Perform the following octal additions. a. 770 + 665 b. 101 + 707 c. 202 + 667
The Correct Answer and Explanation is :
Octal Addition Solutions
a. 7708+6658770_{8} + 665_{8}
- Add the digits from right to left:
- Units place: 0+5=50 + 5 = 5.
- Tens place: 7+6=15107 + 6 = 15_{10}. In octal, 151015_{10} becomes 17817_{8}. Write 77 and carry 11.
- Hundreds place: 7+6+1=14107 + 6 + 1 = 14_{10}. In octal, 141014_{10} becomes 16816_{8}. Write 66 and carry 11.
b. 1018+7078101_{8} + 707_{8}
- Add the digits from right to left:
- Units place: 1+7=8101 + 7 = 8_{10}. In octal, 8108_{10} becomes 10810_{8}. Write 00 and carry 11.
- Tens place: 0+0+1=10 + 0 + 1 = 1.
- Hundreds place: 1+7=10101 + 7 = 10_{10}. In octal, 101010_{10} becomes 12812_{8}. Write 22 and carry 11.
c. 2028+6678202_{8} + 667_{8}
- Add the digits from right to left:
- Units place: 2+7=9102 + 7 = 9_{10}. In octal, 9109_{10} becomes 11811_{8}. Write 11 and carry 11.
- Tens place: 0+6+1=70 + 6 + 1 = 7.
- Hundreds place: 2+6=8102 + 6 = 8_{10}. In octal, 8108_{10} becomes 10810_{8}. Write 00 and carry 11.
Explanation
Octal addition works similarly to decimal addition but is based on base 88. If a sum exceeds 77, it “wraps around” (like in decimal with 1010), and the remainder is written, while the quotient becomes a carry for the next higher place.
- Convert to Decimal (Optional): For verification, converting octal numbers to decimal simplifies checking correctness.
- Carry Over: The key difference from decimal arithmetic is that the carry occurs at 88, not 1010.
Benefits of Octal
Octal representation is often used in computing systems as a shorthand for binary, grouping bits in sets of three, simplifying human interaction with binary data.