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Our terminating decimal calculator will teach you how to find the decimal representation of a number, detect the possible presence of repeating decimals, and much more. Keep reading to find out:

  • What is a terminating decimal;
  • What are repeating decimals;
  • How to calculate the decimal representation of a fraction;
  • When you should stop calculating the repeating decimals in a representation;
  • How to convert from repeating decimals to fractions; and more.

Additionally, we have prepared several examples of all the math explained in the text.
We promise it will be interesting (and surprisingly easy!). What are you waiting for?

Decimal representations: what is a terminating decimal and a repeating decimal

The number of numbers is quite big: even natural numbers are infinite, and they are actually the smallest infinite set in math! This number only grows when we deal with real numbers. Real numbers allow for infinitely small variations (compare them to integers, where the smallest variation is 11: 121\rightarrow 2). Between two adjacent integers, we can find infinitely many real numbers: this is when we need to introduce the decimal representation of a number:

14.9138362171...14.9138362171...

Before the decimal separator, we meet the integer part. After the separator, we meet the decimal part.

Among real numbers, we can identify two subsets:

  • Rational numbers; and
  • Irrational numbers.

The difference between the two types is that the format can be represented as a ratio between two integers, while the latter cannot. Let's see a couple of examples to make things clear:

1215=0.8\frac{12}{15} = 0.8

and:

663208=3.1875\frac{663}{208}=3.1875

are rational numbers. On the other hand, Pi and the square roots of 22 are irrational numbers:

π=3.141592653589793...\pi = 3.1415926535 89793...

And:

2=1.41213562373090...\sqrt{2} = 1.41213562373090...

These numbers have an infinite amount of digits. Is this the condition that numbers have to satisfy to be irrational? No! In fact, we can find rational numbers with infinitely many digits:

103=3.33333333333333...\frac{10}{3}=3.33333333333333...

Without going too deep, we can see the difference immediately: in a rational number with an infinite number of digits, we are forced to find a repeating pattern. In the example above, the digit 33 repeats an infinite number of times.

🙋 To write (or store) the decimal representation of an irrational number, you need an infinite amount of information. Conversely, a rational number, even with infinite decimal representation, can be "transmitted" in the finite message, for example, "one point three repeated infinite times".

How to calculate terminating decimals and repeating decimals

To calculate the terminating decimals and repeating decimals from a fraction, you must calculate the decimal representation of the result of the fraction. To do so, we will use the long division procedure. We talked in detail about it in our long division calculator!

Let's get started. Take a fraction — any fraction. Above the line, we find the numerator, below the denominator. These two quantities correspond to the dividend and divisor:

dividenddivisor=dividend/divisor\frac{\mathrm{dividend}}{\mathrm{divisor}} = \mathrm{dividend}/\mathrm{divisor}

🙋 If you need the result of such division directly, you can visit our fraction to decimal converter: specify the number of decimal digits, and let us do the math; in the terminating decimals calculator, you will learn how to calculate all the decimals!

To compute the decimal representation, perform these steps:

  • Check if the first digit of the dividend contains the divisor.

    • If it does, write down the result of the integer division, and keep the remainder of the division in mind.

    • If not, write down 00 or keep an empty position at the beginning of the result. Note the remainder.

  • "Copy" the remainder in front of the next digit of the dividend: you will obtain a number with, possibly, multiple digits. Worry not!

  • Compute the integer division between the number obtained in the previous step and the divisor. Note the result, and save the remainder.

  • Repeat the last two steps until you find one of the following two situations:

    • The dividend is 00: you've found a terminating decimal; or
    • You've already met the dividend (after the decimal separator): you've found a repeating decimal.

Let's follow the steps of the long division with an example. Take 1717 as dividend and 1414 as divisor. Write them down like this:

1417.00000000014|17.000000000

Consider the first digit of the dividend:

1417.000014|\textcolor{blue}{1}7.0000

And compute:

1/14=0 rem. 11/14=0\ \mathrm{rem.}\ 1

17/14=017/14=0 will become the first digit of our result, while we carry 11 to the next digit. If the operations are too tricky, you can use our remainder calculator. We then perform the division between the number obtained by carrying 11 and the divisor:

14117.00000000017/14=1 rem. 3\begin{split} &14|1^{\textcolor{blue}{1}}\textcolor{blue}{7}.000000000\\ &17/14 = 1\ \mathrm{rem.}\ 3 \end{split}

Update the result and append the decimal separator: 17/14=0117/14=01. Now carry over that 33, and proceed with the divisions:

1417.300000000030/14=2 rem. 2\begin{split} &14|17.^{\textcolor{blue}{3}}\textcolor{blue}{0}00000000\\ &30/14 = 2\ \mathrm{rem.}\ 2 \end{split}

Update the result: 17/14=01.217/14=01.2 and proceed.

