Converting Base Numbers Directly Without Using Base 10Date: 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/ |
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