Binary computing principles

Binary computing is based on the principles of using the binary number system (base-2) to perform calculations and logical operations in digital systems. Here are the key principles:

1. Binary Number System

• Uses only two digits: 0 and 1 (unlike decimal, which uses 0-9).

• Each digit is called a bit (binary digit).

• Numbers are represented using powers of 2 (e.g., ${1011}_2=1\times 2^3+0\times 2^2+1\times 2^1+1\times 2^0={11}_{10}$).

2. Boolean Logic

• Binary computing relies on Boolean algebra, where:

  • 0 represents False.

  • 1 represents True.

• Basic logic gates perform operations:

  • AND (both inputs are 1 → output 1)

  • OR (at least one input is 1 → output 1)

  • NOT (inverts input: 0 → 1, 1 → 0)

  • XOR (output 1 if inputs differ).

3. Binary Arithmetic

• Addition: Follows rules similar to decimal but with carry-over at 2 (e.g., $1+1={10}_2$).

• Subtraction: Uses borrows or two’s complement for negative numbers.

• Multiplication/Division: Shift-and-add methods (e.g., left-shift = multiply by 2).

4. Data Representation

• All data (numbers, text, images) is encoded in binary.

• Integers: Stored in fixed-width formats (e.g., 8-bit, 32-bit) using:

  • Unsigned: Only positive numbers (0 to $2^n-1$).

  • Signed: Two’s complement for negative numbers.

• Floating-Point: IEEE 754 standard splits into sign, exponent, and mantissa.

5. Memory and Addressing

• Memory stores data in binary-addressable units (bytes, words).

• Addresses are binary numbers pointing to memory locations.

6. CPU Operations

• A CPU executes instructions encoded in binary (machine code).

• Registers hold binary data temporarily.

• Clock cycles synchronize operations (e.g., fetch-decode-execute).

7. Binary Circuits

• Transistors act as switches (on=1, off=0) to build logic gates.

• Combinational (e.g., adders) and sequential (e.g., flip-flops) circuits process binary data.

Example: Adding Two 4-Bit Numbers

text

   A: 0101 (5)  
+  B: 0110 (6)  
──────────────  
Sum: 1011 (11)  

• Performed using an adder circuit with carry propagation.

Why Binary?

• Reliability: Two distinct states (0/1) are easy to implement electronically (e.g., voltage levels).

• Simplicity: Simplifies hardware design (e.g., transistors as switches).

• Universality: All digital systems, from calculators to supercomputers, use binary.

Binary computing is the foundation of all modern digital technology, enabling everything from smartphones to AI systems.