Unit 3: Data Representation - AL ICT
This unit covers how information is converted into binary for computer processing. Here are the core components included in the syllabus:
1. Number Systems and Conversions
- Systems: Decimal (Base 10), Binary (Base 2), Octal (Base 8), and Hexadecimal (Base 16).
- Conversions: Moving between any of these bases (e.g., Binary to Hexadecimal).
2. Integer Representation
- Unsigned Integers: Positive whole numbers.
- Signed Integers: Techniques for negative numbers:
- Sign-and-Magnitude: MSB acts as the sign bit.
- 1’s Complement: Inverting all bits.
- 2’s Complement: The standard for subtraction and negative values.
3. Floating Point Representation (IEEE 754)
How real numbers (fractions) are stored using Sign bit, Exponent, and Mantissa in Single (32-bit) and Double (64-bit) precision.
4. Character Encoding
- BCD & ASCII: Standard codes for digits and English characters.
- EBCDIC: Legacy IBM mainframe encoding.
- Unicode: Modern global standard (UTF-8, UTF-16) for all languages.
5. Binary Arithmetic & Logic
- Addition and subtraction using 2’s complement.
- Bitwise operations: AND, OR, NOT, XOR.
6. Multimedia Representation
- Images: Pixels, resolution, and color depth.
- Audio/Video: Sampling rates and bit rates.
IEEE 754 Standard of Floating Point Arithmetic
Sri Lanka GCE A/L ICT - Data Representation
1. Introduction
In the GCE A/L ICT syllabus, representing real numbers (numbers with decimal points) is a crucial topic. Computers cannot store numbers like 10.5 or -3.14 directly in integer format. Instead, they use the IEEE 754 Standard.
For the A/L examination, you are primarily required to understand the Single Precision (32-bit) format.
2. Structure of Single Precision (32-bit)
A 32-bit floating-point number is divided into three parts:
| Part | Size (Bits) | Position | Function |
|---|---|---|---|
| Sign Bit (S) | 1 bit | Bit 31 (Leftmost) | 0 = Positive (+), 1 = Negative (-) |
| Exponent (E) | 8 bits | Bits 30 - 23 | Stores the power of 2 (with a Bias) |
| Mantissa (M) | 23 bits | Bits 22 - 0 | Stores the fractional part of the number |
Value = (-1)S × (1.M) × 2(E - 127)
Key Concept: The Bias
Since the exponent can be negative (e.g., $2^{-3}$), computers use a Bias of 127 to store it as a positive integer.
- Stored Exponent = Real Exponent + 127
- Real Exponent = Stored Exponent - 127
3. Normalization
Before converting a binary number to IEEE 754, it must be Normalized. This means shifting the binary point so that there is only one '1' to the left of the binary point.
Format: $1.xxxxx \times 2^y$
Note: In IEEE 754, the leading '1' is implied (hidden) and is not stored in the Mantissa bits to save space.
4. Step-by-Step Conversion Guide
Method A: Decimal to IEEE 754 (Single Precision)
- Determine the Sign: If positive, S=0. If negative, S=1.
- Convert to Binary: Convert the absolute value of the decimal number to binary (integer part and fractional part).
- Normalize: Shift the binary point to get the form $1.xxxxx \times 2^E$.
- Calculate Exponent: Add 127 to the real exponent ($E_{stored} = E + 127$). Convert this result to 8-bit binary.
- Determine Mantissa: Take the bits after the binary point from the normalized form. Pad with zeros to make it 23 bits.
- Combine: Arrange as [Sign] [Exponent] [Mantissa].
Method B: IEEE 754 to Decimal
- Identify Parts: Split the 32 bits into Sign (1), Exponent (8), and Mantissa (23).
- Check Sign: Is it positive or negative?
- Calculate Real Exponent: Convert Exponent bits to decimal, then subtract 127.
- Reconstruct Mantissa: Add the implied '1.' before the Mantissa bits ($1.M$).
- Calculate Value: Apply the formula: $(-1)^S \times 1.M \times 2^{RealExp}$.
5. Worked Example
Question: Convert -10.25 to IEEE 754 Single Precision.
Solution:
- Sign: Negative, so S = 1.
