What is a hexadecimal number system?
The hexadecimal system is a positional numeral system with a base of 16. It uses sixteen individual symbols to represent values:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.
The letters correspond to the decimal values A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15.
It is widely used in computer science and digital electronics because it offers a compact representation of binary data.
Every four binary digits (bits) correspond directly to one hexadecimal digit, which simplifies reading and writing binary values.
Example of interpretation
For instance, the hexadecimal number 3F8₁₆ can be expanded as:
So 3F8₁₆ = 1016₁₀ in decimal form.
What is the octal number system?
The octal system is a base-8 numeral system, using the digits 0 through 7 to represent all possible values.
Each digit represents a power of eight, similar to how each digit in the decimal system represents a power of ten.
This system is particularly important in older computing systems and digital devices, where octal numbers were used to simplify binary input and output.
Example of interpretation
For the octal number 113₈, its decimal equivalent is found as:
Formula
To convert from hexadecimal to octal, follow a two-step process through the decimal system:
- Convert hexadecimal → decimal.
- Convert decimal → octal.
Step 1. Convert hexadecimal to decimal
where:
- is the numeric value of the hexadecimal digit (from 0 to 15),
- is the position index starting from 0 for the least significant digit.
Step 2. Convert decimal to octal
Divide the resulting decimal number repeatedly by 8, writing down each remainder until the quotient becomes 0. Then, read the remainders in reverse order to obtain the octal value.
Example
Let’s convert 4B₁₆ to the octal system.
Step 1. Convert 4B₁₆ → decimal
Each digit is expressed as a decimal value:
Then,
Step 2. Convert 75₁₀ → octal
Perform repeated division by 8:
| Division | Quotient | Remainder |
|---|---|---|
| 75 ÷ 8 | 9 | 3 |
| 9 ÷ 8 | 1 | 1 |
| 1 ÷ 8 | 0 | 1 |
Now write the remainders in reverse order: 113₈.
Thus,
Alternate method using binary
Take 4B₁₆:
- Convert each hex digit to binary:
- 4 → 0100
- B → 1011
So, 4B₁₆ = 01001011₂.
- Divide this binary number into groups of 3 bits (from right): 01001011 → 001 001 011 (adding leading zeros where needed to bring the value to a multiple of 3 bits).
- Convert each group to octal:
- 001 = 1
- 001 = 1
- 011 = 3
So, 01001011₂ = 113₈ (same result).
Conversion table of 4-bit groups
| Hexadecimal | Binary |
|---|---|
| 0 | 0000 |
| 1 | 0001 |
| 2 | 0010 |
| 3 | 0011 |
| 4 | 0100 |
| 5 | 0101 |
| 6 | 0110 |
| 7 | 0111 |
| 8 | 1000 |
| 9 | 1001 |
| A | 1010 |
| B | 1011 |
| C | 1100 |
| D | 1101 |
| E | 1110 |
| F | 1111 |
Conversion table of 3-bit groups
| Binary | Octal |
|---|---|
| 000 | 0 |
| 001 | 1 |
| 010 | 2 |
| 011 | 3 |
| 100 | 4 |
| 101 | 5 |
| 110 | 6 |
| 111 | 7 |
Notes
- To convert larger numbers more efficiently, you can skip the decimal step by using binary as an intermediate. Since each hexadecimal digit equals 4 binary bits, and each octal digit equals 3 binary bits, conversions can be made directly through binary grouping.
- The converter automatically handles these steps internally, giving you an accurate octal representation within seconds.
Frequently Asked Questions
How to convert hexadecimal number 1F₁₆ to octal step by step?
First, convert to decimal:
Now convert decimal 31 to octal:
31 ÷ 8 = 3 remainder 7,
3 ÷ 8 = 0 remainder 3.
Reverse the remainders: 37₈.
Can a hexadecimal number with a fraction be converted to octal?
Yes. Convert both the integer and fractional parts separately using the same principle. The integer part is divided by the base; the fractional part is multiplied by the new base.
Why are octal and hexadecimal systems important in computing?
Because they represent binary data in a compact, human-readable form. Octal groups bits in sets of three, and hexadecimal in sets of four, making them indispensable for programming, debugging, and digital circuit design.