Fraction to Decimal Calculator

What the converter returns

A fraction is a division that has not been carried out yet. Three eighths means three divided by eight, and the decimal form is the answer to that division. This page does the division, keeps every digit that matters, and tells you whether the digits stop or repeat.

Enter the numerator and the denominator, and add a whole number first if you have a mixed number such as 2 1/4. The output gives the decimal, the number of decimal places for a terminating value, the length of the repeating block for a repeating value, the fraction in lowest terms, and the same value written as an improper fraction. There is a second box for the trip back: type a decimal and the page returns the fraction with the smallest denominator that fits the digits you typed.

The arithmetic is exact. A terminating decimal is printed with all of its digits, not a rounded version, so 1/64 comes out as 0.015625 and not 0.016. A repeating decimal is printed in bracket notation, where the digits inside the brackets repeat forever: 1/6 is 0.1(6), meaning 0.16666 and so on.

Both inputs have to be whole numbers, and the denominator cannot be zero. A zero denominator is not a number at all, because it asks how many zeros fit into the numerator, and no answer satisfies that.

Terminating decimals: when the division stops

A fraction turns into a terminating decimal when its denominator, after the fraction is in lowest terms, is built only from 2s and 5s. Those are the prime factors of 10, and a denominator made of them can always be scaled up to a power of 10 by multiplying the top and bottom by the missing factors.

Eight is 2 x 2 x 2, so 3/8 can be scaled by 5 x 5 x 5 to give 375/1000, which is 0.375. Twenty five is 5 x 5, so 1/25 scales by 4 to give 4/100, which is 0.04. Forty is 2 x 2 x 2 x 5, and the highest power of either prime is three, so 1/40 needs three decimal places: 0.025.

That gives a rule you can use without dividing. Factor the denominator, count the 2s and the 5s, and the number of decimal places is the larger of the two counts. A denominator of 8 has three 2s and no 5s, so three places. A denominator of 64 has six 2s, so six places, and 1/64 is 0.015625. A denominator of 125 has three 5s and no 2s, so three places, and 1/125 is 0.008.

The same rule explains the fractions that do not terminate. Add a single 3 to the denominator and the scaling cannot reach a power of 10, so the division keeps producing digits. Six is 2 x 3, and 1/6 never stops.

Repeating decimals and the bracket notation

When the denominator has a prime factor other than 2 or 5, the long division eventually produces a remainder it has seen before, and from that point the digits repeat in a cycle. The remainder is the fingerprint: with a denominator of 7 there are only six possible nonzero remainders, so within seven steps the division has to revisit one.

Writing the repeat with brackets is the compact form. One third is 0.(3). Five sixths is 0.8(3), which means the 8 does not repeat and the 3 does. One seventh is 0.(142857), a six digit block that runs forever. Twenty two sevenths is 3.(142857), the same block after the integer part.

Handwritten work usually puts a dot over the first and last digit of the repeat instead of brackets. Both notations say the same thing, and both are shorter than writing a dozen digits and a row of dots.

The repeating block can be long. One seventeenth has a period of 16 digits, 0.(0588235294117647), and 1/19 has a period of 18. The period always divides the number of possible remainders, which is at most one less than the part of the denominator left after the 2s and 5s are removed. A denominator of 13 can give a period of at most 12, and 1/13 delivers 6.

Long division by hand, step by step

The decimal form is not a different operation from division, it is the same one with the remainder carried down. Five sixteenths shows the whole method in four steps.

• 5 is smaller than 16, so the decimal starts at 0. and the first digit comes from 50
• 50 / 16 = 3, remainder 2, so the first digit is 3
• Bring down a zero: 20 / 16 = 1, remainder 4
• 40 / 16 = 2, remainder 8
• 80 / 16 = 5, remainder 0, so the decimal is 0.3125 and it stops

One seventh shows what happens when the remainder never reaches zero.

• 10 / 7 = 1, remainder 3
• 30 / 7 = 4, remainder 2
• 20 / 7 = 2, remainder 6
• 60 / 7 = 8, remainder 4
• 40 / 7 = 5, remainder 5
• 50 / 7 = 7, remainder 1

The next step is 10 / 7 again, which is where the page started. That repeat is the proof that the block 142857 cycles forever, and it is why the bracket notation is honest rather than lazy.

Two checks catch most hand calculation errors. The digits of a terminating answer should multiply back to the original fraction, so 0.3125 x 16 = 5. And a repeating answer should show the same remainder twice, which is the signal to stop writing digits and draw the line.

