To convert the decimal number 4054 to binary by first converting it to octal, follow these steps:
1. Convert the decimal number to octal:
- First, we need to transform the decimal number 4054 into an octal number.
- The octal representation of the decimal number 4054 is 7726.
2. Convert the octal number to binary:
- Next, convert the octal number 7726 directly into its binary equivalent.
- Each octal digit can be represented as a 3-bit binary number:
- [tex]\(7_{8} = 111_{2}\)[/tex]
- [tex]\(7_{8} = 111_{2}\)[/tex]
- [tex]\(2_{8} = 010_{2}\)[/tex]
- [tex]\(6_{8} = 110_{2}\)[/tex]
- Put together the binary equivalents of each octal digit:
- [tex]\(7_{8} \rightarrow 111_{2}\)[/tex]
- [tex]\(7_{8} \rightarrow 111_{2}\)[/tex]
- [tex]\(2_{8} \rightarrow 010_{2}\)[/tex]
- [tex]\(6_{8} \rightarrow 110_{2}\)[/tex]
- Combining these binary values gives:
- [tex]\(111 \, 111 \, 010 \, 110\)[/tex]
Therefore, the binary representation of the decimal number 4054 is [tex]\(111111010110\)[/tex].