To convert the decimal number [tex]\(360_{10}\)[/tex] to its octal (base-8) equivalent, follow these steps:
1. Understand the relationship between decimal and octal systems:
- The decimal system is base-10, using digits from 0 to 9.
- The octal system is base-8, using digits from 0 to 7.
2. Find the highest power of 8 that fits into 360:
- The powers of 8 are [tex]\(8^0 = 1\)[/tex], [tex]\(8^1 = 8\)[/tex], [tex]\(8^2 = 64\)[/tex], [tex]\(8^3 = 512\)[/tex], etc.
- In this case, [tex]\(8^3 = 512\)[/tex] is too large, so we use [tex]\(8^2 = 64\)[/tex].
3. Divide 360 by the highest applicable power of 8:
- Divide 360 by 64 (which is [tex]\(8^2\)[/tex]): [tex]\(360 \div 64 = 5\)[/tex] remainder [tex]\(360 - (5 \times 64) = 40\)[/tex].
- So, [tex]\(5\)[/tex] is the coefficient for [tex]\(8^2\)[/tex].
4. Repeat for the next highest power of 8:
- Now we need to convert the remainder [tex]\(40\)[/tex].
- Divide 40 by 8 (which is [tex]\(8^1\)[/tex]): [tex]\(40 \div 8 = 5\)[/tex] remainder [tex]\(40 - (5 \times 8) = 0\)[/tex].
- So, [tex]\(5\)[/tex] is the coefficient for [tex]\(8^1\)[/tex].
5. Combine the coefficients:
- After dividing and considering remainders at each step, the coefficients from highest to lowest powers are used to form the final octal number.
Therefore, the octal equivalent of the decimal number [tex]\(360_{10}\)[/tex] is:
[tex]\[
360_{10} = 550_{8}
\][/tex]
