1/8 as a Decimal

One eighth is 0.125, and the decimal stops

One eighth is one part of a whole cut into eight equal pieces. In decimal form it is 0.125, and that is the entire answer. There is no repeating tail, no rounding, and no reason to write an ellipsis. The decimal ends after three places because the division comes out even.

The same value has three common costumes. As a fraction it is 1/8, as a decimal it is 0.125, and as a percent it is 12.5 percent. All three describe one slice of eight. In money terms, one eighth of a dollar is 12.5 cents, so two dollars split eight ways leaves 25 cents per share.

The fraction 0.125 is also 125/1000 before it is reduced. Read the digits by place value: the 1 is one tenth, the 2 is two hundredths, and the 5 is five thousandths. Add them up and you get 125 thousandths, which reduces to one eighth.

That tidiness is rare among fractions. Most denominators, like 3 or 7, produce decimals that never end. Eight is one of the easy ones, and the next sections show the two standard ways to prove 0.125 is right.

Two ways to convert a fraction to a decimal

There are two reliable routes from a fraction to a decimal, and both work for 1/8.

Method one is scaling to a power of ten. Multiply the top and bottom of the fraction by the same number until the bottom is 10, 100 or 1000. For 1/8, multiply by 125, since 8 x 125 = 1000: the top becomes 125 and the bottom becomes 1000. Then read the answer straight off: 125/1000 is 0.125. This method works whenever the denominator can be turned into a power of ten.

Method two is long division. Divide 1.000 by 8 and the work stops after three digits with nothing left over. This method works for every fraction, including ones like 1/3 where no power of ten ever fits.

The table explains the split. A denominator built only from 2s and 5s can always reach a power of ten, so its decimal ends. Anything else repeats. Eight is 2 x 2 x 2, which is why it lands in the tidy group.

The long division in three steps

Long division shows where each digit of 0.125 comes from. Set up 1.000 divided by 8. Eight does not fit into 1, so the whole part is 0 and the decimal point goes in. From there, each step multiplies the remainder by 10 and divides again.

The third step leaves a remainder of 0, so the division is finished and the quotient reads 0.125. Compare that with 1/7, where the remainder cycles through six values and never hits zero. Hitting zero is what makes a decimal terminate.

The same three steps convert any number of eighths. Three eighths runs 30, then 60, then 40: 8 fits into 30 three times with 6 left over, into 60 seven times with 4 left over, and into 40 five times with nothing left. The digits 3, 7, 5 give 0.375.

One reading note: the decimal point goes in during the first step, when you write 1 as 1.0. Forgetting the point is how answers like 125 or 12.5 appear. The fraction is less than 1, so the answer has to be too.

Why eighths are binary fractions

Eight is 2 x 2 x 2, so an eighth is a power of two in the denominator. That makes the eighths a family of halvings: start at 1/2, halve to 1/4, halve again to 1/8. Each halving adds exactly one digit to the decimal, which is why 0.5, 0.25 and 0.125 have one, two and three places.

Powers of two show up in computing, where numbers are written in base 2. In that system one half is written 0.1, one quarter is 0.01, and one eighth is 0.001, with each place worth half the one before it. Wolfram MathWorld's binary number entry covers how place value works in base 2.

Anything divided by a power of two comes out exact in binary. That is why 1/8 is one of the few common fractions a computer stores without error. One tenth, by contrast, needs an endless binary tail, and spreadsheets compensate for it with rounding behind the scenes.

The everyday version of the same fact sits inside a byte. A byte holds 8 bits, so one eighth of a byte is one bit, and one eighth of 16 gigabytes is 2 gigabytes. When memory or storage divides by 8, the share is always a clean whole number, which is the same property that makes 0.125 terminate.

Every eighth from 1/8 to 8/8

Because every eighth terminates, the whole family fits in one table with exact decimals and percents. The kitchen column uses the US measuring rule that a cup holds 16 tablespoons, so each eighth of a cup is 2 tablespoons.

Two rows reduce: 2/8 is 1/4 and 6/8 is 3/4, which is why their decimals match the quarters. The odd numerators keep the eighth: 3/8, 5/8 and 7/8 are already in lowest terms, and their decimals all end in the digit 5.

Every decimal in the table is exact, so none of them needs a rounding rule. That is the practical difference between eighths and sevenths, where every entry in the table would need an ellipsis.

Cups, inches and money

In the kitchen, one eighth of a cup is 2 tablespoons, which is also 1 fluid ounce under US measures. Half of that, one sixteenth of a cup, is a single tablespoon. Recipes that call for 3/8 of a cup are asking for 6 tablespoons, which avoids digging out the odd-sized cup.

On a ruler, inches are divided into sixteenths, and the eighth marks sit halfway between the quarter marks. One eighth of an inch is 3.175 millimeters, because the inch is 25.4 millimeters and 25.4 / 8 = 3.175. Five eighths of an inch is 15.875 millimeters.

With money, one eighth of $240 is $30, one eighth of $1,000 is $125, and one eighth of $80 is $10. A discount of 12.5 percent is exactly one eighth off, so an $80 item marked 12.5 percent off costs $70. The half-percent tail is why 12.5 percent feels awkward in conversation and one eighth feels natural on paper.

Time works the same way in payroll. One eighth of an hour is 7.5 minutes, so a quarter hour is two eighths. A shift of 8 hours split into eighths gives 1 hour per piece, which is the cleanest division in the whole system.

For land and rooms, one eighth of a 2,000 square foot lot is 250 square feet. One eighth of a 40-acre parcel is 5 acres, a common size for a small hay field. NIST Special Publication 811 is the reference for the inch, the mile and the other US units used in these conversions.

Rounding, reading and checking 0.125

0.125 is exact, but it still gets rounded when a form or a price needs fewer digits. To two places it rounds to 0.13, because the third digit is 5 and the rule pushes the hundredths up. To one place it is 0.1. The percent 12.5 rounds to 13 percent on a whole-number form. Each rounding step throws away accuracy, so keep 0.125 in the math and round only the final answer.

Reading slips are worth a pass. The value 0.0125 is one eightieth, ten times smaller than 0.125. The value 1.25 is five quarters, since 1.25 x 8 = 10, ten times larger. And 1/18 is not an eighth at all: it is 0.0555, with the 5 repeating. The denominator, not the numerator, decides how the decimal behaves.

Checking the answer takes one multiplication. Multiply 0.125 by 8 and 0.125 x 8 = 1, which is the original whole. The same check works for any eighth: 0.375 x 8 = 3, and 0.875 x 8 = 7. If the product misses the numerator, a digit was copied wrong.

The place-value check runs in the other direction. 0.125 is 100 thousandths plus 20 thousandths plus 5 thousandths, which is 125/1000. Dividing top and bottom by 125 leaves 1/8. Both checks are one line, and together they make 0.125 one of the easiest conversions to trust.