NBC 101 • Unit 118 min read★ Very High Exam Frequency

Number Systems & Binary Data Representation (Radix 2, 8, 10, 16)

Q: Why do modern computers use 2's Complement representation for negative numbers rather than sign-magnitude?

2's Complement simplifies CPU ALU hardware because addition and subtraction use the identical adder circuitry, and it eliminates the problem of having two distinct representations for zero (+0 and -0). (Examiner Tip: In 2's complement, zero has only one unique representation (all 0s).)

Unit 1: Computer Evolution, Architecture & Number Systems • Fundamentals of Computer

👨‍🏫 Professor's Mental Model: The Language of Light Switches

Computer ki har memory cell ek electronic transistor switch hai jo ya toh ON hota hai (1) ya OFF hota hai (0). Jaise hum insaan 10 ungliyon ki wajah se Decimal (0-9) number system use karte hain, waise hi computer switches ki wajah se Binary (0 and 1) use karta hai. Hexadecimal (Base 16) lambe 32-bit binary codes ko chota karke likhne ka shortcut hai!

Interactive Radix Lab & Step-by-Step Notebook

Learn Radix Conversions, Successive Division Ladders, Nibble Mapping & 2's Complement Arithmetic.

NEP 2020 Study Sheet

Decimal Input (Integer or Float):

Target Base:

Exam Numericals:

DECIMAL (Base 10)(25.625)₁₀

BINARY (Base 2)(11001.101)₂

OCTAL (Base 8)(31)₈

HEXADECIMAL (Base 16)(19)₁₆

Conversion Solution for (25.625)₁₀ → Binary

PART 1: INTEGER CONVERSION(25)₁₀ → Binary

Repeatedly divide the value by Base 2 until quotient is 0. Collect remainders in reverse order: Bottom to Top (MSB ↑ to LSB).

DivisorQuotientRemainder

1LSB

1MSB

(Quotient = 0, Stop)

Read Upwards (Bottom to Top)

Start with the MSB (Last remainder) at the bottom and read up to the LSB (First remainder) at the top:

11001

Integer Result = (11001)₂

PART 2: FRACTIONAL CONVERSION(0.625)₁₀ → Binary

Repeatedly multiply the fractional part by Base 2. Extract the integer parts in order: Top to Bottom (MSB ↓ to LSB).

MultiplicationExtracted Integer

0.625 × 2 = 1.250

1MSB

0.250 × 2 = 0.500

0.500 × 2 = 1.000

1LSB

Read Top to Bottom

Fractional binary sequence:(.101)₂

Final Combined Radix Conversion Result:Integer Part (11001) + Fractional Part (.101)

(25.625)₁₀ = (11001.101)₂

Academic Lecture Notes & Solved Study Pages

Unit 1 • Core Concepts, Step-by-Step Proofs & Notebook Solutions

3 Notebook Pages

NOTEBOOK PAGE 1 OF 3• Lecture Study Sheet

1. Positional Number Systems Overview

💡 Core Principle: Every positional number system is governed by its Radix / Base (r), which denotes the total count of unique digital symbols available.

r = 10Base 10

Decimal Number System

Allowed Digits:0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Weights:10⁰, 10¹, 10², 10³...

Natural human number system with 10 fingers. Standard positional base.

r = 2Base 2

Binary Number System

Allowed Digits:0, 1 (Bits)

Weights:2⁰, 2¹, 2², 2³, 2⁴...

Native language of computers. 0 = OFF (Low Voltage), 1 = ON (High Voltage).

r = 8 (2³)Base 8

Octal Number System

Allowed Digits:0, 1, 2, 3, 4, 5, 6, 7

Weights:8⁰, 8¹, 8², 8³...

Compact shorthand. Exactly 1 Octal digit maps to 3 binary bits (2³ = 8).

r = 16 (2⁴)Base 16

Hexadecimal Number System

Allowed Digits:0-9 and A(10), B(11), C(12), D(13), E(14), F(15)

Weights:16⁰, 16¹, 16², 16³...

Industry standard for RAM addresses. Exactly 1 Hex digit maps to 4 binary bits (1 Nibble).

NOTEBOOK PAGE 2 OF 3• Lecture Study Sheet

2. Step-by-Step Radix Conversion Formulas

Decimal to Other Bases (Successive Division & Multiplication Method)

1. Integer Part Rule:

Continuously divide by the target base (r) until quotient becomes 0. Collect remainders from Bottom to Top (MSB to LSB ↑).

2. Fractional Part Rule:

Continuously multiply fractional part by base (r). Extract and collect integer parts from Top to Bottom (↓).

Solved University Example: Convert (25.625)₁₀ to BinaryExam Classic

STEP 1: Integer Part (25)₁₀ → Successive Division

25 ÷ 2 = 12Remainder: 1 (LSB)

12 ÷ 2 = 6Remainder: 0

6 ÷ 2 = 3Remainder: 0

3 ÷ 2 = 1Remainder: 1

1 ÷ 2 = 0Remainder: 1 (MSB)

Read Bottom-to-Top:(11001)₂

STEP 2: Fractional Part (0.625)₁₀ → Successive Multiplication

0.625 × 2 = 1.250Take Integer: 1 (MSB)

0.250 × 2 = 0.500Take Integer: 0

0.500 × 2 = 1.000Take Integer: 1 (LSB)

Read Top-to-Bottom:(.101)₂

Combine Integer and Fractional parts at the Radix point:(25.625)₁₀ = (11001.101)₂

Other Bases to Decimal (Positional Weight Expansion Method)

Multiply each digit by its positional weight (Base^Position) and sum them up:

Example: Convert (1101.1)₂ to Decimal

= (1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰) + (1 × 2⁻¹)

= 8 + 4 + 0 + 1 + 0.5

= (13.5)₁₀

Fast Binary ↔ Octal (3-bit) & Hexadecimal (4-bit) Shortcuts

Octal ↔ 3-Bit Binary:

Replace each octal digit with its 3-bit binary equivalent:

(75)₈ = (111 101)₂

Hexadecimal ↔ 4-Bit Nibble:

Replace each hex digit with its 4-bit binary nibble:

(A3)₁₆ = (1010 0011)₂

NOTEBOOK PAGE 3 OF 3• Lecture Study Sheet

3. 1's Complement & 2's Complement Arithmetic

💡 In digital computer hardware, subtraction is done using adder circuitry via complement arithmetic:M - N = M + 2's Complement of N

1's Complement Rule:

Invert every bit: change 0 → 1 and 1 → 0.

Example: 1's comp of 101100 is 010011

2's Complement Rule:

Add 1 to the 1's complement: 1's Comp + 1.

Example: 010011 + 1 = 010100

4-Step Subtraction Algorithm (M - N):

1Compute the 2's complement of the subtrahend (N).

2Add it directly to the minuend (M) using binary addition.

3If Carry = 1: Discard the carry bit. The result is positive (+).

4If Carry = 0: Result is negative (-). Take 2's complement of answer and attach minus sign.

4-Bit Binary, Octal & Hexadecimal Equivalence Table

Decimal	4-Bit Binary	Octal	Hexadecimal
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

🎯 University Exam Scoring Blueprint

Solve number system numericals with neat step-by-step division ladders.
Write the rule for 2's complement subtraction (Carry discarded = Positive result; No carry = Negative 2's complement).
Remember: Hex digits A to F represent values 10 to 15.