Hex Calculator

Perform hexadecimal arithmetic, convert between hex and decimal, and explore how base‑16 numbers work in programming and electronics.

Hex Arithmetic

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Hex to Decimal

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Decimal to Hex

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Understanding Hexadecimal: Complete Guide with Rules, Examples, and Applications

Hexadecimal (base‑16) represents numbers using digits 0–9 and letters A–F, making binary data concise and readable for humans in computing, electronics, and networking.

This section mirrors the same educational structure as the Exponent page—boxes, subtitles, examples, tables, and practice—so users get a consistent learning experience while working with the Hex Calculator above.

What Is Hexadecimal?

Hexadecimal is a positional number system with base 16. Each position represents a power of 16, from 160 on the right, then 161, 162, and so on.

Key Components
  • Digits: 0–9 and A–F, where A=10, B=11, C=12, D=13, E=14, F=15.
  • Place value: Each digit’s value is digit × 16position.
  • Binary grouping: One hex digit equals 4 bits, enabling compact binary visualization.
Example 1: Reading a hex number
3F16 = 3 × 161 + 15 × 160 = 48 + 15 = 6310.
Example 2: Larger value
1A316 = 1 × 256 + 10 × 16 + 3 = 256 + 160 + 3 = 41910.
Hex Decimal Binary (4-bit) Note
991001Digit
A101010Letter digit
F151111Max single hex
1A260001 1010Two digits
FF2551111 11118-bit max

The Seven Essential Rules of Hexadecimal

These rules summarize how to convert and calculate in hex. Mastering them lets you move between hex, decimal, and binary quickly while performing accurate arithmetic.

1. Hex → Decimal (powers of 16)

Multiply each digit’s decimal value by 16 raised to its position and sum the results.

Example:
7B16 = 7 × 16 + 11 × 1 = 112 + 11 = 12310.
2. Decimal → Hex (repeated division)

Divide by 16, collect remainders, and read them from last to first to form the hex number.

Example:
15610 → 9C16 (remainders: C then 9).
3. Hex ↔ Binary (nibble mapping)

Each hex digit maps to 4 bits; convert groups of 4 bits to single hex digits and back.

Example:
9E16 → 1001 11102.
4. Hex Addition

Add digits right to left; if a column ≥ 16, write remainder and carry 1 to the next column.

Example:
1A3 + 2F = 1D216.
5. Hex Subtraction

Subtract digits; if needed, borrow 1 (which equals 16) from the next column to proceed.

Example:
7F − 2B = 5416.
6. Hex Multiplication

Multiply like decimal long multiplication, but with base 16 and letter digits A–F for 10–15.

Example:
1C × 4 = 7016.
7. Hex Division

Use integer division in base 16 or convert to decimal, divide, and convert back to hex.

Example:
F0 ÷ 10 = F16.
Rule Method Example Result
Hex→DecSum of 16-powers2F47
Dec→HexRepeated division1569C
Hex↔Bin4‑bit groups9E1001 1110
AddCarry on 161A3 + 2F1D2
SubBorrow 1=167F − 2B54

Special Hex Topics Explained

Beyond basic conversion and arithmetic, hex appears with signed formats, prefixes, and data widths used throughout programming and digital design.

Signed values and two’s complement

Hex often displays signed integers stored in two’s complement; interpretation depends on bit width.

Example 1:
FF16 as 8‑bit two’s complement represents −1.
8016 as 8‑bit two’s complement represents −128.
Prefixes and notation

Common hex notations include 0xFF, FFh, and #FF0000 (for colors).

Example 2:
0x3E8 = 100010, #00FF00 is pure green in web colors.
Case and formatting

Hex digits may be uppercase or lowercase; values are identical (A=a).

Example 3:
The calculator accepts a–f or A–F and returns uppercase for consistency.
Bit width awareness

The same hex string can represent different values depending on width and signedness assumptions.

Example 4:
FFFF16 is 65535 unsigned (16‑bit) or −1 as 16‑bit two’s complement.
Expression Type Meaning Result
0xFFUnsigned 8‑bitMax 8‑bit value255
0x80Signed 8‑bitMost negative−128
0x7FSigned 8‑bitMax positive127
0xFFFFUnsigned 16‑bitMax 16‑bit value65535
#FF0000RGB hexRed channel max(255, 0, 0)

Step-by-Step Calculation Examples

Work through these examples and verify each answer using the Hex Calculator above for instant feedback.

Example 1: Addition with carry

Add 1A3 and 2F in hex.

Solution:
Step 1: 3 + F = 1810 → 1216, write 2 carry 1

Step 2: A(10) + 2 + 1 = 1310 → D16

Step 3: 1 + 0 = 1

Result: 1D216.
Example 2: Subtraction with borrow

Compute 7F − 2B in hex.

