docs(notes): add 2025-03-23-data-representation-en.md

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notes/2025-03-23-data-representation-en.md

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+---
+title: Data Representation and Operation
+lang: en
+layout: post
+audio: false
+translated: false
+generated: true
+---
+
+Below is a comprehensive tutorial that breaks down the key topics in the “Data Representation and Operation” chapter. This tutorial covers binary, hexadecimal, and floating-point number systems, Boolean algebra, and arithmetic operations. It’s designed to build your understanding step by step with clear explanations and examples.
+
+---
+
+## 1. Number Systems
+
+### 1.1 Binary Number System
+
+**Concepts:**
+
+- **Base-2 System:** Uses only two digits: 0 and 1.
+- **Place Value:** Each digit represents a power of 2. For a binary number \( b_n b_{n-1} \dots b_1 b_0 \), the value is  
+  \[
+  \sum_{i=0}^{n} b_i \times 2^i
+  \]
+  where \( b_i \) is either 0 or 1.
+
+**Example:**
+
+Convert binary \( 1011_2 \) to decimal:
+- \( 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 11_{10} \)
+
+**Practice Exercise:**
+
+- Convert the binary number \( 110010_2 \) to decimal.
+
+---
+
+### 1.2 Hexadecimal Number System
+
+**Concepts:**
+
+- **Base-16 System:** Uses sixteen symbols: 0–9 and A–F (where A=10, B=11, …, F=15).
+- **Place Value:** Each digit represents a power of 16. For a hexadecimal number \( h_n h_{n-1} \dots h_1 h_0 \), the value is  
+  \[
+  \sum_{i=0}^{n} h_i \times 16^i
+  \]
+
+**Conversion from Binary to Hexadecimal:**
+
+1. Group the binary number into 4-bit chunks (starting from the right).
+2. Convert each 4-bit group to its hexadecimal equivalent.
+
+**Example:**
+
+Convert binary \( 1011011101_2 \) to hexadecimal:
+- Group into 4-bit groups: \( 10 \, 1101 \, 1101 \) (pad left with zeros if needed → \( 0010 \, 1101 \, 1101 \))
+- \( 0010_2 = 2_{16} \)
+- \( 1101_2 = D_{16} \)
+- \( 1101_2 = D_{16} \)
+- Final hexadecimal: \( 2DD_{16} \)
+
+**Practice Exercise:**
+
+- Convert the binary number \( 11101010_2 \) into hexadecimal.
+
+---
+
+### 1.3 Floating-Point Number Representation
+
+**Concepts:**
+
+- **Purpose:** To represent real numbers that can have very large or very small magnitudes.
+- **IEEE Standard:** Most computers use IEEE 754 for floating-point arithmetic.
+- **Components:**
+  - **Sign Bit:** Determines if the number is positive (0) or negative (1).
+  - **Exponent:** Represents the scale or magnitude.
+  - **Mantissa (Significand):** Contains the significant digits of the number.
+
+**Representation:**
+
+For single precision (32-bit):
+- 1 bit for sign.
+- 8 bits for exponent.
+- 23 bits for mantissa.
+
+For example, the value is represented as:
+\[
+(-1)^{\text{sign}} \times 1.\text{mantissa} \times 2^{(\text{exponent} - \text{bias})}
+\]
+where the bias for single precision is 127.
+
+**Example Walk-Through:**
+
+Suppose you have a 32-bit binary string representing a floating-point number:
+- **Sign Bit:** 0 (positive)
+- **Exponent Bits:** e.g., \( 10000010_2 \) → Decimal 130. Subtract bias: \( 130 - 127 = 3 \).
+- **Mantissa Bits:** Suppose they represent a fractional part like \( .101000... \).
+
+Then the number would be:
+\[
++1.101000 \times 2^3
+\]
+Convert \( 1.101000 \) from binary to decimal and then multiply by \( 2^3 \) to get the final value.
+
+**Practice Exercise:**
+
+- Given the following breakdown for a 32-bit floating-point number: sign = 0, exponent = \( 10000001_2 \) (decimal 129), and mantissa = \( 01000000000000000000000 \), compute the decimal value.
+
+---
+
+## 2. Boolean Algebra
+
+### 2.1 Basic Boolean Operations
+
+**Key Operations:**
+- **AND (·):** \( A \land B \) is true only if both \( A \) and \( B \) are true.
+- **OR (+):** \( A \lor B \) is true if at least one of \( A \) or \( B \) is true.
+- **NOT (’ or \(\neg\)):** \( \neg A \) inverts the truth value of \( A \).