- Binary Conversion:
- Integer 10 = $1010_2$
- Fraction 0.25 = $0.01_2$ ($0.25 \times 2 = 0.5 \to 0$, $0.5 \times 2 = 1.0 \to 1$)
- Combined: $1010.01_2$
- Normalization: Shift point 3 places to the left.
$1.01001 \times 2^3$
Real Exponent = 3. - Exponent Calculation:
$3 + 127 = 130$
Binary of 130 = 10000010 - Mantissa: Take bits after the point ($01001$) and pad to 23 bits.
01001000000000000000000 - Final Result:
1 | 10000010 | 01001000000000000000000
Hex: C1240000
6. Practice Questions (A/L Style)
Question 1
Convert the decimal number 6.75 into its IEEE 754 Single Precision binary representation.
Question 2
The following 32-bit binary sequence represents a number in IEEE 754 Single Precision format. Find its decimal value.
0 10000001 01000000000000000000000
Question 3 (MCQ Style)
In the IEEE 754 Single Precision standard, what is the binary value stored in the exponent field if the actual exponent is -2?
A) 00000010
B) 11111101
C) 01111101
D) 10000001
IEEE 754 Decimal to Floating Point Conversion
Sri Lanka G.C.E A/L ICT – Unit 3 (Data Representation)
IEEE 754 is the standard used by computers to store floating point numbers (decimal numbers).
IEEE 754 Single Precision (32-bit)
| Part | Bits | Description |
|---|---|---|
| Sign | 1 bit | Positive or Negative number |
| Exponent | 8 bits | Power of 2 |
| Mantissa (Fraction) | 23 bits | Significant digits |
Sign | Exponent (8 bits) | Mantissa (23 bits)
Step-by-Step Conversion Method
Step 1 – Determine the Sign Bit
- Positive number → 0
- Negative number → 1
Example: +25.5
Sign = 0
Step 2 – Convert Decimal to Binary
Integer Part
25 ÷ 2 25 = 11001
Fraction Part
0.5 × 2 = 1.0
Binary Result:
25.5 = 11001.1
Step 3 – Normalize the Binary Number
11001.1 = 1.10011 × 2⁴
Step 4 – Calculate the Exponent
IEEE 754 uses a Bias value of 127
Exponent = Actual Exponent + Bias Exponent = 4 + 127 = 131
Convert 131 to binary:
131 = 10000011
Step 5 – Find the Mantissa
Take the digits after the decimal point:
1.10011
Mantissa:
10011000000000000000000
Final IEEE 754 Representation
| Part | Value |
|---|---|
| Sign | 0 |
| Exponent | 10000011 |
| Mantissa | 10011000000000000000000 |
Final 32-bit IEEE 754: 0 10000011 10011000000000000000000
Example 2 – Convert 10.25 to IEEE 754
Step 1 – Sign
Positive → 0
Step 2 – Decimal to Binary
10 = 1010
0.25 × 2 = 0.5 0.5 × 2 = 1.0
10.25 = 1010.01
Step 3 – Normalize
1010.01 = 1.01001 × 2³
Step 4 – Exponent
3 + 127 = 130 130 = 10000010
Step 5 – Mantissa
01001000000000000000000
Final Result
0 10000010 01001000000000000000000
Quick Exam Trick
- Find Sign
- Convert Decimal to Binary
- Normalize (1.x × 2ⁿ)
- Add Bias (127)
- Find Mantissa (23 bits)
7. Answers & Explanations
Answer to Question 1
Step 1: Sign
Positive, so S = 0.
Step 2: Binary
6 = $110_2$
0.75 = $0.11_2$ ($0.75 \times 2 = 1.5 \to 1$, $0.5 \times 2 = 1.0 \to 1$)
Result: $110.11_2$
Step 3: Normalize
$1.1011 \times 2^2$
Real Exponent = 2.
Step 4: Exponent Field
$2 + 127 = 129$
Binary of 129 = 10000001
Step 5: Mantissa
Bits after point: $1011$
Pad to 23 bits: 10110000000000000000000
Final Answer:
0 10000001 10110000000000000000000
Answer to Question 2
Step 1: Split
Sign: 0 (+)
Exponent: 10000001
Mantissa: 010000...
