Decimals to Thousandths: Place Value, Comparing, Rounding and Powers of Ten

Imagine splitting one whole into 1,000 equal parts. Each part is one thousandth. You write it as 0.001, or as the fraction $\frac{1}{1000}$.

Make smaller equal parts

You can reach thousandths in three steps. Split a whole into 10 equal parts to make tenths. Split each tenth into 10 equal parts to make hundredths. Split each hundredth into 10 equal parts to make thousandths.

Picture a large cube built from 1,000 small cubes. The large cube is one whole. A flat layer of 100 small cubes is one tenth. A rod of 10 small cubes is one hundredth. One small cube is one thousandth.

Ten thousandths fill one hundredth. One hundred thousandths fill one tenth. All 1,000 thousandths fill the whole. Although 1,000 is greater than 10, a thousandth is smaller than a tenth. You have cut the same whole into many more pieces.

Write a count of thousandths

The thousandths place is the third place after the decimal point. Put the final digit of your count there:

• 7 thousandths is 0.007.
• 45 thousandths is 0.045.
• 625 thousandths is 0.625.

In 0.007, the zeros keep the 7 in the third place. Writing 0.7 would mean seven tenths, a much larger amount. You can also have wholes: 2.045 means 2 wholes and 45 thousandths, or $2\frac{45}{1000}$.

Different names, same amount

Three tenths can be split into 30 hundredths or 300 thousandths. So $0.3=0.30=0.300$. These are equivalent decimals. Zeros added at the end of a decimal leave its value unchanged.

Some familiar fractions have useful decimal names:

Find thousandths in measurements

A meter contains 1,000 millimeters. One millimeter is 0.001 m, so a 175 mm pencil is 0.175 m long. A gram is 0.001 kg, and a milliliter is 0.001 L. One eighth of a kilogram is 125 g: 125 thousandths of a kilogram.

The digit 5 can mean five ones, five tenths, or five thousandths. Its place value tells you which amount it means.

Read the chart

The decimal point sits between the ones and tenths. Whole-number places continue to its left. Smaller parts continue to its right.

Read 4.352 one place at a time. The 4 is worth 4 ones. The 3 is worth 0.3, the 5 is worth 0.05, and the 2 is worth 0.002. A digit's value is its digit multiplied by the size of its place.

To find the value of the 8 in 12.384, start at the point. The 3 counts tenths, and the 8 counts hundredths. The 8 is worth 0.08, or $\frac{8}{100}$.

Follow the same digit

Each place is worth 10 times the place just to its right. One one is 10 tenths. One tenth is 10 hundredths. One hundredth is 10 thousandths.

In 7.777, the three 7s after the point mean 0.7, 0.07, and 0.007. Each is 10 times the one just to its right. Going right makes the value one tenth as large.

Listen for the ending “-ths.” Thousands and thousandths are very different. In 3,000.003, the first 3 is worth 3,000. The last 3 is worth 0.003. The first place after the point is tenths; there is no “oneths” place.

Give zero its space

A placeholder zero keeps other digits in their places. In 6.045, there are no tenths. Remove that zero and you get 6.45: the 4 now means tenths. The amount has changed. A zero before the point, as in 0.625, shows that there are no whole ones.

Count past a 9

Adding one thousandth to 2.456 gives 2.457. Adding one hundredth instead gives 2.466. These steps change different places.

Watch what happens after 2.459. One more thousandth makes 10 thousandths. Regroup those as one hundredth:

The hundredths digit increases, and the thousandths digit becomes 0.

You can name a decimal in words or show its parts. Both ways help you see what the number contains.

Say the place name

Read 4.352 as “four and three hundred fifty-two thousandths.” Say the whole-number part first. Say “and” at the point. Read the decimal digits as one number, then name the last place.

So 15.208 is “fifteen and two hundred eight thousandths.” The number 0.07 is “seven hundredths.” In this reading, “and” marks the point: the whole number 208 is “two hundred eight.”

You can also read digits one at a time. “Four point three five two” names the same number as 4.352.

Turn words into a numeral

Write “six and forty-two thousandths.” The whole-number part is 6. The word “thousandths” tells you the last digit belongs in the third decimal place. Fit 42 into those places: 042. The number is 6.042. Writing 6.42 would mean forty-two hundredths.

The same idea gives “five and three thousandths” as 5.003, or $5\frac{3}{1000}$. Keep the place name when reading: 0.245 is “two hundred forty-five thousandths,” not “two hundred forty-five.”

