Introduction to Octal Numbers

By this point you should be quite familiar with binary (and obviously decimal) numbers. Octal is going to work the same but instead of 0 and 1 we have 0 to 7 for each digit. Likewise each place value is $$2^n$$ where $$n$$ is the position (starting at 0 for rightmost digit).
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The place value for the nth digit from the right of an octal number (n beginning at 0) is given by $$2^n$$.
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Octal numbers are written with the subscript $$_8$$ when it might not be clear what base the number is in.
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The first 6 place values for octal numbers are:
Place543210
Value3276840965126481
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To convert an octal number to decimal just multiply each digit by its place value and add the results.
example

Convert $$23_8$$ to decimal

Breaking this number into place values: $$23_8 = 2\times 8 + 3\times 1 = 16 + 3 = 19_{10}$$
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Convert $$101_8$$ to decimal

\begin{align} 101_8 & = 1\times 64 + 0\times 8 + 1\times 1 \\ & = 65_{10} \\ \end{align}
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To convert a decimal number to octal:
1. Divide decimal number by 8
2. Write remainder of division as next octal digit (moving right to left)
3. Repeat Step 1 with the integer result of the division until we're left with 0
example

Convert $$100_{10}$$ to octal.

It's usually easiest to do this in a table as you work: I like to call this the SOAR table which stands for: S: Step O: Operation A: Answer R: Remainder
SOAR
1$$\frac{100}{8}$$124
2$$\frac{12}{8}$$14
3$$\frac{1}{8}$$01
Now the remainder at the top is the LSB and the remainder at the bottom is the MSB so our answer is: $$100_{10} = 144_8$$
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To convert an octal number to binary first convert it to decimal and then convert that decimal number into binary
example

Convert $$23_8$$ into binary

As we found above $$23_8 = 19_{10}$$ Now we just convert $$19_{10}$$ to binary to get: $$19_{10} = 10011_2$$ So we get: $$23_8 = 10011_2$$
There's a handy quick little shortcut to converting an octal number into binary. It's not necessary to know it but it can save you some valuable time when you're taking a test.
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As a shortcut to convert an octal number to binary convert each octal digit to a 3-bit binary number, placing them next to each other in order will give you the binary value of the number.
example

Convert $$240_8$$ to binary using the shortcut method.

We'll right out the 3-digit binary value of each digit below the number: \begin{align} \phantom{0}2\phantom{00}4\phantom{0\,0}0_8\, & \\ 010\;100\;000 & \end{align} And we can read the answer right off of this: $$240_8 = 010100000_2$$
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A shortcut to convert binary to octal is to:
1. Divide binary number into groups of 3 (starting from right)
2. Convert each 3-digit binary number to an octal digit (it'll be the same as the decimal value)
3. Write these binary digits in order
example

Convert $$10101_2$$ to octal using the shortcut method.

Our two groups are $$010$$ and $$101$$ (we added a leading zero to the leftmost group to make it 3-digits) Now we convert each of these two groups to octal. $$010_2 = 2_8$$ $$101_2 = 5_8$$ So our octal number is: $$10101_2 = 25_8$$
practice problems