The octal number 73 is equivalent to the decimal number:
(a) 47
(b) 59
(c) 111
The Correct Answer and Explanation is:
Correct Answer: (b) 59
To convert an octal number to its decimal equivalent, you must understand the base system involved. Octal is a base-8 number system, meaning each digit can range from 0 to 7. In contrast, decimal is a base-10 number system, the standard system we use every day.
Let’s convert the octal number 73 to decimal:
The octal number “73” consists of two digits:
- The leftmost digit is 7 (which is in the 8¹ position)
- The rightmost digit is 3 (which is in the 8⁰ position)
To convert to decimal, multiply each digit by 8 raised to the power of its position (starting from the right with position 0), and then sum the results:
$$
\text{Decimal equivalent} = (7 \times 8^1) + (3 \times 8^0)
$$
$$
= (7 \times 8) + (3 \times 1)
$$
$$
= 56 + 3 = \boxed{59}
$$
So, the octal number 73 is equal to the decimal number 59.
Why This Works
Each position in an octal number represents a power of 8, just as each position in a decimal number represents a power of 10. For example:
- In decimal: 53 = (5 × 10¹) + (3 × 10⁰) = 50 + 3 = 53
- In octal: 73 = (7 × 8¹) + (3 × 8⁰) = 56 + 3 = 59
This positional value system applies to all number bases.
Conclusion
Option (b) 59 is the correct answer. The conversion of base-8 (octal) to base-10 (decimal) involves multiplying each digit by powers of 8 and summing the results. Understanding base conversions is critical in computer science and digital electronics, where different base systems like binary, octal, and hexadecimal are frequently used.
