To convert the octal number [tex]\(133_8\)[/tex] to its binary equivalent, we can follow a step-by-step process:
1. Convert the octal number to its decimal (base-10) equivalent.
Each digit in an octal number represents a power of 8. The octal number [tex]\(133_8\)[/tex] can be expanded as follows:
[tex]\[
133_8 = 1 \cdot 8^2 + 3 \cdot 8^1 + 3 \cdot 8^0
\][/tex]
Compute each term:
[tex]\[
1 \cdot 8^2 = 1 \cdot 64 = 64
\][/tex]
[tex]\[
3 \cdot 8^1 = 3 \cdot 8 = 24
\][/tex]
[tex]\[
3 \cdot 8^0 = 3 \cdot 1 = 3
\][/tex]
Add these values together to get:
[tex]\[
64 + 24 + 3 = 91
\][/tex]
Therefore, the decimal equivalent of [tex]\(133_8\)[/tex] is [tex]\(91_{10}\)[/tex].
2. Convert the decimal number [tex]\(91_{10}\)[/tex] to its binary (base-2) equivalent.
To convert from decimal to binary, we repeatedly divide the number by 2 and record the remainders. Starting with 91:
[tex]\[
91 \div 2 = 45 \quad \text{remainder} \, 1
\][/tex]
[tex]\[
45 \div 2 = 22 \quad \text{remainder} \, 1
\][/tex]
[tex]\[
22 \div 2 = 11 \quad \text{remainder} \, 0
\][/tex]
[tex]\[
11 \div 2 = 5 \quad \text{remainder} \, 1
\][/tex]
[tex]\[
5 \div 2 = 2 \quad \text{remainder} \, 1
\][/tex]
[tex]\[
2 \div 2 = 1 \quad \text{remainder} \, 0
\][/tex]
[tex]\[
1 \div 2 = 0 \quad \text{remainder} \, 1
\][/tex]
Reading the remainders from bottom to top, we get the binary number:
[tex]\(
1011011_2
\)[/tex]
Therefore, the binary equivalent of the octal number [tex]\(133_8\)[/tex] is:
[tex]\[
133_8 = 1011011_2
\][/tex]
