Decimals, rounding and conversion

Decimal places count tenths, hundredths and thousandths

A decimal extends the same base-ten place-value system used for whole numbers. To the left of the decimal point are ones, tens and hundreds; to the right are tenths, hundredths and thousandths. Each place is one tenth of the place immediately to its left. Thus 2.407 = 2 + 4/10 + 0/100 + 7/1000. The zero holds the hundredths place so that the 7 continues to mean seven thousandths.

Write a zero before the decimal point for values with no whole units: 0.5 means five tenths, while 0.05 means five hundredths. The location of a digit matters as much as the digit itself. English examples here use a decimal point; Lithuanian examples use a decimal comma. These are different notations for the same base-ten values. A decimal point or comma separates whole places from fractional places; it is not an instruction to multiply.

Trailing zeros keep the value; place value determines order

Adding zeros at the end of a terminating fractional part does not change its value: 0.5 = 0.50 = 0.500, because 5/10 = 50/100 = 500/1000. A zero inserted before an existing fractional digit can change its place and value: 0.05 is not 0.5. Extending a whole-number numeral from 5 to 50 also changes its value. Ask whether the original digits kept their places.

Compare decimals by their numerical place values, not their written lengths. Rewrite 0.7 as 0.70 to compare it with 0.65: seventy hundredths exceed sixty-five hundredths, so 0.7 > 0.65. For 2.04 and 2.4, the whole parts agree, but zero tenths are less than four tenths, so 2.04 < 2.4. Negative values reverse the order of their positive magnitudes: −0.75 < −0.7, since −0.75 lies farther left on the number line.

A terminating decimal is an exact fraction of a power of ten

Three fractional decimal places name thousandths: 0.375 = 375/1000 = 3/8. For a terminating decimal with k fractional places, use denominator 10 to the power k and the signed integer made by removing the decimal separator as numerator. For example, 1.205 = 1205/1000 = 241/200. Simplifying changes the fraction expression, not the number. Extra trailing zeros give a larger power-of-ten denominator but the same value.

The sign applies to the whole decimal value. Thus −1.25 = −125/100 = −5/4 = −(1 + 1/4), not −1 + 1/4. A whole number is covered by the same idea: 3 = 3/1, with zero fractional places. This conversion is exact for a specified terminating numeral. If that numeral was itself a rounded measurement or a rounded calculation, converting it exactly does not recover the value that existed before rounding.

Long division follows the remainders

A fraction a/b means a divided by b, with b nonzero. For 3/8, there are no complete whole units, so start with 0. Multiply the remainder 3 by ten: 30 ÷ 8 gives digit 3 and remainder 6. Then 60 ÷ 8 gives digit 7 and remainder 4; finally 40 ÷ 8 gives digit 5 and remainder 0. The successive digits occupy tenths, hundredths and thousandths, giving 3/8 = 0.375 exactly. A zero remainder ends the division; additional decimal digits would all be zero.

For 1/6, 10 ÷ 6 gives digit 1 and remainder 4. The next step, 40 ÷ 6, gives digit 6 and remainder 4 again. Returning to the same remainder returns to the same next digit and remainder, so the six repeats indefinitely: 1/6 = 0.1666…. We write this as 0.1(6), where parentheses identify the repeating block rather than multiplication. For a negative fraction, divide its magnitude and apply the sign to the complete result.

Test the denominator after reducing the fraction

A fraction in lowest terms with a positive denominator has a terminating base-ten decimal exactly when its denominator has no prime factors other than 2 and 5. Denominator 1 qualifies too: it has no prime factors and gives an integer. For 3/8, the denominator is 2 × 2 × 2; for 7/20, it is 2 × 2 × 5. Both terminate: 0.375 and 0.35. The fraction 3/6 first reduces to 1/2 and therefore terminates, even though its unreduced denominator contained a factor 3. Zero reduces to 0/1 and terminates as well.

The reason is the base itself: 10 = 2 × 5, so every power-of-ten denominator contains only those prime factors. A reduced denominator with only 2s and 5s can be enlarged to a power of ten by supplying the missing factors. For 3/8, multiply numerator and denominator by 125 to obtain 375/1000. Conversely, reducing an exact fraction over a power of ten cannot introduce a different prime factor. A remaining factor such as 3 in 1/3 or 1/6 therefore prevents termination; division instead produces a recurring expansion.

