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As long as the numbers are exactly representable (fractions whose
denominator is a power of 2), and intermediate results do not require
rounding, then floating-point arithmetic is *exact*. It is easy
to predict how many digits are needed for the results of arithmetic
operations:

- addition and subtraction of two
`n`-digit values with the*same*exponent require at most

digits, but when the exponents differ, many more digits may be needed;`n`+ 1 - multiplication of two
`n`-digit values requires exactly 2`n`digits; - although integer division produces a quotient and a remainder of
no more than
`n`-digits, floating-point remainder and square root may require an unbounded number of digits, and the quotient can need many more digits than can be stored.

Whenever a result requires more than `n` digits, rounding
is needed.