Decimal to Other Bases

The Decimal to Other Bases Converter allows you to convert numbers represented in the decimal numeral system (base-10) into equivalent representations in different numeral systems or bases. You provide a decimal number as input, including integer and fractional parts, and select the target base or numeral system to which you want to convert the decimal. You can choose from a range of typical bases, including binary (base-2), octal (base-8), hexadecimal (base-16), and various other bases up to 36.

Decimal to Other Bases Converter

Decimal Number

base₁₀

Target Base

Custom Number of Digits

Results:

Base Output
Complement Output

Calculator

enter a decimal number
from the drop-down menu, select the base to which you want to convert the decimal number
Custom Number of Digits: If you do not check the "Custom Number of Digits" checkbox, the converter will automatically calculate the minimum number required to represent the complement of the number in the selected base. If you check the "Custom Number of Digits" checkbox, you can manually specify the number of digits used for the complement calculation.

Results

the equivalent representation of the decimal number in the chosen base
the complement in the calculated base (is the same as the "base result" with a positive decimal input)

Two's Complement

Two's complement is typically discussed in the context of binary (base 2) numbers because it is a standard method for representing signed integers in digital systems. However, the two complements can be extended to other bases, though they are less common in practice. Here's how it works:

Two's Complement in Other Bases

General Steps:

Positive Numbers: Any base's representation of positive numbers remains the same.
Negative Numbers:
- Start with the absolute value of the number in the desired base.
- Invert all the digits (find each digit's "base complement").
- Add 1 to the least significant digit (the rightmost digit).

Example in Base 10 (Decimal):
Convert -47 to two's complement in a base 10 system using 3 digits (which is uncommon but useful for illustration):

Absolute value: The representation of 47 in base 10 is 047 (we use leading zeros to fit into 3 digits).
Invert the digits: Subtract each digit from 9 (since 10 - 1 = 9).
- 9 - 0 = 9
- 9 - 4 = 5
- 9 - 7 = 2
- So, 047 becomes 952.
Add 1: Add 1 to the least significant digit:
- 952 + 1 = 953.

So, the two's complement representation of -47 in a 3-digit base 10 system is 953.

Example in Base 16 (Hexadecimal):

Convert -26 to two's complement in a 2-digit base 16 system:

Absolute value: The representation of 26 in hexadecimal is 1A.
Invert the digits: Subtract each digit from F (since 16 - 1 = F in hexadecimal).
- F - 1 = E
- F - A = 5
- So, 1A becomes E5.
Add 1: Add 1 to the least significant digit:
- E5 + 1 = E6.

So, the two's complement representation of -26 in a 2-digit base 16 system is E6.

Key Points:

Base Complement: For any base B, the complement of a digit d is B-1-d.
Carry in Addition: When adding 1, if there's a carry-out from the most significant digit, it wraps around (like in modular arithmetic).
Common Use: While possible, two's complement is primarily used in binary systems due to its direct alignment with digital hardware and ease of arithmetic operations.

Decimal to Other Bases