BinHexDec Explained: Fast Conversion Techniques and Examples

Mastering BinHexDec: A Beginner’s Guide to Converting Between Binary, Hex, and DecimalComputers use several number systems to represent and manipulate data. The three most common are binary (base-2), hexadecimal (base-16), and decimal (base-10). Understanding how to convert among these systems is an essential skill for programmers, hardware engineers, and anyone interested in how computers work. This guide—focused on the practical conversions commonly called “BinHexDec”—explains the fundamentals, step-by-step conversion methods, useful shortcuts, worked examples, and simple programs to automate conversions.

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Why these number systems matter

• Decimal (base-10) is the number system humans use every day. It uses digits 0–9.
• Binary (base-2) is how computers store and process information at the lowest level. It uses digits 0 and 1. Each binary digit is called a bit.
• Hexadecimal (base-16) is a compact way to represent binary values. It uses digits 0–9 and letters A–F (where A=10, B=11, …, F=15). Hex is widely used in programming, memory addresses, color codes, and debugging because each hex digit represents exactly four binary bits (a nibble).

Understanding conversion between these systems helps when reading memory dumps, interpreting bitmasks, debugging low-level code, working with network protocols, or simply learning computer science fundamentals.

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Basics: place values and powers

Every positional number system uses place values determined by powers of the base:

• Decimal: places are 10^0, 10^1, 10^2, …
• Binary: places are 2^0, 2^1, 2^2, …
• Hexadecimal: places are 16^0, 16^1, 16^2, …

For example, the decimal number 237 equals: 237 = 2×10^2 + 3×10^1 + 7×10^0

A binary number 1011 equals: 1011₂ = 1×2^3 + 0×2^2 + 1×2^1 + 1×2^0 = 8 + 0 + 2 + 1 = 11₁₀

A hex number 3B equals: 3B₁₆ = 3×16^1 + 11×16^0 = 48 + 11 = 59₁₀

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Converting decimal to binary

Method 1 — Repeated division (standard):

1. Divide the decimal number by 2.
2. Record the remainder (0 or 1).
3. Replace the number by the integer quotient and repeat until quotient is 0.
4. The binary representation is the remainders read bottom-to-top (last remainder is most significant bit).

Example: Convert 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 Read remainders upwards: 101101₂

Method 2 — Using powers of two:

1. Find the largest power of 2 ≤ number.
2. Subtract and mark a 1 in that place.
3. Repeat with the remainder for next lower power. This is practical for mental conversion of small numbers.

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Converting decimal to hexadecimal

Method — Repeated division by 16:

1. Divide the decimal by 16.
2. Record the remainder (0–15; use A–F for 10–15).
3. Continue with integer quotient until quotient is 0.
4. Read remainders bottom-to-top.

Example: Convert 345 to hex

• 345 ÷ 16 = 21 remainder 9 → least significant hex digit = 9
• 21 ÷ 16 = 1 remainder 5 → next hex digit = 5
• 1 ÷ 16 = 0 remainder 1 → most significant digit = 1 Result: 159₁₆

Shortcut — group binary into nibbles: If you already have the binary form, group bits in sets of four from the right, convert each nibble to a hex digit.

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Converting binary to decimal

Method — sum the powers:

1. Label each bit from right (least significant) position 0 upward.
2. For each bit that is 1, add 2^position to the sum.

Example: 110101₂ = 1×2^5 + 1×2^4 + 0×2^3 + 1×2^2 + 0×2^1 + 1×2^0 = 32 + 16 + 0 + 4 + 0 + 1 = 53₁₀

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Converting binary to hexadecimal and back

Binary to hex:

1. Pad the binary number on the left with zeros so its length is a multiple of 4.
2. Break into 4-bit groups (nibbles).
3. Convert each nibble to its hex digit (0000→0 … 1111→F).

Example: 101111011 → pad to 0010 1111 1011 → 2 FB → 2FB₁₆

Hex to binary:

1. Replace each hex digit with its 4-bit binary equivalent. Example: 4C₁₆ → 0100 1100₂

Because each hex digit maps to exactly four bits, hex is an ideal human-friendly representation of binary data.

