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No, the sum of a repeating decimal and a terminating decimal is never a terminating decimal.

Q: Does the sum of a repeating decimal and a terminating decimal equal a terminating decimal?

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Simply because the solution to your sum produces a repeating decimal. Just as 22/7 (The value of Pi as a fraction) produces the repeating decimal 3.142857

To answer that question we should first talk about why any non-termination decimal number is equal to whatever it is. And to talk about that, we should first talk about the value of ordinary terminating decimals. Consider a terminating decimal, say 0.314. This decimal represents the sum of the fractions 3/10 + 1/100 + 4/1000; and longer (but still terminating) decimals can be computed in a similar way. But how do we decide what value a non-terminating decimal represents, say 0.314159265458979... and so on with a never-ending sequence of digits? By analogy, it should be equal to 3/10 + 1/100 + 4/1000 + 1/10,000 + ... and so on; but how can we figure out what such a never-ending sum adds up to? Well, one way of looking at it is as follows: Whatever value the decimal has, we know that (say)0.314 is off by no more than 0.001, since 0.314159... - 0.314 = 0.000159..., and 0.000159... is clearly < 0.001. Likewise, 0.3141 is off by no more than 0.0001, and 0.31415 is off by no more than 0.00001, and so on. In other words, the sequence of (terminating) decimals, 0.3, 0.31, 0.314, 0.3141, 0.31415, etc. gives us a list of better and better approximations to the ultimate value of the non-terminating decimal; and in fact by taking enough decimal places, the error in the approximation can be made as small as you like. If you've studied calculus, you may recognize this sort of discussion--it means that the value of the non-terminating decimal acts like the limit of the sequence of terminating decimals. In fact, it just *is* the limit of the sequence. So mathematicians have chosen to define the value of a non-terminating decimal as the limit of the sequence of approximations. Now we can talk about the specific case of 0.9 repeating: It turns out that the limit of the sequence 0.9, 0.99, 0.999, ... is just equal to 1, exactly (which should not be too hard to convince yourself of) and therefore the value of the non-terminating decimal 0.9 repeating is, by definition, equal to 1.

111011000 (decimal 472). The sum is 257+215.

Expressed in decimal, the sum of the numbers 1 to 8192 is 33558528 - expressed in binary, this number is equal to 10000000000001000000000000.

Generally speaking, with the exception of trailing zeroes, the number of decimal places in the answer of a multiplication sum is equal to the sum of the number of decimal places in the question. Therefore, 4.97 x 3.456 contains five decimal places in total, therefore the answer contains five decimal places. In this instance, 4.97 x 3.456 = 17.17632.

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To sum this answer up you half to turn the fraction into a decimal and if it ends that is terminating but if it keeps going it is called a repeating decimal EXAMPLES Terminating- 5/10=.5 Repeating- 1/3=.3333 (bar notation over the 3)

Simply because the solution to your sum produces a repeating decimal. Just as 22/7 (The value of Pi as a fraction) produces the repeating decimal 3.142857

None of the items in the list.

Not necessarily. 1/3 = 0.333... 1/6 = 0.166... Their sum is 1/2 or 0.5 certainly not

Infinity.

To answer that question we should first talk about why any non-termination decimal number is equal to whatever it is. And to talk about that, we should first talk about the value of ordinary terminating decimals. Consider a terminating decimal, say 0.314. This decimal represents the sum of the fractions 3/10 + 1/100 + 4/1000; and longer (but still terminating) decimals can be computed in a similar way. But how do we decide what value a non-terminating decimal represents, say 0.314159265458979... and so on with a never-ending sequence of digits? By analogy, it should be equal to 3/10 + 1/100 + 4/1000 + 1/10,000 + ... and so on; but how can we figure out what such a never-ending sum adds up to? Well, one way of looking at it is as follows: Whatever value the decimal has, we know that (say)0.314 is off by no more than 0.001, since 0.314159... - 0.314 = 0.000159..., and 0.000159... is clearly < 0.001. Likewise, 0.3141 is off by no more than 0.0001, and 0.31415 is off by no more than 0.00001, and so on. In other words, the sequence of (terminating) decimals, 0.3, 0.31, 0.314, 0.3141, 0.31415, etc. gives us a list of better and better approximations to the ultimate value of the non-terminating decimal; and in fact by taking enough decimal places, the error in the approximation can be made as small as you like. If you've studied calculus, you may recognize this sort of discussion--it means that the value of the non-terminating decimal acts like the limit of the sequence of terminating decimals. In fact, it just *is* the limit of the sequence. So mathematicians have chosen to define the value of a non-terminating decimal as the limit of the sequence of approximations. Now we can talk about the specific case of 0.9 repeating: It turns out that the limit of the sequence 0.9, 0.99, 0.999, ... is just equal to 1, exactly (which should not be too hard to convince yourself of) and therefore the value of the non-terminating decimal 0.9 repeating is, by definition, equal to 1.

111011000 (decimal 472). The sum is 257+215.

Expressed in decimal, the sum of the numbers 1 to 8192 is 33558528 - expressed in binary, this number is equal to 10000000000001000000000000.

For a number to be rational you need to be able to write it as a fraction. To answer your question, it must repeat as a decimal or else terminate which can be thought of as repeating zeroes. Further, every repeating decimal can be written as a fraction and you can find the fraction by using the formula for the sum of an infinite geometric series.

Expressed as a decimal, 7 3/100 is equal to 7.03.

672.72 repeating

Generally speaking, with the exception of trailing zeroes, the number of decimal places in the answer of a multiplication sum is equal to the sum of the number of decimal places in the question. Therefore, 4.97 x 3.456 contains five decimal places in total, therefore the answer contains five decimal places. In this instance, 4.97 x 3.456 = 17.17632.