1/7 as a Decimal

One seventh is 0.142857, and the loop never stops

One seventh is what you get when a whole is cut into seven equal parts and you take one of them. In decimal form that share is 0.142857, and the six digits keep coming back in the same order: 0.142857142857 and on without a stop. There is no final digit, and no amount of rounding makes the value exact.

The division behind the number is short to set up. Put 1 under the division bracket and 7 outside it, then work through the digits one at a time. Seven does not fit into 1, so the answer starts with 0 and a decimal point. Multiply the remainder by 10 and try again, which is the same move you make in any long division problem.

As a percent, one seventh is 14.2857 percent. That number is also endless, because it is the decimal multiplied by 100. A seventh of a dollar is about 14.3 cents, and a seventh of $7 is exactly $1, since seven equal shares of seven dollars leave one dollar each.

The fraction is small, but the decimal is not tidy. That gap between a simple fraction and a long decimal is the whole story of sevenths, and the rest of this page shows where the six digits come from and how to use them without losing money to rounding.

The long division, remainder by remainder

Long division turns 1/7 into digits without a calculator. Set up 1.000000 divided by 7. At each step, take the current remainder, multiply it by 10, divide by 7, write the digit, and keep what is left.

Read the digit column from top to bottom: 1, 4, 2, 8, 5, 7. The sixth step leaves a remainder of 1, which is the remainder you started with. Once a remainder comes back, every later step has to repeat, so the block 142857 runs forever.

The first step is the one people stumble on. Seven does not go into 1, so the digit above the line is 0 and the decimal point goes in before you continue. The 1 becomes 10, and 7 goes into 10 once with 3 left over. From there the remainders run 3, 2, 6, 4, 5 and back to 1.

You can stop as soon as a remainder repeats. For 1/7 that happens after six digits. For 1/3 the first remainder is 1 and it returns right away, so the block is a single 3. The rule is the same in both cases: the digits between the two visits to the same remainder form the repeating block.

Why the block is exactly six digits

There are only six possible nonzero remainders when you divide by 7: 1, 2, 3, 4, 5 and 6. A remainder of 0 would end the decimal, and dividing 1 by 7 never produces one. With six slots and no empty ones, the cycle has to close within six steps, and for 1/7 it uses all six.

A denominator that produces the longest possible block, one digit shorter than the denominator itself, is called a full reptend prime. Seven is the smallest one. Wolfram MathWorld's page on full reptend primes lists the sequence, and 7 sits at the front of it. The next few are 17, 19, 23 and 29, which is why 1/17 has a sixteen-digit block and 1/19 has an eighteen-digit block.

Not every prime behaves this way. One thirteenth has a block of six digits, 076923, which is shorter than the twelve digits a full reptend would give. The length always divides the number one less than the denominator, so for 7 the options are 1, 2, 3 and 6, and the actual length is 6.

The arithmetic behind the six digits is the number 999999. It factors as 3 x 3 x 3 x 7 x 11 x 13 x 37, and the factor 7 is what makes the six-digit block divide out cleanly. That fact becomes useful later, when the decimal has to go back to a fraction.

Every seventh is a rotation of 142857

Every seventh is the same six digits, started at a different point. Two sevenths is 0.285714, three sevenths is 0.428571, and the pattern continues through six sevenths. This is why the sevenths are easy to memorize as a group: learn 142857 once and rotate it.

The rotation shows up in multiplication too. 142857 x 2 = 285714, x 3 = 428571, x 4 = 571428, x 5 = 714285, and x 6 = 857142. Each product is a different starting point in the same loop, which is the signature of a cyclic number.

Percent values follow the decimal. One seventh is 14.29 percent, two sevenths is 28.57 percent, and six sevenths is 85.71 percent. The rounded percentages add to 99.99 rather than 100, and the missing 0.01 percent is the price of rounding each share separately.

Split bills are the everyday use. A $84 check split seven ways is $12 a person, because 84 / 7 = 12. A $100 check split seven ways is $14.29 a person, with a penny lost in the rounding and usually handed to the server.

What rounding the loop costs

Rounding a repeating decimal is unavoidable on paper and on a screen. Two places gives 0.14, three places gives 0.143, four places gives 0.1429, and six places gives 0.142857. Each extra digit moves the estimate closer, but none of them reaches the exact value.

The size of the error depends on the amount. On $1,000, one seventh is $142.857, which rounds to $142.86. Using 14 percent instead gives $140.00, short by $2.86. Using 14.29 percent gives $142.90, four cents over. The rounded percent is much closer, but it still misses by a few cents.

Scale the amount up and the gap grows. On $3,000, one seventh is $428.571. The one-decimal percent of 14.3 gives $429.00, which is 43 cents too high. The four-digit percent of 14.29 gives $428.70, which is 13 cents low. For a payroll or a tax line, those cents matter when they repeat across hundreds of rows.

The safe habit is to keep the full decimal until the last step. Multiply the amount by 1, divide by 7, and round only the final answer. Rounding the percent first and multiplying second bakes the error into every calculation that follows.

A calculator display adds one more layer. Many screens show twelve digits, 0.142857142857, and then stop. The screen has rounded for you, so copying that number into a spreadsheet carries a tiny error even though the display looks complete.

Turning 0.142857 back into 1/7

The repeating block can be turned back into the fraction with one subtraction. Call the decimal x. Multiply by 10 six times, which shifts the point past one full block, so 1,000,000x sits at 142857.142857 and the tail continues as before. Subtract the original x from both sides, and the repeating tails cancel.

What is left is 999,999x = 142,857. Divide both sides by 999,999 and the fraction is 142857/999999. Both numbers divide by 142,857, because 142857 x 7 = 999999, so the fraction reduces to 1/7.

The same trick works for any repeating decimal. For 0.333, ten times the value is 3.333, so nine times the value is 3 and the fraction is 1/3. For 0.1666, ten times the value is 1.666 and nine times the value is 1.5, which gives 1/6. The number of nines matches the length of the block, which is why a six-digit block produces 999999.

This is a good check on a calculator answer. If a tool reports 0.142857142857, run it through the method above with six digits and the result should land on 1/7. If it lands on something else, the tool rounded before the last step or the block was copied short.

Sevenths in weeks, recipes and miles

Sevenths turn up wherever a week, a mile or a cup gets divided. A week has seven days, so one seventh of a week is one day and three sevenths is three days. A month with 28 days is exactly four weeks, which makes one seventh of it four days.

In the kitchen, one seventh of a cup is 2.29 tablespoons, because a cup holds 16 tablespoons and 16 / 7 = 2.2857. That is close to 2 tablespoons plus a teaspoon. One seventh of a teaspoon is about 0.14 teaspoons, smaller than the usual pinch of 1/16 teaspoon.

On the road, one seventh of a mile is 754.3 feet, since a mile is 5,280 feet and 5280 / 7 = 754.2857. Seven equal legs of a 5,280-foot walk are 754 feet each, with 2 feet left over at the end.

Money examples are the easiest to check. One seventh of $350 is $50, one seventh of $84 is $12, and one seventh of $700 is $100. A discount of one seventh is the same as 14.29 percent off, so a $70 item drops by $10 and a $350 item drops by $50.

When a price is quoted as 14 percent off, it is not exactly one seventh. On $700 the difference is $2, because 14 percent takes $98 and one seventh takes $100. The gap is small on a single purchase and grows with the amount. The inch, the mile and the cup used above are defined in NIST Special Publication 811, the federal guide to units and conversion factors.