Comparing Numbers in Different BasesDate: 07/28/99 at 00:40:19 From: Agnes Morrison Subject: Bases other than 10 If I am given a number in base 4 and asked whether it is bigger or smaller than another number in another base, should I always convert both to base 10 to compare, or is there an easier way? Thank you. Date: 07/28/99 at 12:41:18 From: Doctor Rick Subject: Re: Bases other than 10 Hi, Agnes. If you know how to convert a number in the other base to base 4 without converting it first to base 10, that _might_ be faster; you can compare any two numbers in the _same_ base the same way you compare numbers in base 10. But you'd have to know your base-4 multiplication tables really well for that to be easy. I would stick with converting both numbers to base 10. In some cases you might not have to do the whole conversion. Let's say I need to compare 331 (base 4) with 304 (base 5). I know that 331 (base 4) < 1000 (base 4) = 64 (base 10) I also know that 304 (base 5) > 300 (base 5) = 3*25 = 75 (base 10) Putting these facts together, I know that 331 (base 4) < 64 (base 10) < 75 (base 10) < 304 (base 5) so 331 (base 4) < 304 (base 5) Also, I don't have to do _any_ conversion to know that 331 (base 5) > 304 (base 4) Why? Because I know that 331 (base 5) > 331 (base 4) Any numeral in one base is always greater than the same numeral in a smaller base. If this isn't clear to you, write out the numbers in expanded form: 331 (base 5) = 3*25 + 3*5 + 1 331 (base 4) = 3*16 + 3*4 + 1 Once you see this, it's obvious that 331 (base 4) > 304 (base 4) Putting the facts together, I know that 331 (base 5) > 331 (base 4) > 304 (base 4) so 331 (base 5) > 304 (base 4) Does that help? These "shortcut" observations will help sometimes, but converting both numbers to base 10 will always give you the answer. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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