Base Converter — Free Online Number Base Converter

How to Use the Base Converter

1. Enter the number: Type the number you want to convert in the input field. Use only valid digits for the selected source base. For binary, use 0 and 1 only. For hexadecimal, use 0-9 and A-F. The input is case-insensitive for letter digits.
2. Select the source base: Choose the base of the number you entered from the "From Base" dropdown. Common options include binary (2), octal (8), decimal (10), and hexadecimal (16). For other bases, select "Custom Base" and enter any value from 2 to 36.
3. Select the target base: Choose the base you want to convert to from the "To Base" dropdown. This supports the same common bases and custom option as the source.
4. Review the results: The main result shows your number in the target base. Below that, the conversion steps explain the intermediate decimal conversion if applicable. The "All Common Bases" section shows the number in bases 2, 8, 10, 16, 32, and 36 simultaneously, which is useful for quick cross-reference.

The conversion happens in real time as you type. If the input contains invalid digits for the selected base (like the digit 3 in binary), the result area will indicate invalid input. This helps you quickly correct any mistakes.

Base Conversion Formulas

Variables Explained

• b (Base/Radix): The number of unique digits in the number system. Decimal has base 10, binary has base 2, and hexadecimal has base 16. The base determines the positional value of each digit.
• d (Digit): Each position in a number holds a digit value from 0 to (base - 1). In hexadecimal, digits include 0-9 and A-F (where A=10 through F=15). In base 36, digits extend through Z=35.
• n (Position): The position of each digit, counted from right to left starting at 0. The rightmost digit is at position 0, the next at position 1, and so on. Each position represents b^n in value.

Step-by-Step Example

Convert 1A3 from hexadecimal (base 16) to binary (base 2):

1. Convert each hex digit to decimal: 1=1, A=10, 3=3
2. Calculate decimal value: 1 × 16^2 + 10 × 16^1 + 3 × 16^0 = 256 + 160 + 3 = 419
3. Convert 419 to binary by repeated division: 419/2=209 r1, 209/2=104 r1, 104/2=52 r0, 52/2=26 r0, 26/2=13 r0, 13/2=6 r1, 6/2=3 r0, 3/2=1 r1, 1/2=0 r1
4. Read remainders from bottom to top: 110100011

Alternatively, you can convert each hex digit directly to its 4-bit binary equivalent: 1=0001, A=1010, 3=0011, giving 000110100011 or 110100011 without leading zeros. This shortcut works because 16 is a power of 2.

Practical Examples

Example 1: Web Developer Color Code Conversion

Tom, a web developer, needs to understand the RGB values of the CSS color #3CB371 (Medium Sea Green). He uses the base converter to convert each hex pair to decimal:

• Red: 3C (hex) = 3 × 16 + 12 = 60 (decimal)
• Green: B3 (hex) = 11 × 16 + 3 = 179 (decimal)
• Blue: 71 (hex) = 7 × 16 + 1 = 113 (decimal)

So #3CB371 equals RGB(60, 179, 113). Understanding this conversion helps Tom adjust colors programmatically and communicate precise color values to designers who may prefer decimal notation.

Example 2: Networking IP Address Analysis

Aisha, a network engineer, needs to calculate the subnet mask for a /20 CIDR network. She knows the subnet mask has 20 ones followed by 12 zeros in binary. Using the converter to convert each octet:

• First octet: 11111111 (binary) = 255 (decimal)
• Second octet: 11111111 (binary) = 255 (decimal)
• Third octet: 11110000 (binary) = 240 (decimal)
• Fourth octet: 00000000 (binary) = 0 (decimal)

The subnet mask is 255.255.240.0. Binary-to-decimal conversion is essential in networking for subnetting, CIDR calculations, and understanding IP addressing schemes.

Example 3: Computer Science Homework

Kevin, a computer science student, needs to convert the decimal number 1000 to bases 2, 8, and 16 for an assignment. Using the base converter:

• Decimal 1000 → Binary: 1111101000
• Decimal 1000 → Octal: 1750
• Decimal 1000 → Hexadecimal: 3E8

Kevin can verify each conversion by converting back to decimal. 3E8 in hex = 3 × 256 + 14 × 16 + 8 = 768 + 224 + 8 = 1000. The "All Common Bases" feature shows all conversions simultaneously, saving time on multi-base assignments. For handling numbers in word form, try our number to words converter.

Example 4: Unix File Permissions

Laura, a system administrator, sets file permissions using chmod 755 and needs to understand what each digit means. She converts each octal digit to binary:

• Owner (7): 111 in binary = read + write + execute
• Group (5): 101 in binary = read + execute
• Others (5): 101 in binary = read + execute

Each octal digit represents a 3-bit binary value encoding read (4), write (2), and execute (1) permissions. Understanding octal-to-binary conversion is essential for Unix and Linux system administration.

Base Conversion Reference Table

Tips and Complete Guide

Quick Binary-Hex Conversion Shortcut

Because hexadecimal base 16 is 2^4, each hex digit maps exactly to four binary digits. Memorize the 16 mappings (0=0000, 1=0001, ... 9=1001, A=1010, B=1011, C=1100, D=1101, E=1110, F=1111) and you can convert instantly between hex and binary without going through decimal. Group binary digits in sets of four from right to left for binary-to-hex. This is the fastest way to read memory dumps, analyze network packets, or debug machine code.

Powers of Two Every Programmer Should Know

Knowing key powers of 2 accelerates base conversion and programming: 2^8 = 256 (one byte), 2^10 = 1,024 (1 KB), 2^16 = 65,536, 2^20 = 1,048,576 (1 MB), 2^32 = 4,294,967,296 (max 32-bit unsigned integer), and 2^64 = 18,446,744,073,709,551,616 (max 64-bit unsigned integer). These values appear constantly in memory allocation, data type limits, array sizing, and performance optimization. For scientific number representation, see our scientific notation calculator.

Base Conversion in Real Applications

Beyond computer science, base conversion appears in many areas. ISBN numbers use a base-11 check digit. Airline booking codes use base 36 (0-9, A-Z). Ancient Babylonians used base 60, which is why we have 60 seconds in a minute and 60 minutes in an hour. The Mayan civilization used base 20. Understanding different bases provides insight into both modern technology and the history of mathematics.

Common Mistakes to Avoid

• Using invalid digits for the base: Each base only allows digits from 0 to (base-1). Binary only uses 0 and 1 — the digit 2 is invalid in binary. Octal uses 0-7 — the digits 8 and 9 are invalid. Always verify your input digits are valid for the source base.
• Confusing the direction of conversion: When converting binary to hex, group digits from right to left, not left to right. 10110 grouped as 1-0110 gives 16 in hex, but incorrectly grouped as 1011-0 gives B0, which is wrong.
• Forgetting that hex letters represent numbers: In hexadecimal, A=10, B=11, C=12, D=13, E=14, F=15. The hex number "10" equals decimal 16, not decimal 10. This confusion is especially common for beginners.
• Ignoring case sensitivity: While most systems treat hex letters as case-insensitive (A=a=10), some contexts are case-sensitive. Our converter accepts both cases, but be consistent in your own notation.
• Dropping leading zeros: When converting hex to binary for bitwise operations, maintaining leading zeros in each 4-bit group (0001 instead of just 1) ensures proper alignment and prevents bit-shifting errors.

Frequently Asked Questions