Hexadecimal Converter

Hexadecimal (base-16) exists in computing because it's the most human-readable compact representation of binary data. Sixteen symbols — 0 through 9 and A through F — map perfectly onto groups of 4 binary bits (a nibble), making each hex digit an exact representation of 4 bits. A byte (8 bits) becomes exactly two hex digits. A 32-bit integer becomes exactly 8 hex digits. This clean mapping between binary and hexadecimal is why hex appears everywhere computers need to display binary data to humans: memory addresses, color codes, cryptographic hashes, machine code, file signatures, MAC addresses, and error codes all use hex because it's far more readable than the equivalent binary while maintaining an exact, lossless relationship to the underlying bits.

The Binary-Hex Bridge

The most important property of hexadecimal: each hex digit corresponds exactly to 4 binary bits, making conversion between the two trivial once you memorize the 16 hex-to-binary mappings. Hex 0-9: 0000-1001. A=1010, B=1011, C=1100, D=1101, E=1110, F=1111.

Convert binary to hex by grouping bits in sets of 4 from the right. Binary 101110011010₂: Group: 1011 1001 1010. Convert each group: 1011=B, 1001=9, 1010=A. Result: B9A₁₆. The reverse — hex to binary — just expand each hex digit to its 4-bit equivalent. 7B3₁₆: 7=0111, B=1011, 3=0011. Binary: 011110110011₂. This direct mapping is why programmers use hex to represent binary data rather than decimal — converting between hex and binary is mechanical, while decimal-to-binary conversion requires division and remainders.

The Hexadecimal Number System

Hex uses 16 symbols where decimal uses 10. The symbols 0-9 represent the same values as in decimal. The letters A through F extend the count: A=10, B=11, C=12, D=13, E=14, F=15. And like all positional systems, each position represents a power of the base: 16^0=1, 16^1=16, 16^2=256, 16^3=4,096, 16^4=65,536.

Hexadecimal numbers are typically prefixed with 0x (in programming contexts: 0xFF) or subscripted with 16 (FF₁₆) to distinguish them from decimal numbers. The value FF₁₆: (15 × 16) + (15 × 1) = 240 + 15 = 255₁₀. The maximum value of one byte: two hex digits, both at their maximum (FF), equals 255 in decimal — the well-known byte maximum.

Converting Decimal to Hexadecimal

Use repeated division by 16, recording the remainder (as a hex digit) at each step. Convert 58,620₁₀ to hex: 58,620 ÷ 16 = 3,663 remainder 12 (C). 3,663 ÷ 16 = 228 remainder 15 (F). 228 ÷ 16 = 14 remainder 4. 14 ÷ 16 = 0 remainder 14 (E). Reading remainders bottom to top: E4FC₁₆. Verify: (14 × 4,096) + (4 × 256) + (15 × 16) + 12 = 57,344 + 1,024 + 240 + 12 = 58,620₁₀. Correct.

Memory Addresses and Debugging

Jason, 29, a software developer in Austin, Texas, uses hex constantly while debugging. Memory addresses in 64-bit systems like 0x7FFD4B2C8A10 are 48-bit addresses (12 hex digits = 48 bits). CPU error codes like 0x0000007B (Windows stop code for inaccessible boot device) are hex values identifying specific error conditions in the Windows kernel. File signatures — the "magic bytes" at the start of files that identify their format — are documented in hex: JPEG files start with FF D8 FF, PNG files with 89 50 4E 47 (which is 89 PNG in ASCII), PDF files with 25 50 44 46 (which is %PDF in ASCII). Knowing how to read hex makes these identifying patterns immediately decipherable rather than opaque sequences of numbers.

Network MAC addresses are written as six hex byte pairs separated by colons or hyphens: 00:1A:2B:3C:4D:5E. Each pair represents one byte of the 6-byte hardware address burned into every network interface. The first three bytes identify the manufacturer (organizationally unique identifier, OUI); the last three are the device's unique serial. Converting any pair to decimal — 1A₁₆ = 26₁₀, 2B₁₆ = 43₁₀ — immediately shows the numerical value of each byte in the address.

Converting Hexadecimal to Decimal

Multiply each hex digit by its positional power of 16 and sum. Convert 3A7F₁₆ to decimal: F = 15, 7 = 7, A = 10, 3 = 3. Position values: 16^0=1, 16^1=16, 16^2=256, 16^3=4,096. Calculation: (3 × 4,096) + (10 × 256) + (7 × 16) + (15 × 1) = 12,288 + 2,560 + 112 + 15 = 14,975₁₀.

A useful shortcut for larger hex numbers: convert each hex digit to 4 binary bits, concatenate all bits, then convert the full binary number to decimal. This is often faster than computing powers of 16 for numbers larger than 4 hex digits. For 3A7F₁₆: 3=0011, A=1010, 7=0111, F=1111. Binary: 0011 1010 0111 1111 = 0011101001111111₂. Binary to decimal: 2^13+2^11+2^9+2^6+2^5+2^4+2^3+2^2+2^1+2^0 = 8192+2048+512+64+32+16+8+4+2+1 = 14,879. Wait — let me recompute: 00111010 01111111. 0+0+8192+4096+2048+0+512+0 + 0+64+32+16+8+4+2+1 = 14,975₁₀. Confirmed.

Color Codes: Hex in Everyday Design

CSS and HTML color codes are the most widely encountered hexadecimal outside of programming. A color like #4A9FD2 encodes three bytes of RGB color: 4A = red, 9F = green, D2 = blue. 4A₁₆ = 74₁₀ means the red channel is 74 out of a possible 255. 9F₁₆ = 159₁₀ means the green channel is 159 out of 255. D2₁₆ = 210₁₀ means the blue channel is 210 out of 255. This is a medium-saturation blue-green color with moderate red component.

The shorthand 3-digit hex notation (#4A9FD2 → #4AD with doubles) only works when both hex digits of each channel are identical: #AABBCC shortens to #ABC, but #4A9FD2 cannot be shortened because no channel has matching digits. Pure red is #FF0000 (255 red, 0 green, 0 blue). Pure green: #00FF00. Pure blue: #0000FF. White: #FFFFFF (255,255,255). Black: #000000. Gray is any equal value in all three channels: #808080 is medium gray (128,128,128).