A repeating block gives an exact rational value

For a fixed positive integer denominator b, long division has only b possible remainders: 0 through b − 1. Either a remainder reaches zero, or a nonzero remainder eventually repeats. Repeated remainders force the same subsequent digits, so every rational number has a terminating or eventually recurring decimal expansion. The repeating block need not start immediately after the separator: 1/6 = 0.1(6), while 1/3 = 0.(3). Parentheses specify the entire infinite repeating pattern, making these exact expressions; 0.333 is only a finite approximation to 1/3.

A recurring decimal can be converted back to a fraction. If x = 0.(27), then 100 × x = 27.(27); subtracting x removes the identical repeating tail, so 99 × x = 27 and x = 27/99 = 3/11. Irrational numbers, such as √2, have decimal expansions which neither terminate nor become periodic. Their irrationality needs mathematical reasoning: inspecting a finite run of digits cannot prove that no repeating pattern ever appears. An ellipsis alone does not specify a pattern, and a calculator’s finite display cannot decide whether an unknown exact number is rational.

Rounding chooses a neighboring multiple of a place unit

Choose the place before changing the value. Rounding to a whole unit chooses among integers; rounding to one decimal place chooses among tenths; two places choose among hundredths. The exact value 0.374 lies between 0.37 and 0.38. Their midpoint is 0.375, so 0.374 is closer to 0.37. At the midpoint, the two choices are equally near and a tie convention is needed. Rounding is a replacement of the original value by a selected nearby value, not an equivalent-fraction rewrite.

This lesson and its explorer use nearest rounding with ties away from zero. At an exact midpoint, choose the neighbor with the greater absolute value: 0.375 becomes 0.38 to two decimal places. This is an explicit convention, not the only rule used in mathematics, software or scientific reporting. Ties to even is another common convention; follow whichever rule a task states. Use = when values really agree, as 0.35 = 0.350, and ≈ when rounding changes the value, as 3/8 ≈ 0.38.

A negative midpoint moves away from zero under our rule

The exact number −53/40 = −1.325 is midway between −1.33 and −1.32. Under ties away from zero, it rounds to −1.33 at two decimal places, because that neighbor has the greater magnitude. It does not round to −1.32 by a blanket instruction to increase a signed number. Away from a tie, simply choose the closer neighbor: −1.324 rounds to −1.32. One reliable way to apply this convention is to round the magnitude by nearest, ties away, then restore the original sign.

Carrying can cross the separator and change the whole part. The exact number 1999/200 = 9.995 is midway between 9.99 and 10.00, so it rounds to 10.00 at two decimal places. The two trailing zeros display the selected hundredths place; numerically, 10.00 = 10. They do not establish that a measured quantity was known accurately to a hundredth. If rounding a small negative value selects zero, the numerical result is simply zero; an extra negative-zero sign is not needed in this lesson.

Round from the original value; distinguish truncation

Rounding in stages can change the final answer. From the exact value 1.249, direct rounding to tenths gives 1.2: it is below the midpoint 1.25. But rounding first to hundredths gives 1.25; rounding that new value to tenths creates a tie, and our convention gives 1.3. The intermediate rounding lost the information that the original value was below the midpoint. Keep the exact fraction or sufficient original digits for calculations, then round once to the final requested place.

Truncation removes later fractional digits and moves toward zero; it does not select the nearest value. At two decimal places, truncating −1.328 gives −1.32, while nearest rounding gives −1.33. For nearest rounding to a place unit u, the absolute rounding error is at most u/2, including exact ties. Thus nearest rounding to hundredths has error at most 0.005 in the quantity’s unit. Truncation to that place has error less than 0.01 but need not satisfy the smaller nearest-rounding bound. These statements concern replacing a known exact number, not every source of error in a measurement.

