Question

What is the calculator display for 1/3 and is the exact decimal equivalent?

Answer 1

Calculator display = 0.33333333

Is this the exact decimal equivalent? No

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Step-by-step explanation:

1/3 = 0.33333333.... where the 3's go on forever. The calculator of course can't display infinitely many 3's. At some point, it will stop even if they theoretically go on forever.

We can prove we have this pattern by using long division.

To shorten things, we write
`0.\overline{3}` to tell the reader "the 3s go on forever". The horizontal bar tells us which digit(s) repeats forever. This is the exact decimal version.

The calculator can only display a finite number of digits, so it is impossible to have the approximate calculator display be the same as the exact decimal equivalent.

Some other examples:

`0.\overline{12} = 0.12121212\ldots\\\\0.\overline{687} = 0.687687687\ldots\\\\0.1\overline{6} = 0.166666\ldots\\\\0.00\overline{83} = 0.0083838383\ldots`