1417.020000000020/14=1 rem. 617/14=01.211417.006000000060/14=4 rem. 417/14=01.2141417.000400000040/14=2 rem. 1217/14=01.21421417.00001200000120/14=8 rem. 817/14=01.214281417.000008000080/14=5 rem. 1017/14=01.2142851417.00000010000100/14=7 rem. 217/14=01.21428571417.000000020020/14=\begin{split} &14|17.0^{\textcolor{blue}{2}}\textcolor{blue}{0}0000000\\ &20/14 = 1\ \mathrm{rem.}\ 6\\ &--------\\ &17/14=01.21\\ &--------\\[-1.2em] &--------\\ &14|17.00^{\textcolor{blue}{6}}\textcolor{blue}{0}000000\\ &60/14 = 4\ \mathrm{rem.}\ 4\\ &--------\\ &17/14=01.214\\ &--------\\[-1.2em] &--------\\ &14|17.000^{\textcolor{blue}{4}}\textcolor{blue}{0}00000\\ &40/14 = 2\ \mathrm{rem.}\ 12\\ &--------\\ &17/14=01.2142\\ &--------\\[-1.2em] &--------\\ &14|17.0000^{\textcolor{blue}{12}}\textcolor{blue}{0}0000\\ &120/14 = 8\ \mathrm{rem.}\ 8\\ &--------\\ &17/14=01.21428\\ &--------\\[-1.2em] &--------\\ &14|17.00000^{\textcolor{blue}{8}}\textcolor{blue}{0}000\\ &80/14 = 5\ \mathrm{rem.}\ 10\\ &--------\\ &17/14=01.214285\\ &--------\\[-1.2em] &--------\\ &14|17.000000^{\textcolor{blue}{10}}\textcolor{blue}{0}00\\ &100/14 = 7\ \mathrm{rem.}\ 2\\ &--------\\ &17/14=01.2142857\\ &--------\\[-1.2em] &--------\\ &14|17.0000000^{\textcolor{blue}{2}}\textcolor{blue}{0}0\\ &20/14 = \\ \end{split}

Wait a minute! We've already computed this division: the result is 11, with remainder 66. This would throw us in a repeating pattern with the following digits: 142857142857. As you can see, the 22 we've found at the beginning of the decimal representation doesn't make it in the repeating pattern!

We can stop our calculations for the repeating decimal part here. The result of the division of 1717 by 1414 is:

17/14=1.2142857142857...17/14 = 1.2142857142857...

We can write the same result in a more succinct way, highlighting the repeating sequence:

17/14=1.214285717/14 = 1.2\overline{142857}

We calculated the non-periodic part as 22, and we calculated the repeating decimals as 142857142857. 11 is the result of the integer division.

🙋 Fun fact: the number of digits in the repeating pattern can't be greater than the divisor (after you make it an integer by multiplying by the appropriate power of 1010).

Why? Because all the possible remainders are contained in the sequence going from 11 to the divisor, then repeating. In the limited case of meeting all those digits, you will necessarily find a remainder you've already met!

Calculate from repeating decimals to fraction

Now that you know how to calculate the terminating decimals and the repeating decimals from a fraction, we can teach you how to do the opposite and calculate from the repeating decimals to the fraction that generated that result. We can identify three cases:

  • Result with terminating decimals (only non-periodic part);
  • Result with only repeating decimals; and
  • Mixed result (both non-periodic and periodic parts).

The first case is the easiest to analyze. To find the generating fraction:

  1. Make the result integer by multiplying by the appropriate power of 1010. E.g., 0.23×100=230.23 × 100 =23 (We multiply by 100100 because there are 22 decimal places). This will be the numerator of the fraction.
  2. The power of 1010 that you used in the previous step (100100) will be the denominator of the fraction.
  3. Find the greatest common divisor of the two results, and divide them both by this quantity.

The result is the most reduced fraction that gives you the original decimal representation.

The calculations become slightly more complex in the presence of repeating decimals. Consider the real number 3.183.\overline{18}. Divide the integer and the decimal part:

3.18=3+0.181818...3.\overline{18} = 3+0.181818...

