Number Systems and Operations

TL;DR

You'll learn about different ways computers represent numbers, focusing on binary, denary (decimal), and hexadecimal. We'll cover how to convert between these systems and perform basic operations like addition. Understanding these concepts helps you grasp how computers process information at a fundamental level.

1. The Mental Model

Think of numbers as being spoken in different languages. Denary is your everyday language. Binary is what computers speak. Hexadecimal is a helpful shortcut for humans to understand binary.

2. The Core Material

Computers use binary (base 2) because their circuits are either on or off, representing 1 or 0. Humans are used to denary (base 10), using digits 0-9. Hexadecimal (base 16) is a compact way to represent binary, using digits 0-9 and letters A-F.

Converting Between Number Systems

Binary to Denary:
Each digit in a binary number represents a power of 2, starting from 2^0 on the far right. You multiply each binary digit by its corresponding power of 2 and add the results.

Example: Convert 1011 (binary) to denary.
1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 1 * 2^0
1 * 8 + 0 * 4 + 1 * 2 + 1 * 1
8 + 0 + 2 + 1 = 11 (denary)

Denary to Binary:
You can use a repeated division method. Divide the denary number by 2, record the remainder, then divide the quotient by 2, and so on, until the quotient is 0. Read the remainders from bottom to top.

Example: Convert 13 (denary) to binary.
13 / 2 = 6 remainder 1
6 / 2 = 3 remainder 0
3 / 2 = 1 remainder 1
1 / 2 = 0 remainder 1
Reading remainders bottom-up: 1101 (binary)

Hexadecimal to Binary:
Each hexadecimal digit directly corresponds to a 4-bit binary number.

Example: Convert A5 (hex) to binary.
A = 1010
5 = 0101
So, A5 (hex) = 10100101 (binary)

Binary to Hexadecimal:
Group the binary digits into sets of 4, starting from the right. Convert each 4-bit group into its corresponding hexadecimal digit. Add leading zeros if your leftmost group has fewer than 4 bits.

Example: Convert 11010110 (binary) to hex.
Group: 1101 0110
1101 = D
0110 = 6
So, 11010110 (binary) = D6 (hex)

Hexadecimal to Denary / Denary to Hexadecimal:
It's often easiest to convert via binary. For example, to go from Hex to Denary, convert Hex to Binary, then Binary to Denary.

Binary Addition

Binary addition follows similar rules to denary addition, but you only have 0 and 1.

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 0 (carry 1 to the next column)
• 1 + 1 + 1 = 1 (carry 1 to the next column)

graph TD
    A["Start with Number (Denary, Binary, Hex)"] --> B{{"What's the Target System?"}};
    B -->|Denary| D{{"From Binary or Hex?"}};
    B -->|Binary| E{{"From Denary or Hex?"}};
    B -->|Hexadecimal| F{{"From Binary or Denary?"}};

    D --> DB["Binary to Denary Conversion (Powers of 2 summation)"];
    D --> DH["Hex to Denary (Hex to Binary then Binary to Denary)"];

    E --> EB["Denary to Binary Conversion (Repeated division by 2)"];
    E --> EH["Hex to Binary (Each Hex digit to 4-bit Binary)"];

    F --> FB["Binary to Hexadecimal (Group 4-bits, convert)"];
    F --> FD["Denary to Hexadecimal (Denary to Binary then Binary to Hex)"];

    FB --> G["End (Target System Number)"];
    FD --> G;
    EH --> G;
    EB --> G;
    DB --> G;
    DH --> G;

3. Worked Example

Let's convert the denary number 197 to hexadecimal.

1. Denary to Binary (repeated division by 2):
   197 / 2 = 98 remainder 1
98 / 2 = 49 remainder 0
49 / 2 = 24 remainder 1
24 / 2 = 12 remainder 0
12 / 2 = 6 remainder 0
6 / 2 = 3 remainder 0
3 / 2 = 1 remainder 1
1 / 2 = 0 remainder 1
   Reading remainders bottom-up: 11000101 (binary)

2. Binary to Hexadecimal (group into 4-bit chunks):
   1100 0101
   Convert each chunk:
   1100 = C
0101 = 5

So, 197 (denary) is C5 (hexadecimal).

4. Key Takeaways

• Binary (base 2) uses 0s and 1s, essential for computers.
• Denary (base 10) is our everyday number system.
• Hexadecimal (base 16) is a convenient shorthand for binary, using 0-9 and A-F.
• Each digit in a number system holds a "place value" based on powers of its base (e.g., powers of 2 for binary).
• To convert Denary to Binary, use repeated division by 2, reading remainders upwards.
• To convert Binary to Hex, group binary digits into fours from the right and convert each group.
• Binary addition follows base-2 rules, remember carries.

Common mistakes to avoid:
- Forgetting to read remainders from bottom to top when converting Denary to Binary.
- Mixing up hex digits A-F with their denary values (e.g., thinking A is 16 instead of 10).
- Grouping binary digits from the left instead of the right when converting to hexadecimal, which changes the value.
- Not adding leading zeros to the leftmost binary group if it's less than four bits when converting to hex.

5. Now Try It

Convert the denary number 214 to an 8-bit binary number, then convert that 8-bit binary number to hexadecimal. Your success means you'll have the denary, binary, and hexadecimal representation of 214.