By Dr. Sohel Rana

Chapter-1

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1$!)

📊 The Master Cheat Sheet (0-15)

This table covers everything you will ever need.

Decimal (Base 10)	Binary (Base 2)	Octal (Base 8)	Hexadecimal (Base 16)
0	0000	0	0
1	0001	1	1
2	0010	2	2
3	0011	3	3
4	0100	4	4
5	0101	5	5
6	0110	6	6
7	0111	7	7
8	1000	10	8
9	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F

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🧠 How to Remember (Without Memorizing!)

You only need to remember two magic codes:

1. The “4-2-1” Trick (For Octal) 🐙

Octal digits go from 0 to 7. Notice that $4+2+1 = 7$.

This is all you need to find any Octal number.

• Question: What is 4 in Octal?
  • Look at your code: 4 – 2 – 1
  • Do you need the “4”? YES (Put a 1)
  • Do you need the “2”? NO (Put a 0)
  • Do you need the “1”? NO (Put a 0)
  • Result: 100 in binary is 4 in Octal.
• Question: What is 6 in Octal?
  • How do you make 6? You need 4 + 2.
  • 4 (Yes) – 2 (Yes) – 1 (No)
  • Result: 110

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2. The “8-4-2-1” Trick (For Hexadecimal) 🔶

Hex goes up to 15. Notice that $8+4+2+1 = 15$.

This is the code for Hex.

• Question: What is C (12) in Binary?
  • How do you make 12? 8 + 4.
  • 8 (Yes) – 4 (Yes) – 2 (No) – 1 (No)
  • Result: 1100
• Question: What is F (15) in Binary?
  • How do you make 15? 8 + 4 + 2 + 1.
  • Result: 1111 (All switches ON!)

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✍️ Exam Hack: Build the Table in 10 Seconds

If you sit in an exam, draw this immediately on rough paper. You don’t need to calculate anything. Just follow this pattern:

1. Column 1 (The “1s” place): Alternating 0, 1, 0, 1, 0, 1...
2. Column 2 (The “2s” place): Alternating 00, 11, 00, 11...
3. Column 3 (The “4s” place): Alternating 0000, 1111...
4. Column 4 (The “8s” place): Alternating 00000000, 11111111...

Try it:

Write 0-1-0-1 down the page.

Next to it, write 00-11-00-11.

Next to that, 0000-1111.

Boom! You just wrote the entire binary table without doing a single math calculation. 😎

🛑 Common Confusion Check

User asked: “What is 4 equivalent in octal?”

• In Decimal: It is 4.
• In Octal: It is also 4.
• In Hex: It is also 4.

Why? Because all number systems are the SAME until you run out of digits!

• Binary runs out at 1 (so 2 becomes 10).
• Octal runs out at 7 (so 8 becomes 10).
• Decimal runs out at 9 (so 10 becomes 10).
• Hex runs out at F (so 16 becomes 10).

So, 0, 1, 2, 3, 4, 5, 6, 7 are the identical in Octal, Decimal, and Hex! The confusion only starts at number 8.

Hi ! Think of me as your study partner. I know Digital Logic Design (DLD) can feel like a maze of 1s and 0s, but let’s break down Signed and Unsigned numbers as if we’re talking about a simple bank ledger or a cricket score.

Here is a simplified guide you can copy and paste into your notes.

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🔢 Understanding Signed vs. Unsigned Numbers

In the world of computers, everything is stored in “boxes” called bits. Usually, we use 8 bits (1 Byte) to represent a number.

1. Unsigned Numbers (The “Positive Only” Club)

Unsigned numbers are like a tape measure. They start at zero and only go up. There is no concept of “negative” here.

• Rule: All bits are used to represent the value of the number.
• Range (for 8 bits): $0$ to $2^8 – 1$ (which is 0 to 255).
• Example: * Binary 00001010 = Decimal 10
  • Binary 11111111 = Decimal 255

Real-life Application: Think of your Age or the Number of runs a batsman scores. You can’t have -20 runs or be -5 years old!

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2. Signed Numbers (The “Plus and Minus” Club)

Signed numbers are like a Bank Statement. You can have money in the bank (Positive), or you can be in debt (Negative).

To show if a number is positive or negative, we reserve the left-most bit (the Most Significant Bit or MSB) as a “Sign Post.”

• The Sign Bit Rule:
  • If the first bit is 0 $\rightarrow$ The number is Positive (+).
  • If the first bit is 1 $\rightarrow$ The number is Negative (-).

How do we write Negative numbers?

In DLD, we don’t just put a minus sign. We use 2’s Complement.

1. Start with the positive binary.
2. Flip all bits (0 becomes 1, 1 becomes 0).
3. Add 1 to the result.

• Example (using 8 bits):
  • +5 is 00000101 (Starts with 0)
  • -5 is 11111011 (Starts with 1)

Real-life Application: Temperature. It can be $+40^\circ\text{C}$ in Delhi or $-10^\circ\text{C}$ in Leh. We need a sign bit to tell the difference!