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Converting hexadecimal to decimal

Method — expand with powers of 16:

1. Multiply each hex digit by 16^position (0-based from right).
2. Sum the results.

Example: 2FA₁₆ = 2×16^2 + 15×16^1 + 10×16^0 = 2×256 + 15×16 + 10 = 512 + 240 + 10 = 762₁₀

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Signed integers and two’s complement (brief)

Most systems represent signed integers using two’s complement. Converting a negative decimal to binary (two’s complement) for an n-bit width:

1. Convert the absolute value to binary.
2. Pad to n bits.
3. Invert bits.
4. Add 1.

To convert a two’s complement binary to decimal:

• If the most significant bit (MSB) is 0, interpret as positive normally.
• If MSB is 1, compute the value as – (2^n – unsigned_value) or invert/add technique to find magnitude and add the negative sign.

Example: 8-bit 11110110₂ Unsigned = 246. Two’s complement value = 246 – 256 = -10.

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Fractions and fixed-point / floating-point basics

Binary fractions use negative powers of two: 0.101₂ = 1×2^-1 + 0×2^-2 + 1×2^-3 = 0.5 + 0 + 0.125 = 0.625₁₀.

Converting decimal fractions to binary may produce repeating fractions (like 0.1₁₀). Use repeated multiplication by the base:

1. Multiply fractional part by 2 (for binary).
2. Integer part becomes next bit; fractional part repeats the process. Stop when fraction becomes zero or when you reach desired precision.

Floating-point (IEEE 754) is more complex — it encodes sign, exponent, and mantissa. For general programming tasks, use language-provided conversions.

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Practical tricks and mnemonics

• Hex digit ↔ binary nibble: memorize A→1010, C→1100, F→1111; grouping of 4 bits makes conversions quick.
• Common hex prefixes:
  • 0x (C/C++/Python) e.g., 0xFF
  • # for web colors (e.g., #FF8800)
• Byte boundaries: 8 bits = 2 hex digits. A 32-bit value = 8 hex digits.
• Quick decimal-to-hex for small numbers: divide by 16 mentally or convert via binary grouping.

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Worked examples

1. Convert 2025₁₀ to hex and binary

• Hex: 2025 ÷ 16 = 126 r9; 126 ÷ 16 = 7 r14 (E); 7 ÷ 16 = 0 r7 → 7E9₁₆
• Binary: 7 = 0111, E = 1110, 9 = 1001 → 011111101001₂ (or 1111101001₂ without leading zeros)

1. Convert 0x3A7 to decimal

• 3×16^2 + 10×16^1 + 7 = 3×256 + 160 + 7 = 768 + 160 + 7 = 935₁₀

1. Convert binary 10011011₂ to decimal

• 1×2^7 + 0×2^6 + 0×2^5 + 1×2^4 + 1×2^3 + 0×2^2 + 1×2^1 + 1×2^0 = 128 + 0 + 0 + 16 + 8 + 0 + 2 + 1 = 155₁₀

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Simple converter code snippets

Python (decimal → binary/hex):

n = 345 binary = bin(n)       # '0b101011001' hexa = hex(n)         # '0x159' # remove prefixes: binary_no_prefix = format(n, 'b')   # '101011001' hex_no_prefix = format(n, 'x')      # '159' 

JavaScript:

let n = 345; n.toString(2);  // binary '101011001' n.toString(16); // hex '159' 

C (print):

#include <stdio.h> int main() {     int n = 345;     printf("%d in hex is %x and in binary is ... ", n, n);     return 0; } 

(For binary in C you typically build the string manually or use bitwise loops.)

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Common pitfalls

• Confusing hex digits (A–F) with decimal digits.
• Forgetting to pad binary when grouping into nibbles.
• Misinterpreting signed two’s complement values as unsigned.
• Expecting decimal fractions to always convert to finite binary fractions.

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Quick reference summary

• Binary uses base 2 (bits). Hex uses base 16 (nibbles). Decimal uses base 10.
• To convert decimal → binary/hex: use repeated division by target base.
• To convert binary/hex → decimal: expand using powers of the base.
• Hex ↔ binary: group binary in 4-bit chunks; convert each chunk to a hex digit.

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Mastering BinHexDec conversions builds both practical skills and intuition about how computers represent numbers. With the rules above, a few practiced examples, and simple code utilities, you’ll be comfortable switching between binary, hex, and decimal in programming and debugging tasks.