Keep the fraction exact while changing displayed precision

Begin with 3/8 and two decimal places in the explorer. Follow the reduced fraction, denominator factors and division steps to its exact decimal 0.375. Compare the neighboring hundredths 0.37 and 0.38 and locate their midpoint. The chosen rounding rule gives 0.38, so the rounded statement uses ≈. Switch to three decimal places: the display can now show 0.375 exactly, and the statement uses =. The source fraction has stayed fixed throughout.

Try 1/3 and 1/6 to inspect a repeating block and any nonrepeating prefix. Increasing finite precision gives a closer rounded description; it does not turn either recurring expansion into a terminating exact value. Try 7/20 to see that trailing zeros preserve an exact value, −53/40 for a negative midpoint, and 1999/200 for a carry into the next whole unit. The zero fraction 0/7 reduces to 0/1 and remains zero at every selected place. Explain separately what the exact number is, how division represents it, and what the selected rounding display changes.

Decimal arithmetic follows the same fractional units

Align equal places for addition and subtraction. For 0.4 + 0.35, rewrite 0.4 as 0.40: forty hundredths plus thirty-five hundredths make seventy-five hundredths, so the sum is 0.75. The alignment follows the units, rather than the lengths of the numerals. For multiplication, the units combine: 0.2 × 0.3 = (2/10) × (3/10) = 6/100 = 0.06. The product of two positive values below one is smaller than either, which checks its scale.

Division preserves a quotient when both dividend and divisor are multiplied by the same nonzero factor: 0.75 ÷ 0.5 = 75 ÷ 50 = 3/2 = 1.5. The divisor must still be nonzero. Measurements follow the previous lesson’s quantity rules: 0.4 m + 0.35 m = 0.75 m; multiplying a length by a length gives area; dividing a volume by an equal-unit portion size gives a portion count. Keep exact terminating values or fractions during these operations, respect parentheses and operation order, and round the final result only when the question calls for it.

Written precision and measurement certainty are different

If an exact mathematical length is 0.375 m and we report 0.38 m, the rounding changed it by 0.005 m, or 5 mm. This is exactly the half-hundredth bound for rounding in metres. Changing the reporting unit changes the place unit: rounding to a hundredth of a metre is different from rounding to a hundredth of a centimetre. Always attach the measurement unit to an error bound. Error from rounding a known value is only one possible contribution to the uncertainty of an actual measurement.

A ruler, observation or calculation may have its own limitations. Writing 2.00 m instead of 2 m can identify a reporting place, but cannot by itself improve the measurement or establish its accuracy. Check that an exact conversion used =, a changed value used ≈, and rounding used the stated convention on the original value. Compare the result’s magnitude with the source fraction and verify simple arithmetic through fractions or inverse operations. The next lesson uses quantities and their units to explain ratios, rates and proportional reasoning.

Worked examples

Try it yourself

1. Write 0.407 as a sum of fractional place values. Which equalities hold: 0.4 = 0.400 and 0.4 = 0.04? Compare 0.7 with 0.65 and −0.75 with −0.7.

   Show the answer and reasoning

   0.407 = 4/10 + 0/100 + 7/1000. The first equality holds; the second does not. 0.7 > 0.65 and −0.75 < −0.7.

   The zero preserves the hundredths place. Trailing zeros leave the existing digits in place, while the extra zero in 0.04 makes 4 count hundredths instead of tenths. Compare 0.70 with 0.65 using hundredths. For negative numbers, the greater positive magnitude lies farther left, so −0.75 is smaller.

2. Convert 1.205 and −1.25 to exact fractions in lowest terms. Does −1.25 mean −1 + 1/4? If 1.205 was already a rounded reading, does fraction conversion restore the unrounded reading?

   Show the answer and reasoning

   1.205 = 241/200 and −1.25 = −5/4. The negative value means −(1 + 1/4). Converting a rounded numeral cannot restore the earlier value.

   Use 1205/1000 and simplify by 5; use −125/100 and simplify by 25. The negative sign applies to the complete magnitude, while −1 + 1/4 would equal −3/4. The conversion expresses the provided numeral exactly; any information lost before it was provided is still lost.

3. Convert 7/20 and 1/6 to decimals. State which division terminates and which repeats. For 1/6, identify the nonrepeating prefix, repeating block and repeated remainder.

