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The BaseConversion Method: Why Does it Work? Consider first the binary case: When you divide a number by two, the remainder will be either a zero or a one. If the remainder is 0, then the baseten number must have been even (that is, a multiple of two), so there will be no ones, and therefore the rightmost digit will be "0". If the remainder is 1, then the baseten number must have been odd (that is, one more than a multiple of two), so there will be a "1" as the rightmost digit in the units column. Copyright © Elizabeth Stapel 20022011 All Rights Reserved
Now divide again by two. If the remainder is zero, then, after getting rid of any extra 1 (from being an odd number), the number that was left must be a multiple of four, so there won't be any 2 left over. Otherwise, there was a multiple of two left over, so there will be a 1 in the twos column. Continue in like manner. Each time you divide, you're asking "Does the original number contain a multiple of this power of two?", and the remainder is either telling you "yes" (with a "0") or "no" (with a "1"). This reasoning carries
through with other bases. If, say, you're converting to base eleven and
you divide by 11,
the remainder will tell you how many more than a multiple of eleven the
given number is. Since this remainder value must be between 0
(if the number is an exact multiple of eleven) and 10
(one less than a multiple of eleven), then the digits in a baseeleven
number will never contain one single solitary digit for "eleven".



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