Fractions

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A fraction is one whole number over another, like 1/3. Solve keeps such a quotient exact rather than turning it into a decimal straight away, so a chain of fractions adds up to the answer it should instead of drifting by a tiny amount.

A quotient of two whole numbers is kept as an exact fraction, so a chain of fractions adds up the way it should rather than drifting the way the underlying doubles do. 1/49 * 49 is exactly 1, not 0.9999999999999999.

1/3 + 1/3 + 1/3 // 1
2/7 * 14 // 4
1/49 * 49 // 1

A fraction is shown as a decimal by default, so a result still reads as a number. Ask for as fraction to see it as a fraction, reduced to lowest terms, and as decimal for the decimal.

1/3 as fraction // 1/3
10/4 as fraction // 5/2
(1/3 + 1/7) as fraction // 10/21

A decimal written with a point is exact as well (see decimals), and the two meet cleanly. A decimal reads as the fraction it is, so a fraction plus a decimal is still an exact fraction, and a division of decimals that never ends, such as 0.1 / 3, is kept as the fraction 1/30, the same way 1/3 is kept.

(1/3 + 0.1) as fraction // 13/30
(0.1 / 3) as fraction // 1/30
0.1 / 3 * 3 == 0.1 // true

The same holds for a decimal on its own. as fraction writes it as the fraction its digits spell, reduced to lowest terms: 0.333333 is 333,333 millionths, and 3.14159 is 314,159 hundred-thousandths. It does not round a decimal onto a simpler fraction it happens to be close to, so 0.3333333 is 3333333/10000000 and not a third. A third is written 1/3, and is then exactly a third.

0.333333 as fraction // 333333/1000000
3.14159 as fraction // 314159/100000
0.3333333 as fraction // 3333333/10000000
0.125 as fraction // 1/8
$0.25 as fraction // 1/4

Transcendental work (sqrt, sin, a non-integer power) stays floating point, because its answers have no exact fraction to keep. Asked for as a fraction, such a value is given a close one instead, with a denominator of no more than a million, since there is no exact one to write:

sqrt(2) as fraction // 47321/33461