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BvU

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Is a number with a terminating decimal in a different category of number (in the way "real" or "natural" are categories) from a number with a repeating decimal at the end ?

I would say no, both are rational numbers.

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Mark44

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Note that I am not saying that 0.5000 is the same as 0.4999. The ellipsis (...) that I included for 0.4999... signifies that the representation continues in the same pattern.

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phyzguy

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Mark44

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And in base 3, 1/3 is 0.1.

This might seem arcane to some, but it has ramifications in how computers do calculations. We, as humans, are very comfortable with decimal (base-10) fractions, especially in situations involving money, such as $5.53. Many decimal fractions such as 0.1, 0.2 and many others have termination representations in base 10, but have infinitely repeating forms in binary (base 2), which is predominantly used in computers for real number calculations. As fractions 1/5 + 1/5 + 1/5 + 1/5 + 1/5 = 1, but if we add .2 + .2 + .2 + .2 + .2 on a computer, the result is slightly different from 1.

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mathman

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HallsofIvy

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In base six only fractions whose denominator is a product of powers of 3 and 2 are terminating "decimals".

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How would you represent .5 minus .499... ? Is that done in two ways also?

Note that I am not saying that 0.5000 is the same as 0.4999. The ellipsis (...) that I included for 0.4999... signifies that the representation continues in the same pattern.

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Mark44

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It's very simple .5 - .499... = 0, exactly. There are not two ways to represent zero, unless you want to consider .000... as somehow different from just plain 0.How would you represent .5 minus .499... ? Is that done in two ways also?

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