2.02a - Decimals and fractions

2.02a - Decimals and fractions

Why can one fraction fit into a short decimal while another seems to go on for ever? Learn to read decimal place value and move between fractions and decimals, using equivalent fractions and division to explain the digits.

Decimal place value

Is 0.40.4 the same number as 0.400.40? Imagine a metre strip: four tenths of it covers exactly the same length as forty hundredths. Changing the size of the pieces changes how many you count, not the total length.

A decimal point separates whole-number places from fractional places. Each move one column right divides the value of a digit by ten.

TensUnitsTenthsHundredthsThousandths
101011110\frac1{10}1100\frac1{100}11000\frac1{1000}

Decimal place value

A digit's place value is the value of its position. In 3.4073.407, the 44 means four tenths, the 00 records no hundredths, and the 77 means seven thousandths.

Zeros keep other digits in the right columns. You can add or remove zeros at the end of the fractional part: 0.4=0.40=0.4000.4=0.40=0.400. Removing the zero inside 0.4070.407 would move the 77, changing the number to 0.470.47.

Worked example 1: a digit in the tenths column

Question. Write the value of the digit 66 in 5.625.62.

Plan. Find the digit's column, then multiply its place value by six.

first place after the point=110tenthsvalue of the 6=6×110=0.6six tenths\begin{aligned} \text{first place after the point}&=\frac1{10} &&\text{tenths}\\ \text{value of the 6}&=6\times\frac1{10}=0.6 &&\text{six tenths} \end{aligned}

Answer. 0.60.6, or 610\frac6{10}.

Check. 5.62=5+0.6+0.025.62=5+0.6+0.02, so the three place values rebuild the number.

Now move the 66 one place right by inserting a zero. Its value becomes ten times smaller.

Worked example 2: a placeholder zero

Question. Write the value of the digit 66 in 5.0625.062.

Plan. Count places from the decimal point, including the zero.

second place after the point=1100hundredthsvalue of the 6=6×1100=0.06six hundredths\begin{aligned} \text{second place after the point}&=\frac1{100} &&\text{hundredths}\\ \text{value of the 6}&=6\times\frac1{100}=0.06 &&\text{six hundredths} \end{aligned}

Answer. 0.060.06, or 6100\frac6{100}.

Check. 5+0.06+0.002=5.0625+0.06+0.002=5.062; using 0.60.6 would instead give 5.6025.602.

Read every place, even when its digit is zero.

Watch out

0.06≠0.60.06\ne0.6: six hundredths is not six tenths. But 0.060=0.060.060=0.06, because an extra zero thousandths adds nothing.

Try a digit further to the right.

Terminating decimals and fractions

A decimal that stops can be written as a fraction by counting its places, then simplifying.

For example, 0.320.32 is 3232 hundredths, so 0.32=32100=8250.32=\frac{32}{100}=\frac8{25}. A negative sign stays with the value: −0.32=−825-0.32=-\frac8{25}. A number above one works in exactly the same way: 1.32=132100=33251.32=\frac{132}{100}=\frac{33}{25}.

Terminating decimal

A terminating decimal has a finite number of decimal places. For example, 0.3750.375 terminates; 0.333…0.333\ldots, with threes continuing for ever, does not.

To go the other way without a calculator, try making the denominator 1010, 100100 or 10001000. Multiply numerator and denominator by the same number, so the fraction's value stays fixed.

Worked example 3: converting eighths

Question. Write 38\frac38 as a decimal.

Plan. Make the denominator 10001000, because 8×125=10008\times125=1000.

38=3×1258×125equivalent fraction=3751000=0.375read thousandths\begin{aligned} \frac38&=\frac{3\times125}{8\times125} &&\text{equivalent fraction}\\ &=\frac{375}{1000}=0.375 &&\text{read thousandths} \end{aligned}

Answer. 0.3750.375.

Check. 38\frac38 is between 14\frac14 and 12\frac12, and 0.3750.375 is between 0.250.25 and 0.50.5. Converting back gives 3751000=38\frac{375}{1000}=\frac38.

Change only the numerator. The denominator and the conversion factor remain the same.

Worked example 4: more eighths

Question. Write 58\frac58 as a decimal.

Plan. Use the same thousandths denominator.