Solution:
7F → 127, 2B → 43; 127 − 43 = 84; 84 → 5416.
Example 3: Decimal to hex

Convert 100010 to hex.

Solution:
1000 ÷ 16 = 62 r 8; 62 ÷ 16 = 3 r 14 (E); 3 ÷ 16 = 0 r 3 → 3E816.
Example 4: Combined operations

Convert 4D16 to decimal, add 2510, and convert back to hex.

Solution:
4D16 = 7710; 77 + 25 = 102; 102 → 6616.

Real-World Applications of Hexadecimal

Hex makes binary data compact and readable, so it appears in software engineering, networking, graphics, and embedded systems.

💾 Memory and Pointers

Debuggers and operating systems display addresses and dumps in hex for clarity (e.g., 0x7FFE1234).

🎨 Web Colors

CSS colors use #RRGGBB; #FF0000 is red, #00FF00 green, #0000FF blue.

🔌 Embedded Systems

Registers and bitmasks are documented in hex, matching hardware bit patterns efficiently.

🌐 Networking

Packet analyzers, checksums, and protocol fields are frequently shown in hex for pattern recognition.

🔐 Cryptography

Hashes like SHA‑256 are often rendered in hex for compact textual representation.

🧰 File Formats

Hex editors show binary files as hex for inspection and patching without decoding to text.

Common Mistakes and How to Avoid Them

Hex errors often come from mixing bases or mishandling carries/borrows; clear notation and methodical steps prevent most mistakes.

❌ MISTAKE 1: Ignoring A–F mapping
Using A–F as “invalid” digits causes failed conversions. ✅ Treat A–F as 10–15.
❌ MISTAKE 2: Mixing bases in one step
Switching between decimal and hex mid‑column leads to wrong carries. ✅ Stick to one base per step or convert at boundaries.
❌ MISTAKE 3: Dropping leading zeros early
Removing zeros while grouping bits can shift nibble alignment. ✅ Group binary in 4‑bit blocks first, then simplify.
❌ MISTAKE 4: Misreading signed values
Treating two’s complement as unsigned changes meaning. ✅ Decide width/signedness before interpreting.
💡 Pro Tips for Success:
  • Write base subscripts (e.g., 2F16, 4710) during manual work.
  • Use 4‑bit groups for hex↔binary conversions.
  • Check carries/borrows column by column.
  • Verify results in the calculator to build intuition.
  • Confirm signedness and bit width before interpreting values.

Nibble Grouping and Large Values

Because hex compresses four bits per digit, it maps neatly to typical CPU widths and helps reason about limits and masks.

Understanding nibble alignment

Each nibble (4 bits) maps to one hex digit; fixed widths (8, 16, 32, 64 bits) map to 2, 4, 8, 16 hex digits respectively.

Example 1: 8‑bit
Max 8‑bit value is 0xFF = 25510.
Example 2: 16‑bit
0xFFFF = 6553510.
Example 3: 32‑bit
0xFFFFFFFF = 4,294,967,29510 (unsigned).
Bit width Max hex Description
8FFOne byte (2 hex digits)
16FFFFTwo bytes (4 hex digits)
32FFFFFFFFFour bytes (8 hex digits)
64FFFFFFFFFFFFFFFFEight bytes (16 hex digits)
128…32 hex digits…Common in crypto hashes
Converting with nibble alignment:
  1. Pad binary on the left so length is a multiple of 4.
  2. Split into 4‑bit groups (nibbles).
  3. Map each nibble to a hex digit (0000→0, …, 1111→F).
  4. Join digits; strip extra leading zeros if desired.

Practice Problems with Solutions

Try these problems, then open the solutions to compare your work and verify with the Hex Calculator.

Basic Level
Problem 1: Convert 4D16 to decimal
Show Solution

4 × 16 + 13 = 64 + 13 = 77

Problem 2: Convert 20010 to hex
Show Solution

200 ÷ 16 = 12 r 8 (C, 8); 12 ÷ 16 = 0 r 12 (C) → C8

Intermediate Level
Problem 3: Compute 3A16 + 7F16
Show Solution

3A + 7F = B916

Problem 4: Convert 1110 01012 to hex
Show Solution

1110→E, 0101→5 → E5

Advanced Level
Problem 5: Interpret FF16 as 8‑bit two’s complement
Show Solution

FF is −1 in 8‑bit signed representation

Problem 6: Evaluate (1C16 × 416) − 216 in hex
Show Solution

1C × 4 = 70; 70 − 2 = 6E