3.7: Operations in Different Bases

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).

Figure 2.8.1: Base Six Number Line

Example \(\PageIndex{1}\)

\[\begin{aligned}
5415_{\text {SIX}}+3042_{\text {SIX}} \\
\\
5415_{\text {six}} \\
+3042_{\text {six}} \\
\hline
12501_{\text {six}}
\end{aligned} \nonumber \]

Figure 2.8.2: Base Three Number Line

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.

Figure 2.8.3: Base Seven Number Line

Example \(\PageIndex{3}\)

\[\begin{aligned}
5631234_{\text {SEVEN}} - 2564310_{\text {SEVEN}} \\
\\
5631234_{\text {seven}} \\
-2564310_{\text {seven}} \\
\hline
3033624_{\text {seven}}
\end{aligned} \nonumber \]

Figure 2.8.4: Base Eight Number Line

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:

1. \(65_{\text {nine }}+41_{\text {nine }}=\)

2. \(123_{\text {four }}+301_{\text {four }}=\)

3. \(540_{\text{six}}-23_{\text{six}}=\)

4. \(220_{\text {three }}-122_{\text {three }}=\)

5. \(1001001_{\text{two}}-1011_{\text{two}}=\)

Practice Problems

Add or subtract the following expressions:

1. \(435_{\text{six}}-31_{\text{six}}\)
2. \(10010_{\text{two}}-111_{\text{two}}\)
3. \(2100_{\text {three }}+21_{\text {three }}\)
4. \(57643_{\text {eight }}+24677_{\text {eight }}\)