The radix/base of octal number system is -
8
Understanding the Radix of Number Systems
The question asks about the radix or base of the octal number system. The radix, or base, of a number system defines the number of unique digits used to represent numbers in that system. It also determines the place value of each digit in a number.
What is the Octal Number System?
The octal number system is a positional numeral system that uses eight distinct digits. These digits are 0, 1, 2, 3, 4, 5, 6, and 7.
In any number system, the place value of each digit is a power of the radix. For example, in the decimal system (base 10), the number 123 is interpreted as:
$$1 \times 10^2 + 2 \times 10^1 + 3 \times 10^0$$Similarly, in the octal system, a number like $173_8$ would be interpreted using powers of the octal base:
$$1 \times \text{Base}^2 + 7 \times \text{Base}^1 + 3 \times \text{Base}^0$$Identifying the Radix of the Octal System
Since the octal number system uses eight unique digits (0 through 7), its radix or base is 8. This means that each position in an octal number represents a power of 8.
Let's look at the options provided:
- Option 1: 4 - This is the radix of the quaternary number system.
- Option 2: 8 - This aligns with the definition of the octal number system.
- Option 3: 2 - This is the radix of the binary number system.
- Option 4: 10 - This is the radix of the decimal number system.
Therefore, the radix/base of the octal number system is 8.
Comparing Different Number System Bases
It's helpful to compare the octal system with other common number systems:
| Number System | Radix/Base | Digits Used | Example Number (Base indicated by subscript) |
|---|---|---|---|
| Binary | 2 | 0, 1 | $1011_2$ |
| Octal | 8 | 0, 1, 2, 3, 4, 5, 6, 7 | $375_8$ |
| Decimal | 10 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 | $987_{10}$ |
| Hexadecimal | 16 | 0-9, A-F | $A3F_{16}$ |
This table clearly shows that the octal system corresponds to a radix of 8.
Conclusion
Based on the definition and characteristics of the octal number system, its radix or base is 8. This is because it uses eight unique digits (0-7) to represent numbers.
Revision Table: Octal Number System Radix
| Concept | Description | Value for Octal System |
|---|---|---|
| Radix / Base | Number of unique digits in the system | 8 |
| Digits Used | The set of unique digits | 0, 1, 2, 3, 4, 5, 6, 7 |
| Place Value | Powers of the radix | $\dots, 8^2, 8^1, 8^0$ |
Additional Information on Number Systems
Number systems are fundamental concepts in mathematics and computer science. They provide a way to represent numerical values using a set of symbols and rules.
- Positional Systems: Most common number systems (like decimal, binary, octal, hexadecimal) are positional, meaning the value of a digit depends on its position within the number.
- Base Conversion: It is possible and often necessary to convert numbers between different bases, such as converting an octal number to a decimal number or vice versa.
- Octal in Computing: The octal system was historically used in computing as a compact way to represent binary numbers, as three binary digits map directly to one octal digit ($2^3=8$). While less common now than hexadecimal, it is still encountered in some contexts.
Important Questions from Number System
Consider the following statements :
1. (25)! + 1 is divisible by 26
2. (6)! + 1 is divisible by 7
Which of the above statements is/are correct ?
If the sum S is divided by 8, what is the remainder ?
If the sum S is divided by 60, what is the remainder ?
Find the sum of squares of the greatest value and the smallest value of K in the number so that the number 45082K is divisible by 3.
How many composite numbers are there from 53 to 97 ?
