Everyday math calculators

Exact answers and approximate ones

Some answers can be written down completely and some cannot. One third plus one sixth is exactly one half, and 0.375 is exactly 3/8. But 1/3 as a decimal never ends: 0.3333… goes on forever, so any decimal you write for it is a rounded version. The trouble starts when a rounded value is fed back into more arithmetic. Three times 0.333 is 0.999, not 1, and every further step can carry that small error along.

The calculators in this section avoid the problem by keeping numbers in an exact form for as long as possible. Fractions are held as a whole-number numerator and denominator, repeating decimals are written with the repeating block in parentheses (0.(3) for one third), and rounding works on the decimal digits you typed rather than on a binary approximation. Ordinary floating-point arithmetic, used by most software, works differently: it stores 0.1 and 0.2 in binary, where neither is exact, and adds them to 0.30000000000000004. Where a tool does approximate, as when it rounds a decimal to the nearest sixteenth, it shows the size of the error.

Fractions, decimals and percentages

The three forms are different spellings of the same value. A fraction shows a part of a whole directly, which is why recipes and inch rulers use halves, quarters and eighths. A decimal is easier to compare and to type into other tools. A percentage is a decimal scaled by 100, the usual way to state rates, discounts and shares.

To go from a fraction to a decimal, divide the top by the bottom; to get a percentage, multiply that decimal by 100. Going back, a decimal with three places sits over 1,000, and the fraction is then simplified by the greatest common factor. A fraction in lowest terms gives a decimal that stops when its denominator has no prime factors other than 2 and 5; all others repeat.

A ratio is a close relative: 3:2 splits a whole into 5 parts, so the two shares are 3/5 (60%) and 2/5 (40%). Keeping track of which whole a fraction or percentage refers to prevents most mistakes with any of these forms.