Step-by-Step Guide to Converting Numbers Between Bases

Converting numbers between different bases, such as binary, decimal, hexadecimal, and octal, is an essential skill in programming and digital electronics. Here's a step-by-step guide to help you understand the process:

1. Understanding the Basics

- Binary (Base 2): Uses digits 0 and 1. Each digit represents a power of 2.
- Octal (Base 8): Uses digits 0 to 7. Each digit represents a power of 8.
- Decimal (Base 10): The standard number system we use, using digits 0 to 9.
- Hexadecimal (Base 16): Uses digits 0 to 9 and letters A to F (where A=10, B=11, ..., F=15). Each digit represents a power of 16.

2. Converting from Any Base to Decimal

To convert a number from any base to decimal, use the following formula:

Decimal = dn × bn + dn-1 × bn-1 + ... + d1 × b1 + d0 × b0

Where d represents each digit, and b represents the base. The position of each digit determines the power of the base.

Example: Convert the binary number 1011 to decimal.
1 × 23 + 0 × 22 + 1 × 21 + 1 × 20 = 8 + 0 + 2 + 1 = 11
So, 10112 = 1110.

3. Converting from Decimal to Any Base

To convert a decimal number to another base, repeatedly divide the decimal number by the new base and record the remainders. The base b digits are the remainders read in reverse order.

Example: Convert the decimal number 45 to binary.
45 ÷ 2 = 22 remainder 1
22 ÷ 2 = 11 remainder 0
11 ÷ 2 = 5 remainder 1
5 ÷ 2 = 2 remainder 1
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1
Reading the remainders in reverse, we get 101101.
So, 4510 = 1011012.

4. Converting Directly Between Non-Decimal Bases

To convert directly between two non-decimal bases (e.g., binary to hexadecimal), first convert to decimal, then to the target base. Alternatively, use groupings for bases that are powers of two:

- Binary to Hexadecimal: Group binary digits in sets of four, starting from the right. Convert each group to its hexadecimal equivalent.
- Example: Convert 101101112 to hexadecimal.
Grouping: 1011 (0xB) and 0111 (0x7) = 0xB716.

5. Practice and Use Tools

The best way to master base conversions is through practice. You can use online tools like our number base converter to check your work and understand the process better.