0.125 as a Fraction

Three slots after the point

Start at the decimal point and count to the right. In 0.125 the 1 holds the tenths slot, the 2 holds the hundredths slot and the 5 holds the thousandths slot, so the reading is 125 thousandths before any cancelling happens. That slot count fixes the lower term of the fraction, and it is also what a bench micrometer reports when its barrel shows three digits.

Put the three digits over 1000 and the result is 125/1000, an exact copy of the decimal with nothing rounded away. The copy is clumsy rather than wrong. Both terms carry 125, because 125 = 5 x 5 x 5 and 1000 = 2 x 2 x 2 x 5 x 5 x 5. Strike that shared 5 x 5 x 5 from each side and 1/8 remains, which is why 0.125 as a fraction is one eighth.

Divide 1 by 8 to close the loop. The quotient is 0.125, the reading you started with, and 8 x 0.125 = 1, the product every name for one eighth has to produce.

What each slot weighs

Place value is easier to trust when the slots are laid out one at a time. Each digit in 0.125 contributes a fixed weight, and the three weights add straight back to the decimal.

Read down the last column and the sum is 0.1 + 0.02 + 0.005 = 0.125. Nothing in that addition depends on the digits being friendly, which is the point: the slot sets the weight, and the digit only multiplies it.

Slot count also explains why some readings collapse further than others. A three-slot figure such as 0.008 carries 8/1000, and 8 cancels out to leave 1/125, while 0.984 carries 984/1000 and keeps a large lower term because only 8 comes out. The next table sets four of them side by side.

The 1/8 bit and the thousandths a bench reads

A drill index prints sizes as decimals so that a wall chart, a micrometer and a bit shank all agree, and the 1/8 inch bit is the rung most kits start from. Its figure is 0.125 inch, its metric size is 3.175 mm, and the two names describe one bore at the vise.

Neighboring metric sizes sharpen the picture. A 3 mm bit is 0.1181 inch, so it runs 0.0069 inch under 1/8, and a 3.5 mm bit is 0.1378 inch, which runs 0.0128 inch over. Nothing on the shelf substitutes for the fractional size: the 3 mm leaves a bore too tight for an 1/8 pin, and the 3.5 mm leaves one too loose.

Thousandths are the unit a bench works in, and each of them is a fraction with 1000 underneath. The table lines up the readings that turn up on a gauge with the shortest fraction that fits.

The awkward part of a bit chart is the slack between a fractional size and the closest metric bit on the shelf. A 3.2 mm bit cuts a bore 0.025 mm wider than 1/8 inch, which vanishes into softwood fibers and shows up as slop in a metal fit that allows only hundredths of a millimeter. The honest fix is to buy the fractional bit or ream the metric bore to size rather than to split the difference.

Where the millimeter figure comes from

The metric column is not measured, it is defined. NIST's Guide to the SI, Appendix B lists the inch as exactly 25.4 millimeters, so 25.4/8 = 3.175 and any other eighth follows by multiplying that figure by its top: the 1/8 bit at 3.175 mm is the one a bench checks first, and doubling it gives the next rung up.

Because the factor is exact, 3.175 carries no error and needs no rounding. A gauge that reads 3.17 has rounded its display, not the size. The distinction matters when a print states 3.175 and a bench measures 3.17, since the two numbers differ by 0.005 mm, which sits inside the allowance of most hand tools and outside it for a gauge block.

The same logic covers every rung above it. Each step of 1/8 inch adds 3.175 mm, so a chart rebuilds from one figure instead of being memorized row by row, and a typo in the metric column is easy to spot because the spacing between rows should never change.

A breath test that reads 0.125

Blood alcohol concentration counts grams of alcohol in each deciliter of blood, and a deciliter is 100 milliliters. A reading of 0.125 therefore means 0.125 grams per deciliter, or 125/1000, or one eighth, or 12.5 percent, depending on which form the paperwork prefers.

NHTSA's drunk driving page reports that a BAC of 0.08 or above is illegal in every state except Utah, which uses 0.05. The CDC's impaired driving page adds that impairment starts below the legal line, so the arrest threshold is not a safety threshold. The next table places 0.125 against the numbers a driver meets at a roadside stop.

Moving the point three places turns a reading into milligrams per deciliter, so 0.08 and 80 describe one concentration in two notations, and the fraction column is a third notation for the same amount.

0.125 / 0.08 = 1.5625, so a 0.125 reading runs more than half again the standard limit. Court papers may print the same result as 125 mg/dL, and that milligram figure is the fraction 1/8 rescaled to a different unit rather than a different measurement.

Percent, mixed numbers, and 4/32

A percent is the decimal shifted two places, so 0.125 becomes 12.5 percent. As a fraction that percent is 12.5/100, and multiplying both terms by 10 restores 125/1000, the copy from the first section.

One eighth is smaller than a whole, so the answer has no mixed form. Add a whole inch and one appears: 1.125 = 1125/1000 = 9/8 = 1 1/8. The fractional piece is still 0.125, and the whole number rides along untouched.

Two documents can disagree about the lower term and still describe one hole. A drawing that says 4/32 inch and a caliper that reads 0.125 inch name the same size, and a machinist who treats them as different sizes drills twice. Doubling climbs the family: twice 0.125 = 0.25, three times 0.125 = 0.375, so the same steps read 12.5 percent, 25 percent and 37.5 percent as a rate.

Slips that ruin a fit

Rounding before copying is the first trap. A gauge that shows 0.12 and a habit of rounding up to 0.13 produce 13/100, and 0.13 - 0.125 = 0.005 inch, a real gap on a shaft that has to seat in a bearing.

Miscounting slots is the second. Reading 0.125 as 125/100 makes it 1.25, ten times the original, because three digits after the point always mean a lower term of 1000 however small the digits look.

A trailing zero is harmless but noisy. The decimal 0.1250 has four slots, copies to 1250/10000, and cancels by 1250 back to 1/8. The extra zero claims precision the instrument may not own, so drop it unless the gauge reads to ten-thousandths.

Whole inches go missing in the same way. A length of 1.125 inches copies to 1125/1000 and cancels to 9/8, not to 125/1000, and keeping the whole number beside the fraction prevents the loss. A minus sign travels through unchanged: -0.125 becomes -125/1000 and then -1/8, with the sign on the upper term and a positive lower term.

The last slip is trusting a two-place display. A readout set to hundredths prints 0.13 for 1/8, and 0.13 is a different number from 0.125 even though the two look alike at a glance. The third slot is where the eighth family lives, so a job that needs an eighth needs a readout that shows three.