Converting a Number from One Base to AnotherDate: 02/28/2008 at 17:03:31 From: Shayan Subject: conversion of any base to binary How to convert a number in any base to binary? For example, 673.6 of base 8 to binary and 310.2 of base 4 to binary? I know conversion of decimal to binary but what about conversion of octal or hexadecimal to binary? Now I convert octal 673.6 to decimal then to binary but I want to get direct method to convert octal 673.6 to binary. Date: 02/28/2008 at 22:09:26 From: Doctor Greenie Subject: Re: conversion of any base to binary Hi, Shayan -- Your question asks about the method for converting from any base to binary; but your examples only ask about converting from base 4, base 8, or base 16. Conversion from bases 4, 8, or 16 to base 2 is easy, because those bases are powers of 2. As a result, each digit in base 4 converts directly to two digits in binary; each digit in base 8 converts directly to three digits in binary; and each digit in base 16 converts directly to four digits in binary. So for your examples we have 673.6 (base 8) = 110 111 011 . 110 (binary) 310.2 (base 4) = 11 01 00 . 10 (binary) - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ Date: 03/01/2008 at 08:32:02 From: Shayan Subject: conversion of any base to binary Thank you. Is there any direct method to convert octal to hexa or any base to any other base? Date: 03/01/2008 at 22:57:18 From: Doctor Greenie Subject: Re: conversion of any base to binary Hello, Shayan -- There is a single method that can be used to convert any number from one base directly to another base, without going through base 10. You just need to be able to perform multiplication and addition in the new base. I will demonstrate the process below. However, first, here is a link to a page in the Dr. Math archives where standard methods are described for converting numbers between base 10 and another base: Converting from Base 6 to 10 and Back http://mathforum.org/library/drmath/view/55738.html Here is a section quoted from that page, showing the conversion of a number from base 6 to base 10. We will use this method to convert from any base to any other base; this example of converting to base 10 is useful because it uses our familiar base 10 arithmetic. As noted in my opening paragraph, we can use the same process to convert to any other base, as long as we can do the required arithmetic in the new base. ***************************** *** Base 6 to Base 10, Fast Method ---------------------------------- To convert 1524 (base 6) to base 10 with the fast method, we start with the leftmost digit and move right; at each step we multiply by 6 and add the next base-6 digit, until we have used all the digits. For the conversion of 1524 (base 6) to base 10, the process is as follows: base-6 equivalent of action base-10 value this base-10 value ----------------------------------------------------------------- get first base-6 digit (1) 1 1 multiply by 6 6 10 add next digit (5) 11 15 multiply by 6 66 150 add next digit (2) 68 152 multiply by 6 408 1520 add last digit (4) 412 1524 In this process, we find the base-10 equivalent of our base-6 number by "building" the base-6 number from left to right. We start with the leftmost base-6 digit, 1; when we multiply by 6, we get the base- 10 equivalent of the base-6 number 10, and when we then add the next base-6 digit (5), we get the base-10 equivalent of the base-6 number 15; when we again multiply by 6 and add the next base-6 digit (2), we get the base-10 equivalents of the base-6 numbers 150 and then 152; and, finally, when we multiply again by 6 and then add the last base-6 digit (4), we get the base-10 equivalents of the base-6 numbers 1520 and finally 1524. ***************************** Now let's use exactly the same method to convert 1524 base 6 to base 8 instead of to base 10. We can copy the first column of the table above exactly as it is, because the actions we take to perform the conversion are exactly the same. But the results in the second column will be different, because now all the arithmetic we are doing is in the new base, 8. base-6 equivalent of action base-8 value this base-8 value ----------------------------------------------------------------- get first base-6 digit (1) 1 1 multiply by 6 6 10 add next digit (5) 13 15 multiply by 6 102 150 add next digit (2) 104 152 multiply by 6 630 1520 add last digit (4) 634 1524 We can verify this result by converting it to base 10, again using exactly the same process. The result we get should of course be the original number, 412. This time, since we are converting to base 10, the arithmetic is in our familiar base 10: base-8 equivalent of action base-10 value this base-10 value ----------------------------------------------------------------- get first base-8 digit (6) 6 6 multiply by 8 48 60 add next digit (3) 51 63 multiply by 8 408 630 add next digit (4) 412 634 We could also convert the base-6 number 1524 directly to base 4 using the same method. This time the conversion is a bit more difficult, because some of the base-6 digits are 2-digit numbers in base 4 (4 base 6 is "10" base 4; 5 base 6 is "11" base 4). And the "6" we have to multiply by each time is "12" in base 4. So the conversion looks like this -- with all the arithmetic in base 4: base-6 equivalent of action base-4 value this base-4 value ----------------------------------------------------------------- get first base-6 digit (1) 1 1 multiply by "12" 12 10 add next digit (5 = "11") 23 15 multiply by "12" 1002 150 add next digit (2) 1010 152 multiply by "12" 12120 1520 add last digit (4 = "10") 12130 1524 And finally we can use the same process again (using base 10 arithmetic) to convert this result to base 10 and verify that we get the correct value of 412: base-4 equivalent of action base-10 value this base-10 value ----------------------------------------------------------------- get first base-4 digit (1) 1 1 multiply by 4 4 10 add next digit (2) 6 12 multiply by 4 24 120 add next digit (1) 25 121 multiply by 4 100 1210 add next digit (3) 103 1213 multiply by 4 412 12130 add next digit (0) 412 12130 I hope all this helps. Please write back if you have any further questions about any of this. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ Date: 03/02/2008 at 07:07:15 From: Shayan Subject: Thank you (conversion of any base to binary) Thank you very much for directing me to the right path. |
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