + 3 = 38
  38 * 5 = 190
  190 + 4 = 194

You can convert directly between any two bases using the second method
described above--you just need to be able to do the required
arithmetic in the bases you are working with.  Following is a
demonstration of the conversion

  1234 (base 5) = ??? (base 7)

We use method (2) above, doing the arithmetic in base 7....

  1 * 5 = 5
  5 + 2 = 10  (again, you need to know that 7 in base 7 is 10)
  10 * 5 = 50
  50 + 3 = 53
  53 * 5 = 361
  361 + 4 = 365

Note that the arithmetic is all in base 7 (the "new" base), while the
multiplication at each stage is by 5 (the "old" base).

Checking this result--by evaluating 1234 (base 5) and 365 (base 7)
using our familiar base-10 arithmetic--we have

  1234 (base 5):

  1
  1 * 5 = 5
  5 + 2 = 7
  7 * 5 = 35
  35 + 3 = 38
  38 * 5 = 190
  190 + 4 = 194

  365 (base 7):

  3
  3 * 7 = 21
  21 + 6 = 27
  27 * 7 = 189
  189 + 5 = 194

And one more example: following is a demonstration of the conversion

  1234 (base 7) = ??? (base 5)

We use method (2) above, doing the arithmetic in base 5.  Note that in
this example the (base 5) arithmetic is complicated by the fact that
our multiplier (the old base, 7) is "12" in base 5....

  1
  1 * 12 = 12
  12 + 2 = 14
  14 * 12 = 223
  223 + 3 = 231
  231 * 12 = 3322
  3322 + 4 = 3331

To check this result, we use our familiar base-10 arithmetic to
evaluate both 1234 (base 7) and 3331 (base 5):

  1234 (base 7):

  1
  1 * 7 = 7
  7 + 2 = 9
  9 * 7 = 63
  63 + 3 = 66
  66 * 7 = 462
  462 + 4 = 466

  3331 (base 5):

  3
  3 * 5 = 15
  15 + 3 = 18
  18 * 5 = 90
  90 + 3 = 93
  93 * 5 = 465
  465 + 1 = 466

If you followed the preceding examples with pencil and paper, trying 
to perform the additions and multiplications in the unfamiliar bases,
you probably had considerable difficulties.  For this reason, when
converting between two unfamiliar bases, it is easier to convert first
from the "old" base to base 10 and then from base 10 to the "new"
base, because all the arithmetic in both those steps can be done in
our familiar base 10.

I hope all this helps.  Please write back if you have any further
uestions about any of this.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/ 


Date: 08/22/2015 at 10:53:38
From: Shweta
Subject: Base Conversion of fractional number(321.42) without base10

Hey,

The post above shows how to convert a number from one base to another 
without first converting it to base 10. But I want to know how to convert 
a number from one base to another without converting it to base 10 when 
there's a decimal point in it. 

For example, convert 321.42 (base 6) to base 7. 

I know how to convert from base 6 to base 10, and from base 10 to base 7. 
But I want to know how to convert the 0.42 (base 6) to base 7, bypassing 
the step to convert to base 10.



Date: 08/22/2015 at 11:39:49
From: Doctor Peterson
Subject: Re: Base Conversion of fractional number(321.42) without base10

Hi, Shweta.

Do you know how to convert fractional numbers between bases? You just do 
the same thing you'd do converting either to or from base 10, but do your 
arithmetic in the base you are converting to or from. The same will be 
true for your question.

Show me how you'd convert 0.42_six to base 10, or 0.43_ten to base 7, and 
I'll show you how to adapt that (if you haven't already done it).

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/



Date: 08/22/2015 at 12:15:14
From: Shweta
Subject: Base Conversion of fractional number(321.42) without base10

Thank you for replying.

Yeah, I do know how to convert 0.42(base 6) to base 10, and then base 10 
to base 7. First,

   0.42 (base 6) to base 10 ->
   0.42 (base 6) = 4*6^-1 + 2*6^- 2 
                 = 0.722 (base 10)

From there,

   0.722 (base 10) to base 7 ->
         0.722*7 = 5.054 -> 5
         0.054*7 = 0.378 -> 0
         0.378*7 = 2.646 -> 2
         0.646*7 = 4.522 -> 4

And so on.

So, 0.42 (base 6) = 0.5024 (base 7).

This is what my teacher taught me.

But I am unable to figure out how to convert from base 6 to base 7 
directly.



Date: 08/22/2015 at 13:02:20
From: Doctor Peterson
Subject: Re: Base Conversion of fractional number(321.42) without base10

Hi, Shweta.

There is a quicker way to convert a fractional number to base 10. I show 
several methods for conversions in both directions here:

  Fraction/Decimal Conversion to Other Bases
    http://mathforum.org/library/drmath/view/55744.html

The best method, I think, is to repeatedly divide by the base, then add in 
the next digit, starting at the far end:

   0.42_six, to base ten

   farthest digit is 2: 2/6 = 0.333... 
   add next digit and divide again: 4.333.../6 = 0.722...

To use this method (or yours) to convert to base 7, we would need to do 
the arithmetic, largely division, in base 7. I'd rather avoid dividing in 
any unusual base, because it takes a lot of skill.

Your method for converting FROM base 10 is a good one, and it uses 
multiplication, which is easier to do in an unfamiliar base. So I'd use 
that method for conversion from base 6 to 7, doing the arithmetic in the 
original base, 6.

The method, again, is to multiply by the new base, take the integer part 
as a digit in the answer, and repeat using the fraction part that remains. 

Let's do it:

   0.42_six to base seven, working in base six:

   New base, written in base six, is 11_six

   0.42_six * 11_six = 5.02 --> 5     0.42_six   0.02_six   0.22_six
                                      * 11       * 11       * 11
   0.02_six * 11_six = 0.22 --> 0     ----       ----       ----
                                        42          2         22
   0.22_six * 11_six = 2.42 --> 2      42          2         22
                                      ----       ----       ----
   0.42_six * 11_six = 5.02 --> 5     5.02_six   0.22_six   2.42_six

So our answer is 0.5025..._seven.

This differs from your answer because you rounded the answer in base
ten before converting to base seven. (Using a calculator to help with
the work, I wouldn't have rounded anything, but the calculator would
still be limited in its precision.) 

I could continue this method for as many places as I want, with no loss 
of precision. That's a benefit I'd never thought of in years of advising 
students not to bother converting directly!

This could have been a lot more painful, so thanks for offering a problem 
in which the multiplications were simple!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/



Date: 08/22/2015 at 13:36:00
From: Shweta
Subject: Thank you (Base Conversion of fractional number(321.42) without base10)

Thank you, now I get it. 

All the other students in my class solved it the other way, converting to 
base 10 and back again. But I was just curious to find how to convert it 
directly. 

Thanks again. You were very helpful.
Associated Topics:
High School Number Theory
Middle School Number Sense/About Numbers

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