   Show the answer and reasoning

   7/20 = 0.35 terminates. 1/6 = 0.1(6) repeats, with prefix 1, block 6 and repeated remainder 4.

   For 7/20, 70 ÷ 20 gives digit 3 and remainder 10; then 100 ÷ 20 gives digit 5 and remainder 0. For 1/6, the first digit 1 leaves remainder 4; every subsequent 40 ÷ 6 gives digit 6 and remainder 4. Repeating the remainder repeats the subsequent digit process indefinitely.

4. Which of 6/15, 7/28, 1/12, 5/8 and 0/7 have terminating decimal forms? Reduce first and explain with denominator factors. Can a finite calculator display prove that an unknown exact value is irrational?

   Show the answer and reasoning

   6/15 = 2/5, 7/28 = 1/4, 5/8 and 0/7 = 0/1 terminate. The fraction 1/12 recurs. A finite display cannot prove irrationality.

   The reduced positive denominators 5, 4 = 2 × 2, 8 = 2 × 2 × 2 and 1 contain no prime factors besides 2 and 5. The denominator 12 contains a factor 3 after reduction, so 1/12 does not terminate. A finite sample of decimal digits cannot exclude a repeating pattern farther along; irrationality requires information about the exact number.

5. Using nearest rounding with ties away from zero, round 0.125, −1.325 and 9.995 to two decimal places, and 1/3 to three decimal places. Then write 3/8 at three decimal places. Choose = or ≈ for each relation to its source.

   Show the answer and reasoning

   0.125 ≈ 0.13; −1.325 ≈ −1.33; 9.995 ≈ 10.00; 1/3 ≈ 0.333; 3/8 = 0.375.

   The first three values are exact midpoints, so choose the neighbor with the greater magnitude. The fourth has a repeating exact expansion 0.(3); the fourth digit is 3, so the nearest thousandth is 0.333, still an approximation. The last terminating expansion already fits three places and preserves the exact value. Other tie conventions can change midpoint answers, so the chosen rule matters.

6. Round 1.249 directly to tenths, then through hundredths to tenths, using ties away from zero. Separately, truncate and nearest-round −1.328 to hundredths. Which processes choose the nearest value to their original input?

   Show the answer and reasoning

   Direct rounding gives 1.2, while the staged route gives 1.25 then 1.3. Truncation gives −1.32; nearest rounding gives −1.33.

   The original 1.249 lies below the tenth-place midpoint 1.25; the intermediate hundredth-place result creates a tie that the original value did not have. For −1.328, deleting later fractional digits moves toward zero but leaves an error of 0.008. The nearer hundredth −1.33 has error 0.002, within the nearest-rounding bound 0.005. Direct nearest rounding selects the nearest value to the original input.

7. Calculate 0.4 + 0.35, 0.2 × 0.3 and 0.75 ÷ 0.5 exactly. A known exact length 0.375 m is reported as 0.38 m: find the rounding error in metres and millimetres. Does writing a measurement as 2.00 m prove accuracy to a hundredth of a metre?

   Show the answer and reasoning

   0.75, 0.06 and 1.5. The absolute rounding error is 0.005 m = 5 mm. Writing 2.00 m alone does not prove that measurement accuracy.

   Use shared hundredths for addition, multiply fractional units for the product, and scale both dividend and divisor equally for division. The known length differs from its report by 0.38 − 0.375 = 0.005 m, exactly half the selected 0.01 m place unit. Extra reported digits cannot establish instrument quality or all other measurement uncertainty.

What to carry forward

• Decimal places are powers-of-ten fractional units. Trailing zeros preserve value; compare signed values by their places and number-line order.
• Terminating decimals convert exactly to fractions over powers of ten. A reduced fraction terminates exactly when its positive denominator has only prime factors 2 and 5, with denominator 1 included.
• Repeated long-division remainders explain recurring rational decimals. An explicitly repeating block is exact, while a finite sample cannot prove irrationality.
• Choose the rounding place and state the tie convention. This lesson uses nearest, ties away from zero; round the original value once and distinguish rounding from truncation.
• Use = for equal values and ≈ for changed approximations. Nearest-rounding error is at most half the place unit; reported decimal precision alone does not establish measurement accuracy.