# Difference between terminating and repeating decimals

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

BvU
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2019 Award
You mean

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.

Mark44
Mentor
And to expand on what BvU said, the decimal representation of rational numbers is in general not unique. For example, the fraction 1/2 can be represented as 0.5 (terminating) or 0.4999... (repeating). Both forms represent exactly the same number.

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.

phyzguy
Also note that whether a fraction terminates or repeats depends on the base used. For example 1/3 in base 10 is 0.3333..., but 1/3 in base 12 is 0.4.

Mark44
Mentor
Also note that whether a fraction terminates or repeats depends on the base used. For example 1/3 in base 10 is 0.3333..., but 1/3 in base 12 is 0.4.
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.

mathman
A fraction (in lowest terms) with a denominator consisting only of powers of 2 and 5 will terminate, otherwise repeat. This is simply an artifact of base 10 for our numbers.

HallsofIvy
Homework Helper
In base three only fractions whose denominator is a power of 3 are terminating "decimals" (triminals?).

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

And to expand on what BvU said, the decimal representation of rational numbers is in general not unique. For example, the fraction 1/2 can be represented as 0.5 (terminating) or 0.4999... (repeating). Both forms represent exactly the same number.

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

Mark44
Mentor
How would you represent .5 minus .499... ? Is that done in two ways also?
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.