Show each digit's value

Expanded form writes a number as a sum of its digit values:

You can show the same parts with multiplication and fractions:

You may include the zero part or leave it out. To build a numeral, put each part in its place. The sum 30 + 2 + 0.6 + 0.005 gives 32.605. There are no hundredths, so that place needs a zero. Parts can come in any order: 0.09 + 7 + 0.003 gives 7.093.

Regroup or split the parts

The decimal 0.245 contains 2 tenths, 4 hundredths, and 5 thousandths. Trade the 2 tenths for 20 hundredths. You now have 24 hundredths and 5 thousandths. Trade again to get 245 thousandths, or $\frac{245}{1000}$.

You can also count 3.6 as 36 tenths, 360 hundredths, or 3,600 thousandths.

To partition a number is to split it into parts that add back to it. For example:

Other splits work too: 1 + 0.375 or 1.3 + 0.075. Each split keeps the same total.

Which is greater: 0.6 or 0.587? You need to compare equal-sized parts. Counting the digits alone will not tell you.

Match the places

Line up the decimal points. Add end zeros to make the decimal parts the same length. You can write 0.6 as 0.600 because end zeros keep the value.

Compare from the left. Start with whole numbers, then tenths, hundredths, and thousandths. Stop at the first different place.

For 0.600 and 0.587, both whole-number parts are 0. The tenths digits are 6 and 5. Six tenths is greater, so $0.6>0.587$. You could also compare 600 thousandths with 587 thousandths.

Use the comparison signs

The sign < means “is less than.” The sign > means “is greater than.” Their open side faces the greater number. So $0.587<0.6$ says the same thing as $0.6>0.587$. The sign = means “is equal to”: $0.40=0.4$.

More digits do not always mean more value. Compare 0.125 with 0.3 by writing 0.300. The tenths decide: 1 tenth is less than 3 tenths. But a shorter decimal is not always greater either. Compare 0.40 with 0.45: the tenths match, then 0 hundredths is less than 5 hundredths.

Always check the whole-number part first. The number 2.09 is greater than 1.999 because 2 wholes is more than 1 whole.

Order a list

Put 3.07, 3.7, 3.007, 3.17, and 3.071 in ascending order: least to greatest. Write them as 3.070, 3.700, 3.007, 3.170, and 3.071. The whole-number parts match. Compare the decimal places to get:

Descending order means greatest to least, so you would reverse that list.

With race times, the smallest time wins. Times of 10.451 s, 10.405 s, and 10.415 s rank 10.405 s first, then 10.415 s, then 10.451 s.

Meet three more signs

The sign ≠ means “is not equal to”: $0.5\mathrm{\ne }0.05$. The signs ≤ and ≥ include equality. So $0.70\le 0.7$ is true because the values are equal. And $0.351\ge 0.35$ is true because 0.351 is greater.

Numbers get greater as you move right on a number line. Before you place a decimal, find how much each step is worth.

Count gaps, not marks

Look at two labeled marks and find the difference between them. Share that difference equally among the gaps.

From 2.3 to 2.4, the difference is 0.1. With 10 equal gaps, each gap is 0.01. The inside marks are 2.31, 2.32, and so on through 2.39. There are nine inside marks but 10 gaps. Counting marks instead of gaps gives the wrong step size.

Zoom in

Split the gap from 2.34 to 2.35 into 10 equal gaps. Each smaller step is 0.001. You can now mark 2.341, 2.342, and the other thousandths through 2.349.

To place 2.347, start at 2.340. Count seven steps of 0.001 to the right. The point has seven gaps to its left and three to its right, so it is closer to 2.35.

Counting across the end works just like place value: 2.348, 2.349, 2.350. Ten thousandths make one hundredth.

Check the scale each time

A line from 0 to 1 with 10 equal gaps has steps of 0.1. With four equal gaps instead, it reads 0, 0.25, 0.5, 0.75, 1.

A line from 1.2 to 1.3 with five equal gaps has steps of 0.02. Its marks read 1.2, 1.22, 1.24, 1.26, 1.28, 1.3. The end labels and gap count tell you the scale.

Find a number between two others

Between 4.6 and 4.7, you could choose 4.65, 4.61, or 4.68. To see a smaller gap, write 4.61 and 4.62 as 4.610 and 4.620. The number 4.615 is halfway between them.

There is always another decimal between two different decimals. You can keep zooming into a gap, even when you need more decimal places to name its inside points.

For a quick estimate on a line from 0 to 1, use 0.5 as the middle. The number 0.512 sits just right of the middle. The number 0.09 is near 0, and 0.95 is near 1.

Rounding replaces a number with a nearby, simpler number. To round 6.8 to a whole number, choose between 6 and 7. It is closer to 7, so $6.8\approx 7$. The sign ≈ means “is approximately equal to.”