Consider only the decimal part, and call it xx. Multiply it by the power of 1010 with exponent equal to the length of the repeating decimal part:

x=0.181818...102x=100x=18.181818...\begin{split} x &= 0.181818...\\ 10^2 x &= 100x =18.181818...\\ \end{split}

Now subtract the first number from the second one:

100x=18.181818...x=  0.181818...99x=18.000000\begin{align*} 100x&=18.181818...\\ x&=\ \ 0.181818...\\ 99x&=18.000000 \end{align*}

Hence, we can write xx as:

x=1899x=\frac{18}{99}

Now think again when we split the original number: 3.18=3+0.18=3+x3.\overline{18} = 3+0.\overline{18} = 3+x. Substitute xx with the last result we've found, and sum the integer part and the fraction. You can use our adding fractions calculator if you don't want to spend time with the math!

3.18=3+1899=993+1899=31599=3511\begin{split} 3.\overline{18}& = 3+\frac{18}{99} = \frac{99\cdot3+18}{99}\\[1em] &=\frac{315}{99} = \frac{35}{11} \end{split}

What happens if there is a non-periodic part preceding the repeating decimals? You can add a simple step and reduce the problem to a mixture of the ones we've already met. Consider the number 1.231451.23\overline{145}. We want to separate a strictly periodic and a terminating part. To do so, copy backward (from right to left) the digits of the periodic part over the digits of the non-periodic part (in the same right-to-left fashion) and create a periodic decimal number with the same digits of the period of the original one, but (possibly) a different starting point.

(...).231450.251450.45145\begin{split} (...).&23\overline{145}\\ 0.&2\textcolor{red}{\bold{5}}\overline{145}\\ 0.&\textcolor{red}{\bold{4}}5\overline{145} \end{split}

🙋 Did you see how we replaced the non-periodic part? We started from the last digit of the period (55); we then copied the second-to-last digit (44). We changed the periodic part from 145145 to 451451.

Notice how the digits of the periodic part are the same, even though we changed the starting point: 0.4510.\overline{451} is the decimal we will use to compute the fraction. To find the terminating decimal, subtract the original number and the last result:

1.23145145...0.45145145...=0.78\begin{split} 1.&23145145...-0.45145145...\\ &=0.78 \end{split}

Let's find the fractions corresponding to both numbers. For the terminating decimal we have:

0.78=78100=39500.78 = \frac{78}{100} = \frac{39}{50}

For the repeating decimals, we have:

x=   0.451451451...1000x=451.451451451...\begin{split} x =\: \ \ \ 0.&451451451...\\ 1000x = 451.&451451451...\\ \end{split}

Thus:

1000x=451.451451451...x=   0.451451451...=999x=451.0x=451999\begin{split} 1000x = 451.&451451451...-\\ x = \: \ \ \ 0.&451451451...=\\ 999x=451.&0\\ x=\frac{451}{999}& \end{split}

We can write the original number as the sum of these two fractions:

1.23145=3950+451999=61511499501.23\overline{145} = \frac{39}{50}+\frac{451}{999}= \frac{61511}{49950}

Not the best-looking fraction, but the right one, nevertheless!

🙋 Use our decimal to fraction calculator to see all the passages we explained above!

Another example of terminating decimals and repeating decimals calculations

Let's try again to find the repeating and terminating decimals in an example. Take 13.713.7 as numerator, and 4242 as denominator. Set up the division, and begin.