Step 2: Exponent
Binary $10000001 = 129$ (Decimal)
Real Exponent = $129 - 127 = 2$
Step 3: Mantissa Value
Implied 1 + Fraction = $1.01_2$
Step 4: Calculate
$+1.01_2 \times 2^2$
Shift binary point 2 places right: $101_2$
$101_2 = 5_{10}$
Final Answer: 5.0
Answer to Question 3
Real Exponent = -2.
Stored Exponent = Real Exponent + Bias
Stored Exponent = $-2 + 127 = 125$.
Convert 125 to binary:
125 = 64 + 32 + 16 + 8 + 4 + 1 = 01111101
Correct Option: C) 01111101
🚀 Master GCE O/L A/L ICT | Your IT Degree with Expert Guidance!
Online Individual & Group Classes in English | Sinhala | Tamil
Struggling with assignments, projects, or exams? Get personalized support tailored for BIT (University of Moratuwa), UCSC, and other IT degree students in Sri Lanka.
✨ What You'll Get
- ✅ Live Online Classes (Individual or Group)
- ✅ Sample Projects & Assignments (PHP, MySQL, Java, Python, Web Dev)
- ✅ Past Exam Papers + Model Answers
- ✅ Easy-to-Follow Tutorials & Study Notes
- ✅ Final Year Project Guidance – From Idea to Implementation
- ✅ Doubt-Clearing Sessions & Exam Prep Strategies
🌍 Taught in Your Preferred Language
English | Sinhala | Tamil
📞 Get Started Today!
Call / WhatsApp: +94 72 962 2034
Email: itclasssl@gmail.com
Quick response guaranteed! Share your syllabus or project topic, and we'll craft a learning plan just for you.
🔗 Free Resources & Community Links
🎓 Expert ICT, Coding, School Classes, Digital Marketing & University Project Guidance
Struggling with your university final year project? Want to master coding, upscale your business with expert digital marketing, or learn absolute computer basics from scratch? We offer high-quality individual and group online classes conducted in English, Sinhala, or Tamil mediums. Get guaranteed academic success and professional growth with tailored guidance.
🎓 University Final Year Project Guidance & AI
Get specialized, end-to-end mentoring and technical support to pass your degree or master's program with flying colors:
- 🏫 Targeted Institutes: Expert guidance tailored for BIT UCSC, UoM, SLIIT, NIBM, and other leading universities.
- 🔬 Postgraduate Support: Comprehensive assistance for MSc Software Final Year Projects.
- 🤖 AI & Smart Applications: Step-by-step implementation of AI, Machine Learning (ML), and automation modules.
- ✅ Guaranteed Success: Help with documentation, system architecture, coding, and viva preparation.
🏫 School ICT & Corporate Beginner Classes
- 💻 Non-IT Staff Computer Basics: Absolute beginner-friendly online classes covering essential computer skills, office tools, and internet operations.
- 🎒 Primary & Secondary (Grades 1-10): Interactive online ICT classes tailored to build strong foundations from early ages.
- 📝 Exam Prep: Dedicated training packages for GCE O/L, GCE A/L ICT, and GIT exams.
- 🌍 Global Syllabuses: Complete curriculum coverage for Local, Edexcel, and Cambridge in English & Tamil Mediums.
📢 Software Development & Digital Marketing Services
- ⚙️ Software & Web Development: Professional custom software application and website development built using PHP & MySQL.
- 🎯 Social Media Management: Content creation, publishing, and channel management for Facebook, Instagram, TikTok, and YouTube.
- 📈 Ad Boosting: Highly targeted paid advertising campaigns to drive leads, traffic, and sales to your business.
📞 Connect With Us Instantly
Book your slot for online classes or get a premium tech service quote today!
💬 WhatsApp: +94 729622034
📧 Email: ITClassSL@gmail.com
🌐 Explore Our Resources & Communities
Stay updated with our latest tutorials, project ideas, and student guides across all our official platforms:
- 📺 YouTube Tutorials: Subscribe to our Channel
- 💼 Professional Network: Connect on LinkedIn
- ✍️ Tech Blog: Visit our WordPress Site
- ❓ Project Q&A: Follow our Quora Guide Profile
- 📰 Monthly Updates: Read Our Newsletter
- 🌐 Official Portfolios: Wix Site | Google Business | Strikingly Portfolio
- 🗣️ Student Forum: Join our ElaKiri Discussion Thread
No comments:
Post a Comment