1.4: An Introduction to Measurements and Rounding
- Page ID
- 509523
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)You've probably used a ruler or measure tape to record lengths, widths, and heights of different objects. Not all rulers are created equal, though. Let's say we wanted to measure the piece of wood shown in Figure \(\PageIndex{1}\) with one of the three rulers shown.
Figure \(\PageIndex{1}\): A piece of wood (a) is measured using three different rules. Ruler (b) is the least precise ruler (there are no marks between 1 and 8 cm); ruler (b) is better (there are 1 cm marks between 0 and 8 cm); and ruler (c) is even better, with each centimeter broken up into 0.1 cm marks. (This work is licensed under CC BY-NC 4.0 by Jason D'Acchioli)
Let's use ruler (b), which has measurement marks on it spaced every 1 cm. Using this ruler, we can say the ruler is 4 cm, plus a little extra. It's not quite halfway between 4 and 5 cm, so a good estimation is 4.2 cm.
Now let's use ruler (c), an even better ruler, with even more measurement marks on it. The marks between the 1 cm increments represent 0.1 cm each. Notice that the piece of wood is between 4.3 and 4.4 cm (so it's definitely 4.3 cm!), but closer to 4.3 cm than 4.4 cm. A good estimation of the length is 4.31 cm.
Each ruler was getting better and better in terms of its measuring abilities. Ruler (a) was the worst of the three; we could only approximate its length relative to 0 and 8 cm. Ruler (b), which is slightly better, measured the length as 4.2 cm. Ruler (c) is the best of the three rulers, as we could measure the wood to 4.31 cm. Our goal as chemists is to use the best possible tool to make measurements, and to make sure that we’re reporting those measurements to people looking at data.
Sometimes, making measurements requires rounding numbers to different places (the thousands place, hundreds place, tens place, ones place, tenths place, hundredths place, etc.). To do that, you need to be comfortable with looking at number place values. The following 6 videos review place values, as well as the rules for rounding.
Video 1. Rounding whole numbers, example 1.
Video \(\PageIndex{1}\): Rounding whole numbers.
Video 2. Rounding whole numbers, example 2.
Video \(\PageIndex{2}\): Rounding whole numbers.
Video 3. Rounding to the nearest tens, hundreds, and thousands place.
Video \(\PageIndex{3}\): Rounding to whole numbers.
Video 4. Rounding whole numbers when missing a digit.
Video \(\PageIndex{4}\): Rounding to whole numbers when missing a digit.
Video 5. Rounding decimals to the nearest tenth.
Video \(\PageIndex{5}\): Rounding to the nearest tenths place.
Video 6. Rounding decimals: A number line approach.
Video \(\PageIndex{6}\): Rounding decimals using a number line.
And don’t worry—you’ll have plenty of time to practice when we do some examples involving rounding in class!