From Zero to Engineer

Decimals, rounding and repeating decimals

Significant figures and decimal places, trailing zeros, converting between common fractions and decimals, and writing repeating decimals as common fractions.

A decimal is another way of writing the same number as a common fraction. This lesson covers rounding in two ways - to significant figures and to decimal places - and then converting between common fractions and decimals, including repeating decimals.

Decimals

25 : 8 = 3.125

Rounding decimal fractions

1) Significant figures

They are counted starting from the first non-zero digit beginning from the left-hand side of the number.

2) Decimal places

These places are counted to the right of the decimal point.

$9.4534$ → rounded to 2 significant figures is $9.5$

$9.4534$ → rounded to 3 significant figures is $9.45$

$0.001354$ → rounded to 2 significant figures is $0.0014$

$18.7249$ → rounded to 4 significant figures is $18.72$

$8.1265$ → rounded to 4 significant figures is $8.127$

$123.4467$ → rounded to 1 decimal places is $123.4$

$123.4467$ → rounded to 2 decimal places is $123.45$

$47.0235$ → rounded to 3 decimal places is $47.024$

Trailing zeros

$12645$ → rounded to 2 significant figures is $13000$

$13.1$ → rounded to 3 decimal places is $13.100$

$1515$ → rounded to 2 significant figures is $1500$

$25.13$ → rounded to 4 decimal places is $25.1300$

Common fractions as decimals and decimals as common fractions

\tfrac{7}{4} = 7 : 4 = 1.75

\tfrac{3}{8} = 3 : 8 = 0.375

1.224 = \tfrac{1224}{1000} = \tfrac{153}{125}

0.52 = \tfrac{52}{100} = \tfrac{13}{25}

Non-terminating decimals

\tfrac{1}{3} = 0.3333\ldots = 0{,}(3)

Read as: zero point three repeating.

\tfrac{2}{11} = 0.181818\ldots = 0{,}(18)

\tfrac{1}{7} = 0.142857142857142857\ldots = 0{,}(142857)

Non-terminating decimal fractions as common fractions

0{,}(18) = \tfrac{2}{11}

\begin{aligned}100 \cdot 0{,}(18) &= 18{,}(18) \\100 \cdot 0{,}(18) - 0{,}(18) &= 18{,}(18) - 0{,}(18) \\99 \cdot 0{,}(18) &= 18 \\0{,}(18) &= \tfrac{18}{99} = \tfrac{2}{11}\end{aligned}

0{,}(21) = \tfrac{7}{33}

\begin{aligned}100 \cdot 0{,}(21) &= 21{,}(21) \\99 \cdot 0{,}(21) &= 21 \\0{,}(21) &= \tfrac{21}{99} = \tfrac{7}{33}\end{aligned}

2{,}0(315) = 2\tfrac{7}{222}

\begin{aligned}2.0(315) &= 2 + 0.0(315) \\1000 \cdot 0.0(315) &= 31.5(315) \\999 \cdot 0.0(315) &= 31.5 \\0.0(315) &= \tfrac{31.5}{999} = \tfrac{315}{9990} = \tfrac{7}{222}\end{aligned}

315 : 45 = 7

9990 : 45 = 222

3{,}2(1) = 3\tfrac{19}{90}

\begin{aligned}3.2(1) &= 3.2 + 0.0(1) \\10 \cdot 0.0(1) &= 0.1(1) \\9 \cdot 0.0(1) &= 0.1 \\0.0(1) &= \tfrac{0.1}{9} = \tfrac{1}{90}\end{aligned}

3\tfrac{2}{10} + \tfrac{1}{90} = 3\tfrac{18}{90} + \tfrac{1}{90} = 3\tfrac{19}{90}

Frequently asked questions

What is the difference between significant figures and decimal places?

Significant figures are counted from the first non-zero digit, reading from the left. Decimal places are counted to the right of the decimal point. That is why 0.001354 has two significant figures as 0.0014, but rounds to 0.00 at two decimal places.

How do you turn a repeating decimal into a fraction?

Multiply the number by a power of ten so that one full period shifts, subtract the original number and divide by the difference. For 0.(18) that gives 99 · 0.(18) = 18, so 18/99, which simplifies to 2/11.

Where do trailing zeros in rounding come from?

They show the precision of the figure. 13.1 rounded to three decimal places is 13.100 - the zeros say the result is known to the nearest thousandth, even though its value has not changed.

← Previous

Ratios and percentages

Powers: multiplying, dividing and negative exponents