Converting Bases Without Going through Base 10Date: 09/08/2002 at 12:02:31 From: Sarah Naegele Subject: Base conversions Is there a way to convert between bases without going through base 10? Date: 09/08/2002 at 16:14:42 From: Doctor Jerry Subject: Re: Base conversions Hi Sarah, Yes, but it's a bit awkward. To keep the discussion simple, I'll assume that the number in question is an integer. Suppose x is a number in base b and we want to convert it to a number in base a. So, we originally have x = p_0 + p_1*b + p_2*b^2 + ... + p_n*b^n, where p_0 means p sub 0 and so on. The integers p_0, p_1,...,p_n are chosen from 0,1,...,b-1. This means that the base b representation of x is p_n p_{n-1}...p_1 p_0. We want to express x as x = q_0 + q_1*a + q_2*a^2 + ... + q_m*a^m, where the integers q_0, q_1,...,q_m are chosen from 0,1,...,a-1. We know that q_0 + q_1*a + q_2*a^2 + ... + q_m*a^m = p_0 + p_1*b + p_2*b^2 + ... + p_n*b^n We divide both sides by a (doing this in, say, base b arithmetic). The left side will be q_0/a + (q_1 + q_2*a^1 + ... + q_m*a^{m-1}). The right side (on which we will do the arithmetic in base b) will be, after division, a quotient Q_1 and a remainder f_1. The remainder f_1 is q_0. We repeat this process with the quotient, thus peeling off the "digits" one by one. Suppose we have 102 in base 5 (this is 27 in base 10) and want to convert it to base 8 (base 10), that is, to base 13 (base 5). We divide 102 by 13 and find quotient 3 and remainder 3. So, we're finished (in this particular case) and we find 33 (which is 27 in base 8). - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ Date: 09/09/2002 at 12:34:51 From: Sarah Naegele Subject: Thank you (base conversions) This is exactly what I wanted to know and I followed it completely! Thank you so much for your time! - Sarah |
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