58=5×1258×125scale both parts equally=6251000=0.625read thousandths\begin{aligned} \frac58&=\frac{5\times125}{8\times125} &&\text{scale both parts equally}\\ &=\frac{625}{1000}=0.625 &&\text{read thousandths} \end{aligned}

Answer. 0.6250.625.

Check. 38+58=1\frac38+\frac58=1, and the decimals give 0.375+0.625=10.375+0.625=1 too.

There is a useful connection to prime factors. After a fraction has been fully simplified, its decimal terminates exactly when its positive denominator has no prime factors except 22 and 55. Powers of ten contain only those primes. For instance, eighths terminate because 8=2×2×28=2\times2\times2; thirds cannot be changed to a power-of-ten denominator by multiplying by a whole number. The denominator 11 gives a whole number, which also terminates.

Watch out

0.625=62510000.625=\frac{625}{1000}, not 625100\frac{625}{100}. Count decimal places before simplifying; three places mean thousandths.

Use the same denominator as the last example, then reverse the conversion.

Fractions as decimal divisions

The fraction bar also means division: ab=a÷b\frac ab=a\div b, provided b≠0b\ne0.

Sharing one unit among four gives 0.250.25 each. In written division, if a remainder remains after the units, exchange it for tenths; a remaining tenth becomes ten hundredths, and so on. You may append zeros after the decimal point without changing the starting number.

Worked example 5: division that ends

Question. Use division to write 14\frac14 as a decimal.

Plan. Divide 1.001.00 by 44, exchanging each remainder for smaller units.

1÷4=0 remainder 1no whole units each10÷4=2 remainder 2two tenths each20÷4=5 remainder 0five hundredths each\begin{aligned} 1\div4&=0\text{ remainder }1 &&\text{no whole units each}\\ 10\div4&=2\text{ remainder }2 &&\text{two tenths each}\\ 20\div4&=5\text{ remainder }0 &&\text{five hundredths each} \end{aligned}

Answer. 14=0.25\frac14=0.25. A zero remainder means the division finishes.

Check. Four lots of 0.250.25 make 11.

Change only the divisor, from 44 to 33. The repeated remainder now tells us something that a short calculator screen cannot show fully.

Worked example 6: division that repeats

Question. Use division to write 13\frac13 as a decimal.

Plan. Track the remainders rather than guessing where the digits stop.

1÷3=0 remainder 1no whole units each10÷3=3 remainder 1three tenths each10÷3=3 remainder 1same step for hundredths\begin{aligned} 1\div3&=0\text{ remainder }1 &&\text{no whole units each}\\ 10\div3&=3\text{ remainder }1 &&\text{three tenths each}\\ 10\div3&=3\text{ remainder }1 &&\text{same step for hundredths} \end{aligned}

The remainder is 11 again, so every later step repeats.

Answer. 13=0.333…=0.3˙\frac13=0.333\ldots=0.\dot3, where the dot means that the 33 repeats for ever.

Check. 3331000<13<3341000\frac{333}{1000}<\frac13<\frac{334}{1000}, because multiplying by 33 gives 0.999<1<1.0020.999<1<1.002. The decimal begins 0.3330.333, but does not stop there.

A recurring decimal repeats a digit or block of digits for ever. Dots go above the first and last digit of the repeating block: 0.2˙7˙=0.272727…0.\dot2\dot7=0.272727\ldots. In 0.16˙0.1\dot6, only the 66 repeats. For division of whole numbers by a positive whole-number divisor, there are only finitely many possible remainders. Either zero arrives, or a remainder repeats and the decimal recurs. That is why a fraction cannot have an endless, non-repeating decimal expansion.

We can now answer the opening question: the remainders decide whether the decimal ends or repeats. Keep the recurring notation for an exact value; a finite rounded string is only an approximation.

Watch out

13≠0.33\frac13\ne0.33. The exact decimal is 0.3˙0.\dot3; 0.330.33 is a two-place approximation and is slightly smaller.

Watch for a repeated remainder in your own division.

Quick Recap

  • Each decimal place is one tenth of the place to its left; placeholder zeros matter.
  • Use power-of-ten denominators to convert terminating decimals and fractions.
  • A fraction is a division, with a nonzero denominator.
  • A zero remainder ends the decimal; a repeated remainder starts a recurring pattern.

0.4=0.400.4=0.40, and some fraction divisions continue for ever because the same remainder keeps returning.