1/10 as a Decimal

1 divided by 10 is 0.1

One tenth written as a decimal is 0.1. The 1 sits one place to the right of the decimal point, in the tenths column, so its value is 1 divided by 10. Nothing is rounded and nothing repeats: 0.1 is exact.

The same quantity has several names, and the table lists the ones that show up in homework and on labels. They all describe one part out of ten equal parts of a whole.

0.1 and 0.10 are the same number. The trailing zero changes nothing about the value, it only suggests that a measurement was recorded to two places. In a math answer, 0.1 is the usual form, and 10/100 is useful when you need a common denominator.

The decimal point is the only new idea in the conversion. The fraction 1/10 already tells you the denominator is ten, and a decimal is a shorthand for a fraction with a denominator of ten, one hundred, one thousand, and so on. Writing 0.1 is a way of saying 1/10 without the fraction bar.

The long division, step by step

Division is what a fraction means, so the conversion is one division: 1 divided by 10. Long division makes the steps visible, which is the part a calculator hides.

Set up 1.000 with the decimal point after the 1. Ten does not fit into 1, so the whole part of the answer is 0 and the first digit has to come from the tenths place. Bring down a zero, which turns the 1 into 10 tenths. Ten fits into 10 exactly once, so the first digit after the point is 1, and 10 minus 10 leaves a remainder of 0. A remainder of 0 means the division stops, so the answer is 0.1 with nothing after it.

Two checks confirm the result. Multiply 0.1 by 10 and you get 1.0, which is the number you started with. Write 1/10 over 100 by multiplying top and bottom by 10 and you get 10/100, which reads as 0.10, the same value with an extra zero.

The zero in front of the point matters. Writing .1 is common in engineering notes and spreadsheets, but in school work and in most published text the leading zero is expected, because a bare decimal point is easy to miss on a printed page. The value is identical either way.

Why the digit lands in the tenths place

The decimal point marks the ones place. The first place to its right is tenths, the second is hundredths, the third is thousandths. A 1 in the tenths place means 1 x 1/10, which is 0.1. A 1 in the hundredths place means 1 x 1/100, which is 0.01, ten times smaller.

The SI prefix deci carries the same meaning. NIST lists deci as the prefix for 10^-1, one tenth, so a decimeter is one tenth of a meter and a deciliter is one tenth of a liter. The word decimal and the prefix deci both come from the Latin word for ten, and the arithmetic behind them is the same: cut one unit into ten equal parts and take one.

The table also shows why 0.1 is not 0.01. One is a tenth, the other is a hundredth, and the second is ten times smaller. The extra zero is not decoration. It moves the 1 one place to the right, and each place to the right divides the value by ten.

Read a decimal from left to right and name the last place. The number 0.1 ends in the tenths place, so it is one tenth. The number 0.10 ends in the hundredths place, so it is ten hundredths, which reduces to one tenth. Naming the place is the habit that prevents the 0.1 and 0.01 mix up.

One tenth written as a percent

A percent is a fraction with a denominator of 100, so converting a decimal to a percent means multiplying by 100. For 0.1 that is 0.1 x 100 = 10, and the answer is 10 percent. The reverse works the same way: 10 percent divided by 100 is 0.1. George Mason University's math tutorial describes the shortcut as moving the decimal point two places to the right for decimal to percent, and two places to the left for percent to decimal.

Several rows have a second fraction name. 2/10 reduces to 1/5, 5/10 reduces to 1/2, and 10/10 is 1. Reducing changes the fraction but not the decimal or the percent, because the value stays the same.

The percent form is where the tenths table pays off in daily life. A 10 percent sales tax on a 250 dollar purchase is 0.1 x 250 = 25 dollars, so the total is 275 dollars. A 10 percent tip on a 62 dollar bill is 6.20 dollars. Both are the same multiplication, and both come from reading 10 percent as 0.1 rather than reaching for a calculator.

Taking one tenth of a number

Taking one tenth of a number is the same as dividing it by 10, and dividing by 10 moves every digit one place to the right. The digits themselves do not change. What changes is where the decimal point sits.

Start with 250. Moving the digits one place right gives 25.0, which is written 25. Start with 4,800 and the answer is 480. Start with 1,000,000 and it is 100,000. Each answer can be checked by multiplying back by 10, and the check returns the original number.

The last two rows are where most slips happen. One tenth of 62 is 6.2, not 0.62, because the digits move one place, not two. One tenth of 0.5 is 0.05, because the 5 moves from the tenths place to the hundredths place. The rule is one place for each division by ten, and the decimal point is the marker that keeps count.

The same operation shows up as a discount. Ten percent off a 250 dollar item takes 25 dollars off and leaves 225. Ten percent off 4,800 dollars takes 480 and leaves 4,320. If a price is already given as a decimal, 0.5 dollars becomes 0.05 dollars, which is five cents, and that is the kind of result worth pausing on before you trust it.

Which fractions terminate and which repeat

Whether a decimal stops or repeats depends on the denominator after the fraction is reduced. A fraction in lowest terms terminates exactly when its denominator has no prime factors other than 2 and 5. The number 10 is 2 x 5, so 1/10 terminates after one digit. The number 8 is 2 cubed, so 1/8 is 0.125 and terminates after three digits. The number 3 contains a factor of 3, so 1/3 repeats forever: 0.333 and so on.

The reason is the link between decimals and powers of ten. A terminating decimal is a fraction over 10, 100, 1000 or another power of ten, and every power of ten factors into 2s and 5s: 10 is 2 x 5, 100 is 2 x 2 x 5 x 5, and so on. Any denominator built only from 2s and 5s can be scaled up to a power of ten. Multiply 1/8 by 125/125 and you get 125/1000, which is 0.125. A denominator with a 3 or a 7 in it can never be scaled to a power of ten, so the long division never reaches a remainder of zero and the digits cycle.

The NIST Guide to the SI uses the same base ten structure for units, which is why the prefix table and the decimal table line up. The calculator applies the factor test and reports the type, so a fraction such as 7/40 reduces to a denominator of 40, which is 2 x 2 x 2 x 5, and terminates at 0.175.

Slips that turn 0.1 into the wrong number

The first slip is writing 0.01 when you mean 0.1. One zero after the point is tenths, two zeros push the 1 into hundredths, and the second value is ten times smaller. Say the place name out loud when you write the number and the error disappears.

The second slip is flipping the fraction. The expression 1/10 is one divided by ten, which is 0.1. The expression 10/1 is ten divided by one, which is 10. The bar means division, and the numerator goes first, so the order matters even though both numbers are small.

The third slip is treating 10.0 as ten tenths. The decimal point in 10.0 sits after the zero of the tens digit, so the number is ten, not one. Ten tenths is 1.0, and ten is one hundred tenths. Place value, again, is the whole story.

The fourth slip is a computer artifact rather than a math error. Most calculators store 0.1 in binary floating point, where it cannot be represented exactly, so the stored value is a hair above one tenth and a long display may show 0.10000000000000001. The value is still 0.1 for any practical purpose, and the same storage detail is why adding 0.1 ten times in a spreadsheet can produce 0.9999999999999999 instead of 1.

The fifth slip is mixing up one tenth and one tenth of one percent. One tenth is 0.1, which is 10 percent. One tenth of one percent is 0.001, which is 0.1 percent. The two numbers differ by a factor of 100, and labels on supplements and interest rates use both forms, so read which one is meant before multiplying.