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:

text 0 and 1

Each position represents a power of two.

Example: binary to decimal¶

Convert:

text 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:

text 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:

text 1010111100111100

To make these easier to read, programmers use hexadecimal.

Hexadecimal uses base 16, with digits:

text 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.

text 1 hex digit = 4 bits

So a byte (8 bits) equals:

text 2 hex digits

Binary to hex grouping¶

Binary:

text 1010 1111 0011 1100

Group into 4 bits:

text 1010 1111 0011 1100

Convert each group:

Result:

text 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:

text 0xAF

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

[ 10\times16 + 15 = 175 ]

Larger example¶

text 0x3C7

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

[ = 768 + 192 + 7 = 967 ]

6. Octal¶

Octal (base 8) uses digits:

text 0–7

Octal was historically useful because:

text 1 octal digit = 3 bits

Example:

text 101 110 111

Convert groups:

Result:

text 567

Today, octal is mainly used in Unix permissions.

Example:

text 755

means:

text rwxr-xr-x

7. Binary, Hex, and Bytes¶

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

Example byte:

text 11001010

Group bits:

text 1100 1010

Convert:

text CA

So:

text 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:

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

Converting integers to strings¶

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

Parsing numbers from strings¶

python 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:

```text

3498DB¶

```

Structure:

text RR GG BB

Each pair represents a byte.

Extracting color components¶

```python color = 0x3498DB

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

print(r, g, b) ```

Result:

text 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:

python format(45, '08b')

Output:

text 00101101

This shows the number as 8-bit binary.

Hex formatting¶

python format(255, '02x')

Output:

text ff

11. Bytes and Hex Strings¶

Binary data is frequently represented as hexadecimal strings.

Example:

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

This produces:

text deadbeef

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

Example:

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

12. Worked Examples¶

Example 1¶

Convert 11011010 to hex.

Group bits:

text 1101 1010

Convert:

text D A

Result:

text 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:

text 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?

Exercise 9. The positional value formula says that a number in base \(b\) is \(\sum d_i \cdot b^i\). Using this formula, explain why every hex digit maps to exactly four binary digits, every octal digit maps to exactly three binary digits, but there is no clean grouping of binary digits that maps to one decimal digit. What mathematical property must the base have for such a clean mapping to exist?

Exercise 10. Consider the following Python session:

python x = 0xFF y = int('FF', 16) z = 255 print(x == y == z) # What does this print? print(type(x), type(y), type(z)) # What types are these?

Predict the output. Then explain: when Python encounters 0xFF in source code, does it store the number internally in hexadecimal? What is the relationship between a number's value and its representation in a particular base?

Exercise 11. A programmer writes format(255, '04x') and gets '00ff'. They then try format(-1, '04x') expecting 'ff' (thinking of 8-bit two's complement), but instead get '-1'. Explain why Python's format() with 'x' does not produce two's complement hex for negative numbers. How does this relate to the fact that Python integers have no fixed bit width? What would the programmer need to do to get the 8-bit two's complement hex representation of -1?

Exercise 12. RGB color values are often written as 6-digit hex strings like #3498DB. Explain why hexadecimal is the natural choice for this rather than decimal or binary. Then consider: if humans could comfortably read base-256 (with 256 distinct digit symbols), how would the same color be written? Why don't we use base-256 in practice, and what does this tell us about why hexadecimal became the standard "human-friendly binary shorthand"?

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.

Exercises¶

Exercise 1. Convert the decimal number 42 to binary and hexadecimal.

Exercise 2. Why is hexadecimal commonly used in computing instead of decimal.