Choose a place, then look one place right

1. Find the place you need in the answer.
2. Look at the digit just to its right.
3. If that digit is 0–4, keep the rounding digit. If it is 5–9, increase the rounding digit by one.
4. Drop the decimal digits after the rounding place.

“To 1 decimal place” means “to the nearest tenth.” Your answer has one digit after the point. A whole-number answer has none.

Round the same number two ways

Start with 3.46. For the nearest whole number, look at the tenths digit, 4. Keep the 3: $3.46\approx 3$.

For the nearest tenth, keep your eye on the 4 tenths. The next digit is 6, so increase 4 tenths to 5 tenths: $3.46\approx 3.5$.

When a number is exactly halfway, round up. The number 2.5 is equally close to 2 and 3, so it rounds to 3. For tenths, 4.75 is halfway between 4.7 and 4.8, so it rounds to 4.8.

Carry into a new whole

Round 7.96 to the nearest tenth. The hundredths digit is 6, so increase the 9 tenths. Ten tenths make one whole. The result is 8.0. Keep the zero to show you rounded to tenths.

Use the original number

Three decimal places use the same rule. For 12.749, the tenths digit 7 makes the nearest whole number 13. The hundredths digit 4 makes the nearest tenth 12.7.

Do not round from the last digit in stages. For 5.349 to tenths, the hundredths digit is 4. The answer is 5.3. Rounding first to 5.35 and then to 5.4 changes the result.

“Round down” does not mean subtract one from the rounding digit. For 8.32 to tenths, the 2 hundredths tells you to keep the 3 tenths. The answer is 8.3.

“To 2 decimal places” means to the nearest hundredth. You keep two digits after the point and use the thousandths digit to decide.

Choose the nearer hundredth

The number 2.347 lies between 2.34 and 2.35. Its thousandths digit is 7, so it rounds up to 2.35. The number 2.342 rounds to 2.34 because its thousandths digit is 2. Exactly halfway, 2.345 rounds up to 2.35.

Cutting off digits can miss that decision. The number 6.378 rounds to 6.38, not 6.37. For 3.205, the answer is 3.21. Writing 3.2 would not show two decimal places.

Round through 9s

Try 4.996 to the nearest hundredth. The thousandths digit is 6, so increase 99 hundredths to 100 hundredths. Those make one whole. The result is 5.00, with both zeros kept.

Round to other places

One number can give several rounded answers. Always return to the original:

For tens or hundreds, replace the smaller whole-number places with zeros. Drop the decimal part. So 46.73 rounds to 50 at the nearest ten. The number 253.8 rounds to 300 at the nearest hundred.

Find possible starting numbers

Which numbers with up to three decimal places round to 4.6 at the nearest tenth?

The halfway point below 4.6 is 4.550. It rounds up to 4.6, so it is included. At the other end, 4.649 still rounds to 4.6. But 4.650 is halfway to 4.7 and rounds up to 4.7.

The possible numbers run from 4.550 through 4.649. A number such as 4.58 or 4.612 fits. The rounded result alone does not tell you which starting number was used.

A scale reading of 0.786 kg rounds to 0.79 kg at hundredths, or 0.8 kg at tenths.

A rounded value is an approximation. It is useful when you need a simpler amount, but it usually loses some detail. Write $7.384\text{ m}\approx 7.4\text{ m}$ to show a rounded length.

Keep the detail you need

A race board might show 12.384 seconds as 12.38 s. In everyday talk, you might call a 2.468 kg package “about 2.5 kg.” When a question names a rounding place, use that place.

A cent is one hundredth of a dollar, so money amounts are usually rounded to cents. A US gas price of $3.459 per gallon rounds to $3.46 per gallon at the nearest cent. To the nearest dollar, it is $3.

Estimate before checking a total

An estimate uses simple numbers for a quick, close answer. For 3.86 + 5.08 + 2.9, round each number to a whole number:

The exact total, 11.84, is close to 12. That makes its size reasonable. A total of 118.4 or 7.66 would be far from the estimate and needs another look.

For shopping, $2.95, $4.10, and $0.98 round to $3, $4, and $1. The estimated total is $8. The exact total is $8.03. An estimate need not match every cent to be useful.

Keep close results apart

Race times of 10.405 s and 10.415 s both round to 10.4 s at tenths. That rounded display hides who was faster. Compare the original times: 10.405 s is smaller, so it wins.

Multiplying by 10 makes each digit worth 10 times as much. On a fixed place value chart, every digit moves one place left. Four tenths become four ones: $0.4\times 10=4$.