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📊 Quick Comparison Table

Feature	Unsigned Numbers	Signed Numbers
Range (8-bit)	0 to 255	-128 to +127
Sign Bit	No (All bits are value)	Yes (Left-most bit)
Negatives?	Not possible	Possible
Best For	Counting items, Memory addresses	Calculations, Temperature, Finance

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💡 A Simple Way to Remember

• Unsigned: Think of a Whole Number ($0, 1, 2, 3…$). It’s simple and straightforward.
• Signed: Think of a Direction. If you walk forward, it’s (+). If you walk backward, it’s (-). The first bit is just a “direction arrow.”

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🧠 Check your understanding:

If you see the binary number 10000010:

• In Unsigned, it is a large positive number (130).
• In Signed, it is a Negative number because it starts with 1.

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// — DLD CHAPTER: 2’S COMPLEMENT MASTERY — let dld_advanced_pg = ` `;

⚡ Flashback Revision: The “Sign” Problem

• Computers only know 0 and 1. They don’t have a “+” or “-” key.
• So, we steal the first bit (MSB) to act as the sign.
• 0 = Positive (+) 😄
• 1 = Negative (-) 😠

Feature	Unsigned Numbers	Signed Numbers
Meaning	Only positive numbers (magnitude only).	Can be Positive (+) or Negative (-).
The MSB (First Bit)	It is part of the value (e.g., 128).	It is the Sign Bit (0 is +, 1 is -).
Range (for 8 bits)	0 to 255 (Huge positive range).	-128 to +127 (Split range).
Example (1111)	$8+4+2+1 = \mathbf{15}$	In Signed 2’s comp, this is -1.
Use Case	Counting apples, memory addresses.	Temperature, Bank balance (debt).

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Q1(a) Represent the decimal number 232 in Binary, Gray Code, Octal, BCD, and Excess-3.

1. Decimal to Binary

Target Number: 232

We find the powers of 2 that add up to 232:

• 128 + 64 + 32 + 8 = 232

So, we place a “1” at those positions and “0” at the others:

Binary Answer: 1110 1000

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2. Binary to Gray Code

Rule: Copy the first bit, then XOR (add without carry) the neighbors.

Binary: 1110 1000

• 1 (Copy first) -> 1
• 1 + 1 = 0
• 1 + 1 = 0
• 1 + 0 = 1
• 0 + 1 = 1
• 1 + 0 = 1
• 0 + 0 = 0
• 0 + 0 = 0

Gray Code Answer: 1001 1100

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3. Binary to Octal

Rule: Group binary bits in sets of 3 starting from the right.

Binary: 011 | 101 | 000

• 011 = 3
• 101 = 5
• 000 = 0

Octal Answer: 350 (Base 8)

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4. Decimal to BCD (Binary Coded Decimal)

Rule: Convert each decimal digit separately into 4-bit binary.

Digits: 2 | 3 | 2

• 2 = 0010
• 3 = 0011
• 2 = 0010

BCD Answer: 0010 0011 0010

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5. Decimal to Excess-3

Rule: Add 3 to each decimal digit, then convert to binary.

Digits: 2 | 3 | 2

• First Digit (2): 2 + 3 = 5 -> 0101
• Second Digit (3): 3 + 3 = 6 -> 0110
• Third Digit (2): 2 + 3 = 5 -> 0101

Excess-3 Answer: 0101 0110 0101

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Q1(b) Difference between Signed and Unsigned Binary Numbers

Unsigned Numbers:

• Definition: These numbers can only be positive.
• MSB (Most Significant Bit): The first bit is used for the value (magnitude), just like any other bit.
• Range: Stores larger positive numbers (e.g., 0 to 255).
• Example: 1111 1111 = 255.

Signed Numbers:

• Definition: These numbers can be positive or negative.
• MSB (Most Significant Bit): The first bit is the “Sign Bit”.
  • 0 means Positive (+)
  • 1 means Negative (-)
• Range: The range is split between positive and negative (e.g., -128 to +127).
• Example: 1111 1111 usually represents -1 in Two’s Complement.

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Q1(c) How to perform Addition and Subtraction using 2’s Complement

1. Addition

Addition in 2’s complement is straightforward. You simply add the two binary numbers together. If there is a carry out of the final bit, it is discarded.

Example: 5 + 2

0101 (5)

• 0010 (2)

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0111 (Answer is 7)

2. Subtraction

Computers perform subtraction (A – B) by adding the negative version of the second number (A + (-B)).

Step 1: Find the 2’s Complement of the number to be subtracted (B).

• Example: 7 – 5
• Binary for 5 is 0101.
• Invert digits (1s Complement): 1010
• Add 1: 1011 (This is -5)

Step 2: Add the first number (A) to the 2’s Complement of B.

0111 (7)

• 1011 (-5)

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10010

Step 3: Discard the final carry.

Since we are working with 4 bits, we ignore the extra ‘1’ on the far left.

Final Result: 0010 (Which is equal to decimal 2)