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Q: What causes the difference in terminating decimals and repeating decimals?

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2/5 can be simply represented in the decimal system. It is 1 tenth of 20/5, ie 0.4 . But 1/3 is a bit more tricky, because if you say it is 1 tenth of 10/3 you get 3 and 1 left over to be divided by 3 so you've got nowhere. In the decimal system you can only represent 1/3 by 3/10 + 3/100 + 3/1000 etc forever. This is a geometric series, and the sum of this is (using the formula for a geometric series) just 0.3/(1-1/10) = 1/3 . That's a problem of numbers in "base 10". In base 3, one third is simply 0.1 . A number with repeating decimals (or groups of repeats) can always be turned back to a fraction by summing the series.

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If the denominator in the simplest form of a rational number has any prime factor other that 2 or 5 then it will not divide the numerator without remainder. This results in a repeating decimal.

When you divide by a divisor q, the remainders can only be integers that are smaller than q. If the remainder is 0 then the decimal is terminating. Otherwise, it can only take the values 1, 2, 3, ... ,(q-1). So, after at most q-1 different remainders you must have a remainder which has appeared before. That is where the long division algorithm loops back into an earlier pattern = repeating sequence.

When you divide by a divisor q, the remainders can only be integers that are smaller than q. If the remainder is 0 then the decimal is terminating. Otherwise, it can only take the values 1, 2, 3, ... ,(q-1). So, after at most q-1 different remainders you must have a remainder which has appeared before. That is where the long division algorithm loops back into an earlier pattern = repeating sequence.

the difference is nothing :)

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EDL

the presence of minerals such as sodium cholride,magnesium etc causes the difference in ocean water ....

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