+
+**Truth Tables:**
+
+- **AND:**
+
+  | A | B | A AND B |
+  |---|---|---------|
+  | 0 | 0 | 0       |
+  | 0 | 1 | 0       |
+  | 1 | 0 | 0       |
+  | 1 | 1 | 1       |
+
+- **OR:**
+
+  | A | B | A OR B |
+  |---|---|--------|
+  | 0 | 0 | 0      |
+  | 0 | 1 | 1      |
+  | 1 | 0 | 1      |
+  | 1 | 1 | 1      |
+
+- **NOT:**
+
+  | A | NOT A |
+  |---|-------|
+  | 0 | 1     |
+  | 1 | 0     |
+
+**Practice Exercise:**
+
+- Given the Boolean expression \( \neg(A \land B) \), use a truth table to show it is equivalent to \( \neg A \lor \neg B \) (De Morgan’s Law).
+
+---
+
+### 2.2 Boolean Algebra Laws and Theorems
+
+**Important Laws:**
+
+- **Commutative Law:**
+  - \( A \lor B = B \lor A \)
+  - \( A \land B = B \land A \)
+
+- **Associative Law:**
+  - \( (A \lor B) \lor C = A \lor (B \lor C) \)
+  - \( (A \land B) \land C = A \land (B \land C) \)
+
+- **Distributive Law:**
+  - \( A \land (B \lor C) = (A \land B) \lor (A \land C) \)
+  - \( A \lor (B \land C) = (A \lor B) \land (A \lor C) \)
+
+- **De Morgan’s Laws:**
+  - \( \neg (A \land B) = \neg A \lor \neg B \)
+  - \( \neg (A \lor B) = \neg A \land \neg B \)
+
+**Practice Exercise:**
+
+- Simplify the expression \( A \lor (\neg A \land B) \) using Boolean algebra laws.
+
+---
+
+## 3. Arithmetic Operations in Different Number Systems
+
+### 3.1 Binary Arithmetic
+
+**Key Operations:**
+
+- **Addition:**
+  - Follows similar rules to decimal addition but with base 2.
+  - **Example:** \( 1011_2 + 1101_2 \)
+    - Align and add bit by bit, carrying over when the sum exceeds 1.
+
+- **Subtraction:**
+  - Can be performed by borrowing, or by using the two's complement method.
+  - **Two's Complement:** To represent negative numbers, invert the bits and add 1.
+  - **Example:** To subtract \( 1101_2 \) from \( 1011_2 \), first find the two's complement of \( 1101_2 \) then add.
+
+**Practice Exercise:**
+
+- Perform the binary subtraction \( 10100_2 - 01101_2 \) using two’s complement.
+
+---
+
+### 3.2 Hexadecimal Arithmetic
+
+**Key Operations:**
+
+- **Addition/Subtraction:** Similar to decimal arithmetic but in base-16.
+- **Multiplication/Division:** Also follows the same principles as in decimal; however, you need to convert intermediate results using hexadecimal rules.
+
+**Practice Exercise:**
+
+- Add \( 2A3_{16} \) and \( 1F7_{16} \).
+
+---
+
+### 3.3 Floating-Point Arithmetic
+
+**Challenges:**
+
+- **Rounding Errors:** Due to limited precision, operations may introduce rounding errors.
+- **Normalization:** After an operation, the result must be normalized (i.e., the mantissa adjusted so that it fits the required format).
+
+**Example:**
+
+- **Addition:** When adding two floating-point numbers, you must align the exponents before adding the mantissas.
+
+**Practice Exercise:**
+
+- Outline the steps to add two floating-point numbers represented in IEEE 754 format.
+
+---
+
+## 4. Practical Tips for Mastering the Material
+
+- **Work Through Examples:** Practice converting numbers between systems by hand.
+- **Build Truth Tables:** For Boolean expressions, construct truth tables to understand and verify equivalences.
+- **Use Simulators/Tools:** Many educational websites and software tools allow you to simulate binary arithmetic or IEEE floating-point operations.
+- **Practice Problems:** Look for exercises at the end of your textbook chapters or online problem sets related to computer organization.
+- **Understand the Theory:** Rather than memorizing steps, focus on understanding why a method (like two’s complement) works.
+
+---
+
+## 5. Summary
+
+This chapter lays the foundation for understanding how computers process and represent data. You’ve learned:
+
+- **Binary and Hexadecimal Systems:** How numbers are represented and converted.
+- **Floating-Point Representation:** The structure and function of the IEEE 754 standard.
+- **Boolean Algebra:** The logic underpinning digital circuits and how to manipulate Boolean expressions.
+- **Arithmetic Operations:** How to perform arithmetic in different number systems, including handling issues like rounding in floating-point arithmetic.
+
+By practicing the examples and exercises provided, you’ll build the competence needed for both academic assessments and practical applications in computer organization and architecture.
+
+---
+
+This tutorial should give you a solid start on the chapter. If you have further questions or need additional examples, feel free to ask!