4213.700000000000001/42=0 rem. 113.7/42=042113.7000000000000013/42=0 rem. 1313.7/42=00.4213.1370000000000000137/42=3 rem. 1113.7/42=00.34213.7110000000000000110/42=2 rem. 2613.7/42=00.324213.7026000000000000260/42=6 rem. 813.7/42=00.3264213.70080000000000080/42=1 rem. 3813.7/42=00.32614213.7000380000000000380/42=9 rem. 213.7/42=00.326194213.70000200000000020/42=0 rem. 2013.7/42=00.3261904213.7000002000000000200/42=4 rem. 3213.7/42=00.32619044213.7000000320000000320/42=7 rem. 2613.7/42=00.32619047\begin{split} &42|\textcolor{blue}{1}3.70000000000000\\ &1/42 = 0\ \mathrm{rem.}\ 1\\ &--------\\ &13.7/42 = 0\\ &--------\\[-1.2em] &--------\\ &42|1^{\textcolor{blue}{1}}\textcolor{blue}{3}.70000000000000\\ &13/42 = 0\ \mathrm{rem.}\ 13\\ &--------\\ &13.7/42 = 00.\\ &--------\\[-1.2em] &--------\\ &42|13.^{\textcolor{blue}{13}}\textcolor{blue}{7}0000000000000\\ &137/42 = 3\ \mathrm{rem.}\ 11\\ &--------\\ &13.7/42 = 00.3\\ &--------\\[-1.2em] &--------\\ &42|13.7^{\textcolor{blue}{11}}\textcolor{blue}{0}000000000000\\ &110/42 = 2\ \mathrm{rem.}\ 26\\ &--------\\ &13.7/42 = 00.32\\ &--------\\[-1.2em] &--------\\ &42|13.70^{\textcolor{blue}{26}}\textcolor{blue}{0}00000000000\\ &260/42 = 6\ \mathrm{rem.}\ 8\\ &--------\\ &13.7/42 = 00.326\\ &--------\\[-1.2em] &--------\\ &42|13.700^{\textcolor{blue}{8}}\textcolor{blue}{0}0000000000\\ &80/42 = 1\ \mathrm{rem.}\ 38\\ &--------\\ &13.7/42 = 00.3261\\ &--------\\[-1.2em] &--------\\ &42|13.7000^{\textcolor{blue}{38}}\textcolor{blue}{0}000000000\\ &380/42 = 9\ \mathrm{rem.}\ 2\\ &--------\\ &13.7/42 = 00.32619\\ &--------\\[-1.2em] &--------\\ &42|13.70000^{\textcolor{blue}{2}}\textcolor{blue}{0}00000000\\ &20/42 = 0\ \mathrm{rem.}\ 20\\ &--------\\ &13.7/42 = 00.326190\\ &--------\\[-1.2em] &--------\\ &42|13.700000^{\textcolor{blue}{20}}\textcolor{blue}{0}0000000\\ &200/42 = 4\ \mathrm{rem.}\ 32\\ &--------\\ &13.7/42 = 00.3261904\\ &--------\\[-1.2em] &--------\\ &42|13.7000000^{\textcolor{blue}{32}}\textcolor{blue}{0}000000\\ &320/42 = 7\ \mathrm{rem.}\ 26\\ &--------\\ &13.7/42 = 00.32619047 \end{split}

We've already met the remainder 2626 at the fourth step of our calculations: this means that starting from the fourth reaching up to the last step, we've found our set of repeating digits. We can stop computing divisions and write down our result as:

13.742=0.32619047\frac{13.7}{42} = 0.32\overline{619047}

FAQ

What are the repeating decimals in a number?

The repeating decimals (or recurring decimals) in a number are a set of digits that repeat cyclically in the decimal part of a real number. Real numbers with repeating decimals are always rational; thus, we can express them with the ratio of two integers. For example:

  • 10/3 = 3.333333... = 3.3;
  • 131/88 = 1.4863636363... = 1.48 63; and
  • 4679/1665 = 2.8102102... = 2.8 102.

How do I calculate the repeating decimals from a fraction?

To calculate the repeating decimals from a fraction, follow these easy steps:

  1. Divide the first digit of the numerator by the denominator; note the quotient and memorize the remainder.

  2. Prepend the remainder to the next digit, and perform the division by the denominator again.

  3. Note the decimal separator when you meet it.

  4. Repeat steps 1 and 2 until:

    • The new divisor is a number you've already met in the previous divisions; or
    • The remainder of the division is 0.

The digits between the first and second occurrence of the remainder repeat in the decimal part of the result.

Are numbers with repeating decimals infinite?

Not exactly. Numbers with repeating decimals are not infinite, but the length of their decimal representation is infinite: this means that their decimal part has infinitely many digits. However, we can represent them using a finite number of digits by specifying the periodicity of the repeating part. To do so, we can write a bar over the numbers or enclose them in brackets.

What is the decimal representation of 12/55?

0.218181818.... To calculate this result, follow these steps:

  1. Write 12 as 12.00000.

  2. Divide 12 by 55. The result is 0 with the remainder 12.

  3. Copy the decimal point in the result.

  4. Prepend 12 to 0 and find 120. Divide 120 by 55: the result is 2 with the remainder 10.

  5. Prepend 10 to 0, take the result 100, and divide by 55. The result is 1 with remainder 45.

  6. The next number we divide will be 450. 450/55 = 8 with remainder 10.

  7. We've already met this remainder: the digits 1 and 8 repeat infinitely.

The result is: 12/55 = 0.2181818...

Davide Borchia
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