Follow every digit

Multiplying by 100 is two multiplications by 10, so digits move two places left. Multiplying by 1,000 moves them three places left.

Work out $0.375\times 100$. The 3 tenths become 3 tens. The 7 hundredths become 7 ones. The 5 thousandths become 5 tenths. Put those parts together:

The starting number lies between 0.3 and 0.4. Multiplying those ends by 100 gives 30 and 40. The answer 37.5 fits between them.

Fill empty places

For $4.2\times 1000$, the 4 ones become 4 thousands. The 2 tenths become 2 hundreds. Zeros fill the tens and ones, giving 4,200.

You can also picture the point moving right while the digits stay still. For $2.536\times 100=253.6$, it moves two places right. On a place value chart, the point stays fixed and the digits move left. Both pictures give each digit the same new value.

With whole numbers, multiplying by 10, 100, or 1,000 adds one, two, or three end zeros. For 36, the results are 360, 3,600, and 36,000.

But “just add a zero” fails with decimals. The number 0.30 equals 0.3; it is not 10 times as large. Instead, $0.3\times 10=3$ and $1.25\times 10=12.5$.

Write powers of ten

An exponent counts how many equal factors to multiply. For powers of ten:

The exponent also gives the number of places to move. So $0.058\times {10}^3=58$. The pattern continues: ${10}^4=\mathrm{10,000}$, and $0.027\times \mathrm{10,000}=270$.

To find a missing factor in $0.64\times \mathrm{\square }=640$, follow the 6. It moves from tenths to hundreds, three places left. The factor is 1,000.

Divide 7 by 10. Seven ones become seven tenths, so $7÷10=0.7$. Every digit moves one place right on a fixed place value chart.

Count the places

Dividing by 100 moves digits two places right. Dividing by 1,000 moves them three places right.

For $45÷100$, the 4 tens become 4 tenths. The 5 ones become 5 hundredths. Together they make 0.45.

Decimals follow the same pattern:

You can also picture the point moving left, with the digits staying still. Each digit ends up worth less.

Use zeros to hold empty places

Divide 6 by 1,000. The 6 ones become 6 thousandths. There are no ones, tenths, or hundredths, so write 0.006. For $52÷1000$, the answer is 0.052. The 5 belongs in hundredths and the 2 in thousandths.

A positive number divided by 10, 100, or 1,000 becomes smaller. For $0.6÷100$, the answer is 0.006. An answer of 60 would mean the digits moved the wrong way.

Know when a shortcut works

For a whole number ending in enough zeros, you can remove end zeros: $3400÷100=34$. But 4.5 has no end zero to remove. Follow its place values to get $4.5÷10=0.45$.

With exponents, ${10}^2$ means 100. So $270÷{10}^2=2.7$: the digits move two places right.

Undo the change

Multiplying and dividing by the same power of ten undo each other. For example:

This also finds a missing starting value. If a number multiplied by 100 gives 52.3, divide 52.3 by 100. The missing number is 0.523.

A length of 2.5 m is also 250 cm. Changing the unit changes the number you write, while the length stays the same.

Choose the factor

Metric units have power-of-ten links:

For the same amount, smaller units need a larger count. Multiply to change to smaller units. To change 2.345 kg to grams, multiply by 1,000:

So 2.345 kg = 2,345 g. For meters to centimeters, multiply by 100: 0.85 m = 85 cm.

Larger units need a smaller count. Divide to change to larger units. For 750 mL to liters, divide by 1,000 to get 0.75 L. In the same way, 1,250 m = 1.25 km, and 6 mm = 0.006 m.

Read the thousandths as smaller units

In 3.125 kg, the decimal part is 125 thousandths of a kilogram. Each thousandth is one gram, so this means 3 kg and 125 g. The distance 1.608 km means 1 km and 608 m.

Check the direction before you calculate. Changing 4,500 g to kilograms needs a smaller number because kilograms are bigger units. The answer is 4.5 kg.

US customary units have different links. For example, 12 inches make one foot. A power-of-ten digit shift alone cannot change inches to feet.

Find costs and shares

One dollar equals 100 cents. So 1¢ is $0.01. To change dollars to cents, multiply by 100; to change cents to dollars, divide by 100.

For many items at the same price, multiply the price by the number of items. One hundred erasers at $0.35 each cost $35. Ten pencils at $0.45 each cost $4.50.

For equal shares, divide the total by the number of people. Sharing $45 equally among 100 people gives $0.45 each. In money amounts with cents, keep two decimal places: write $4.50 for four dollars and fifty cents.