Binary and Hexadecimal¶

Computers store and manipulate data using binary numbers—numbers written in base 2. While binary is ideal for hardware, it is often inconvenient for humans to read or write long sequences of bits.

To make binary data easier to work with, programmers frequently use hexadecimal (base 16) and sometimes octal (base 8). These number systems provide compact representations of the same underlying bit patterns.

Understanding how to convert between these bases is an essential skill when working with:

• debugging tools and memory dumps
• binary file formats
• network protocols
• cryptographic data
• low-level systems programming

1. Number Systems and Bases¶

A number system (or base) defines how numbers are represented using digits and positional value.

Each digit position represents a power of the base.

For a base (b):

[ \text{value} = \sum d_i b^i ]

where (d_i) are the digits.

Common bases used in computing¶

Hexadecimal extends decimal with six additional symbols:

Visualization of positional value¶

flowchart LR
    A["Digit × base^position"] --> B["Digit × 16^3"]
    A --> C["Digit × 16^2"]
    A --> D["Digit × 16^1"]
    A --> E["Digit × 16^0"]

Each position represents a power of the base.

2. Binary Numbers¶

Binary uses only two digits:

0 and 1

Each position represents a power of two.

Example: binary to decimal¶

Convert:

101101

[ 1\cdot2^5 + 0\cdot2^4 + 1\cdot2^3 + 1\cdot2^2 + 0\cdot2^1 + 1\cdot2^0 ]

[ = 32 + 8 + 4 + 1 = 45 ]

Visualization¶

flowchart TD
    A["101101"] --> B["1×32"]
    A --> C["0×16"]
    A --> D["1×8"]
    A --> E["1×4"]
    A --> F["0×2"]
    A --> G["1×1"]
    B --> H["45"]
    C --> H
    D --> H
    E --> H
    F --> H
    G --> H

3. Decimal to Binary Conversion¶

To convert a decimal number to binary, repeatedly divide by 2 and record the remainders.

Example: convert 45 to binary¶

Reading remainders bottom to top:

101101

Visualization¶

flowchart TD
    A["45"] --> B["22 r1"]
    B --> C["11 r0"]
    C --> D["5 r1"]
    D --> E["2 r1"]
    E --> F["1 r0"]
    F --> G["0 r1"]
    G --> H["Binary: 101101"]

4. Hexadecimal¶

Binary numbers become long quickly. For example:

1010111100111100

To make these easier to read, programmers use hexadecimal.

Hexadecimal uses base 16, with digits:

0 1 2 3 4 5 6 7 8 9 A B C D E F

Why hexadecimal is useful¶

Hexadecimal aligns perfectly with binary.

1 hex digit = 4 bits

So a byte (8 bits) equals:

2 hex digits

Binary to hex grouping¶

Binary:

1010 1111 0011 1100

Group into 4 bits:

1010 1111 0011 1100

Convert each group:

Result:

AF3C

Visualization¶

flowchart LR
    A["1010111100111100"] --> B["1010"]
    A --> C["1111"]
    A --> D["0011"]
    A --> E["1100"]

    B --> F["A"]
    C --> G["F"]
    D --> H["3"]
    E --> I["C"]

    F --> J["AF3C"]
    G --> J
    H --> J
    I --> J

5. Hexadecimal to Decimal¶

To convert hex to decimal, multiply each digit by the corresponding power of 16.

Example:

0xAF

[ A\times16^1 + F\times16^0 ]

[ 10\times16 + 15 = 175 ]

Larger example¶

0x3C7

[ 3\times16^2 + 12\times16^1 + 7\times16^0 ]

[ = 768 + 192 + 7 = 967 ]

6. Octal¶

Octal (base 8) uses digits:

0–7

Octal was historically useful because:

1 octal digit = 3 bits

Example:

101 110 111

Convert groups:

Result:

567

Today, octal is mainly used in Unix permissions.

Example:

755

means:

rwxr-xr-x

7. Binary, Hex, and Bytes¶

Because hexadecimal aligns perfectly with binary, it is widely used when inspecting raw data.

Example byte:

11001010

Group bits:

1100 1010

Convert:

CA

So:

11001010 = 0xCA

Visualization¶

flowchart LR
    A["11001010"] --> B["1100"]
    A --> C["1010"]
    B --> D["C"]
    C --> E["A"]
    D --> F["0xCA"]
    E --> F

8. Python and Number Bases¶

Python provides convenient syntax for working with different bases.

Literals¶

Python uses prefixes to specify number bases.

Example:

print(0b101101)   # 45
print(0xFF)       # 255
print(0o755)      # 493

Converting integers to strings¶

print(bin(45))   # '0b101101'
print(hex(255))  # '0xff'
print(oct(493))  # '0o755'

Parsing numbers from strings¶

print(int('101101', 2))  # 45
print(int('FF', 16))     # 255

9. Practical Uses of Hexadecimal¶

Hexadecimal appears frequently in programming because it represents binary data compactly.

Common uses include:

• memory addresses
• binary file formats
• color values
• machine instructions
• cryptographic hashes

Example: RGB colors¶

Many graphics systems represent colors using hexadecimal.

Example:

#3498DB

Structure:

RR GG BB

Each pair represents a byte.

Extracting color components¶

color = 0x3498DB

r = (color >> 16) & 0xFF
g = (color >> 8) & 0xFF
b = color & 0xFF

print(r, g, b)

Result:

52 152 219

Visualization¶

flowchart LR
    A["0x3498DB"] --> B["34"]
    A --> C["98"]
    A --> D["DB"]

    B --> E["Red"]
    C --> F["Green"]
    D --> G["Blue"]

10. Fixed-Width Formatting¶

Programs often display numbers using a fixed number of digits.

Example:

format(45, '08b')

Output:

00101101

This shows the number as 8-bit binary.

Hex formatting¶

format(255, '02x')

Output:

ff

11. Bytes and Hex Strings¶

Binary data is frequently represented as hexadecimal strings.

Example:

data = bytes([0xDE, 0xAD, 0xBE, 0xEF])

This produces:

deadbeef

which is a well-known hexadecimal example used in debugging.

Example:

print(data.hex())
print(bytes.fromhex('deadbeef'))

12. Worked Examples¶

Example 1¶

Convert 11011010 to hex.

Group bits:

1101 1010

Convert:

D A

Result:

0xDA

Example 2¶

Convert 0x7B to decimal.

[ 7\times16 + 11 = 123 ]

Example 3¶

Convert 73 to binary.

Break into powers of two:

[ 64 + 8 + 1 ]

Binary:

01001001

13. Exercises¶

1. Convert 10101101 to decimal.
2. Convert 91 to binary.
3. Convert 11100110 to hexadecimal.
4. Convert 0x2F to decimal.
5. Convert 0xA3 to binary.
6. Convert 755 (octal) to decimal.
7. Why does hexadecimal align so well with binary?
8. What hex value corresponds to 11111111?

14. Short Answers¶

1. 173
2. 01011011
3. E6
4. 47
5. 10100011
6. 493
7. Because one hex digit equals four bits
8. FF

15. Summary¶

• Binary (base 2) is the fundamental number system used by computers.
• Hexadecimal (base 16) provides a compact representation of binary data.
• Octal (base 8) groups bits in sets of three and appears in Unix permissions.
• One hex digit represents exactly four bits.
• One byte corresponds to two hex digits.
• Programmers frequently convert between bases when debugging, inspecting memory, or working with binary protocols.

Understanding binary and hexadecimal allows programmers to reason about bit patterns, memory layout, and low-level data formats with precision.