Common fractions in exact decimal form

The tables below are worth keeping open while you work, because the values do not change and the errors people make are usually a digit in the wrong place rather than a wrong method.

Now the fractions that repeat. The last two columns tell you where the repeat starts and how long it runs, which is the part that gets lost when someone writes 0.333 instead of 1/3.

Notice how much a small change of denominator moves the value. One seventh is 0.142857 and one eighth is 0.125, so a seventh is larger by about 0.0179. Two sevenths is 0.285714, which lands between a quarter and a third. Reading a table like this is faster than dividing, once you know the rule from the section above.

Decimal inches: drill bits, wrenches and calipers

In a workshop the fraction and the decimal are the same size written two ways, and the tools speak different languages. Drill bits are sold by number or by fraction, calipers and depth gauges read in decimal inches, and a tap chart lists both. Converting in your head is where sizes get ordered wrong.

The jumps are not evenly spaced, which is the trap. Going from 1/8 to 3/16 adds 0.0625 inch, while going from 3/16 to 1/4 adds the same 0.0625, but going from 15/64 to 1/4 adds 0.015625. A set of bits that looks evenly spaced by fraction is not evenly spaced by diameter.

Wrench sizes have the same problem. Three eighths is 0.375 inch, seven sixteenths is 0.4375 inch, and a 10 mm bolt head is 0.3937 inch, which sits between the two. That is why a 3/8 wrench feels loose on a 10 mm nut and a 10 mm wrench will not fit a 3/8 head at all.

When a drawing gives 0.375 and the shelf is labeled 3/8, the two are identical, and no rounding is involved. When a drawing gives 0.394, the nearest fraction is 25/64 at 0.390625, and the difference of 0.003 inch is about the thickness of a sheet of paper.

Mixed numbers, negatives and improper fractions

A mixed number is a whole part plus a fraction, and the conversion to decimal only adds one step: turn it into an improper fraction first, or divide the fraction part and add it to the whole. Three and five eighths is 3 plus 0.625, which is 3.625. Two and a quarter is 2.25. The whole part does not change the digits after the decimal point.

Negatives follow the same rule with the sign kept in front. Minus seven eighths is -0.875, and the decimal is the same size as the positive version. Mixed negative numbers need one extra thought: -2 1/4 means -2 plus -0.25, which is -2.25, not -1.75. A calculator that treats the whole part as always positive gets this wrong, which is one reason to enter the fraction as -9/4 if you are unsure.

Improper fractions are often easier than mixed numbers. Ten quarters is 2.5, because 10 / 4 = 2.5. Twenty two sevenths is 3.142857, which is a common approximation of pi and the reason 22/7 shows up in older engineering tables. The decimal form of an improper fraction keeps the whole part on the left of the point and the remainder as digits on the right.

Simplifying first saves work but does not change the answer. Sixteenths and thirty seconds are the same value when the numerator doubles with the denominator, so 6/32, 3/16 and 0.1875 are three names for one size. If a decimal has to match a fraction exactly, simplify first and compare the denominators.

Reading the answer backwards: decimal to fraction

The reverse trip is the same arithmetic in a different order. Take the digits after the decimal point, put them over a power of ten with as many zeros as there are digits, and reduce. The decimal 0.375 is 375/1000, and dividing both by 125 gives 3/8. The decimal 0.0625 is 625/10000, and dividing both by 625 gives 1/16.

Terminating decimals come back exactly. Repeating ones do not, because the typed digits are a truncation. Type 0.3333 and the page returns 3333/10000, which is close to 1/3 but not equal to it. Type 0.1666 and it returns 833/5000 rather than 1/6. That is not a bug, it is the honest reading of the digits you supplied, and it is why a measurement recorded to four decimals should be treated as a measurement rather than as an exact fraction.

The number of digits decides how close you can get. A decimal with two places can only land on fractions with denominators that divide 100, so 0.33 cannot become 1/3. With three places the candidates are fractions over 1000, which brings in 1/8 exactly and 1/3 only approximately. If you need the exact fraction back, keep the repeating block, or keep the original fraction and never convert it in the first place.

The NAEP mathematics framework makes the same distinction when it describes grade 8 students working with rational numbers written either as decimal fractions or as common fractions, and NIST Special Publication 811 sets the rules for writing measured values in decimal form. Both point at the same habit: know which form is exact, and say so when you hand the number to someone else.