2.8: Operations in Different Bases
- Page ID
- 50993
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)First, let us set up the number lines. We plan to add in base six and then (separate exercise) base three. We are not adding mixed bases in this book, but yes, it is a thing. Google it.
Second, add it vertically. Rewrite the problem. Carry your numbers that are greater than or equal to 10, just like you did in elementary school. But count using the number (actually count it with your finger, swallow your pride and do it).
Example \(\PageIndex{1}\)
\[\begin{aligned}
5415_{\text {SIX}}+3042_{\text {SIX}} \\
\\
5415_{\text {six}} \\
+3042_{\text {six}} \\
\hline
12501_{\text {six}}
\end{aligned} \nonumber \]
Example \(\PageIndex{2}\)
\[\begin{aligned}
1202_{\text {THREE}}+1022_{\text {THREE}} \\
\\
1202_{\text {three}} \\
+1022_{\text {three}} \\
\hline
10001_{\text {three}}
\end{aligned} \nonumber \]
Now, we will subtract in different bases. For this example, we will use base seven and eight respectively. Below are the corresponding number lines. Again, use your fingers to count on the number line (everyone else is doing it). Borrow, when appropriate, like you did in elementary school.
Example \(\PageIndex{3}\)
\[\begin{aligned}
5631234_{\text {SEVEN}} - 2564310_{\text {SEVEN}} \\
\\
5631234_{\text {seven}} \\
-2564310_{\text {seven}} \\
\hline
3033624_{\text {seven}}
\end{aligned} \nonumber \]
Example \(\PageIndex{4}\)
\[\begin{aligned}
3741_{\text {EIGHT}}- 1465_{\text {EIGHT}} \\
\\
3741_{\text {eight}} \\
-1465_{\text {eight}} \\
\hline
2254_{\text {eight}}
\end{aligned} \nonumber \]
Partner Activity 1
Add or subtract the following numbers:
-
\(65_{\text {nine }}+41_{\text {nine }}=\)
-
\(123_{\text {four }}+301_{\text {four }}=\)
-
\(540_{\text{six}}-23_{\text{six}}=\)
-
\(220_{\text {three }}-122_{\text {three }}=\)
-
\(1001001_{\text{two}}-1011_{\text{two}}=\)
Practice Problems
Add or subtract the following expressions:
- \(435_{\text{six}}-31_{\text{six}}\)
- \(10010_{\text{two}}-111_{\text{two}}\)
- \(2100_{\text {three }}+21_{\text {three }}\)
- \(57643_{\text {eight }}+24677_{\text {eight }}\)