3.5: Converting Between (our) Base 10 and Any Other Base (and vice versa)

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Page ID: 51822

Amy Lagusker
College of the Canyons

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To convert any number in (our base) Base 10 to any other base, we must use basic division with remainders. Do not divide using decimals; the algorithm will not work.

Example \(\PageIndex{1}\)

Convert from (our) Base 10 to (weird) Base _____

Change \(236_{\text {ten}}\) to ______ \(_{\text {five}}\)

Solution

Keep dividing by 5, until your quotient is zero.

\[\begin{aligned}
236 \div 5 &=47 \; r \; \mathbf{1} \\
47 \div 5 &=9 \; r \; \mathbf{2} \\
9 \div 5 &=1 \; r \; \mathbf{4} \\
1 \div 5 &=0 \; r \; \mathbf{1}
\end{aligned} \nonumber \]

Now write your remainders backwards!

Answer: \(1421_{\text {five}}\)

Example \(\PageIndex{2}\)

Convert from (weird) Base ____ to (our) Base 10.

Solution

First, notice how to break down \(602_{\text {ten}}\):

\[602_{\text {ten }}: 602=6\left(10^{2}\right)+0\left(10^{1}\right)+2\left(10^{0}\right) \nonumber \]

Now, use the same approach to change \(602_{\text {eight}}\) into Base 10

\[6\left(8^{2}\right)+0\left(8^{1}\right)+2\left(8^{0}\right)=386_{\text {ten}} \nonumber \]

Example \(\PageIndex{3}\)

Convert \(5361_{\text {seven}}\) into Base 10.

Solution

\[5\left(7^{3}\right)+3\left(7^{2}\right)+6\left(7^{1}\right)+1\left(7^{0}\right)=1905_{\text {ten }} \nonumber \]

Partner Activity 1

Convert the base 10 numbers into base 4
1. \(30_{\text {ten}}\) = _____ \(_{\text {four}}\)
2. \(2103_{\text {ten}}\) = _____ \(_{\text {four}}\)
3. \(16_{\text {ten}}\) = _____ \(_{\text {four}}\)
Convert the base 5 numbers into base 10
1. \(30_{\text {five}}\) = ______ \(_{\text {ten}}\)
2. \(2103_{\text {five}}\) = ______\(_{\text {ten}}\)
3. \(16_{\text {five}}\) = ______ \(_{\text {ten}}\)

Think carefully about 2c!

***For extra practice, click here.

Practice Problems

Write the following Base 10 numbers into the new Base.
1. 5567 into Base 9
2. 12 into Base 4
3. 100 into Base 3
4. 73 into Base 2
Write the following numbers into Base 10.
1. \(64_{\text {seven}}\)
2. \(157_{\text {eight}}\)
3. \(1001001_{\text {two}}\)
4. \(84671_{\text {eleven}}\)