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ICSE • Class 7 • Computer Science • Ch 2
Estimated Time: 45 Mins
Study Progress: In Progress

Number System - An Introduction

In ICSE Class 7 Computer Studies, "Number System - An Introduction" provides an authoritative, mathematically rigorous master study guide investigating positional number systems used in computing: Decimal (Base 10), Binary (Base 2), Octal (Base 8), and Hexadecimal (Base 16), along with inter-base conversions and binary arithmetic operations.

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Why Does a Billion-Dollar Supercomputer Only Count up to ONE Before It Runs Out of Numbers?

Look at the keyboard in front of you: it has ten digits, twenty-six letters, punctuation marks, emojis, and symbols. Yet inside the microscopic silicon chips of your computer, the most powerful processor on Earth is fundamentally blind to every number except TWO: ZERO AND ONE! Digital computers use the BINARY NUMBER SYSTEM (Base 2) because electronic microscopic transistors are physical binary switches—they can only be either OPEN or CLOSED! How do computer engineers read strings like 11111111 without losing their minds? They invented Hexadecimal (Base 16), condensing four binary digits into a single character! Let's master number systems.

Why This Chapter Matters

Binary arithmetic is the native logic of all digital microprocessors, network IP addresses, and cryptography algorithms. Mastering base conversions is essential for ICSE Computer Science.

Before You Begin (Prerequisites)

  • Place value and expanded form in decimal arithmetic.
  • Basic division with remainders.
  • Powers of numbers (2^0, 2^1, 2^2, 2^3...).

What You Will Learn (Core Objectives)

  • Define a positional number system and state the significance of the Base (Radix).
  • Compare Decimal, Binary, Octal, and Hexadecimal number systems.
  • Convert Decimal numbers into Binary using the successive division method.
  • Convert Binary numbers into Decimal using positional power expansion.
  • Perform multi-digit Binary Addition ($1+1=10$) and Binary Subtraction.
  • Convert between Decimal and Hexadecimal systems.

Chapter Roadmap & Progression

1 1. Positional Number Systems & Base...
2 2. Inter-Base Number Conversions
3 3. Octal & Hexadecimal Conversions
4 4. Binary Arithmetic: Addition & Su...

Complete Concept Guide (100% Curriculum Coverage)

1. Positional Number Systems & Base (Radix)

Understand
The Four Core Computing Systems:
  • Decimal (Base 10): Digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Standard human counting.
  • Binary (Base 2): Digits 0 and 1 (Bits). Machine language of computers.
  • Octal (Base 8): Digits 0 to 7. Shorthand for 3-bit binary groups.
  • Hexadecimal (Base 16): Digits 0-9 and letters A-F (A=10, B=11, C=12, D=13, E=14, F=15). Used in memory addressing and web hex color codes (#FFFFFF).

2. Inter-Base Number Conversions

Conversions
A. Decimal to Binary (Successive Division by 2):

Divide the decimal number repeatedly by 2, record remainders (0 or 1), and read remainders from bottom to top (MSB to LSB).
Example: (25)10 = (11001)2.

B. Binary to Decimal:

Multiply each bit by its positional power of 2 starting from the right (20, 21, 22...):
(11001)2 = (1 × 16) + (1 × 8) + (0 × 4) + (0 × 2) + (1 × 1) = (25)10.

3. Octal & Hexadecimal Conversions

Octal & Hex
A. Decimal to Hexadecimal:

Divide successively by 16. Example: (92)10 ÷ 16 = 5 with remainder 12 (C) → (5C)16.

B. Hexadecimal to Decimal:

(3B)16 = (3 × 161) + (11 × 160) = 48 + 11 = (59)10.

4. Binary Arithmetic: Addition & Subtraction

Binary Arithmetic
A. Binary Addition:
  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 0 = 1
  • 1 + 1 = 10 (0 with carry 1)
  • 1 + 1 + 1 = 11 (1 with carry 1)

Example: (1011)2 + (1101)2 = (11000)2 (11 + 13 = 24).

B. Binary Subtraction:

0 - 0 = 0; 1 - 0 = 1; 1 - 1 = 0; 0 - 1 = 1 (with a borrow of 1 from the next bit).
Example: (1101)2 - (1010)2 = (0011)2 (13 - 10 = 3).

Key Programming Syntax, Statements & Translator Rules

Binary Positional Value Formula
$$N_{10} = \sum_{i=0}^{n-1} d_i \times 2^i$$
Converts binary sequence into base-10 decimal.
Binary Addition Carry Rule
$$1_2 + 1_2 = 10_2 \quad (\text{Sum } 0, \text{ Carry } 1)$$
Fundamental digital logic rule.

Digital Logic: Four Number Systems & Binary Addition Mechanics

Computer Science: Number Systems & Binary Arithmetic THE FOUR NUMBER SYSTEMS • Decimal (Base 10): 0 to 9 Standard human counting system • Binary (Base 2): 0 and 1 (Bits) Machine language of computer transistors (Off/On) • Octal (Base 8): 0 to 7 (3-bit groups) • Hexadecimal (Base 16): 0-9 & A-F A=10, B=11, C=12, D=13, E=14, F=15 Memory addresses & Web color hex codes (#FFFFFF) • Radix / Base defines count of unique symbols BINARY ARITHMETIC RULES • Addition Rules: 0 + 0 = 0 • 0 + 1 = 1 • 1 + 0 = 1 1 + 1 = 10 (Sum 0, Carry 1!) 1 + 1 + 1 = 11 (Sum 1, Carry 1!) • Conversion Algorithm: Decimal → Binary: Successive division by 2 Read remainders from BOTTOM TO TOP (MSB to LSB) Binary → Decimal: Multiply by powers of 2 (20, 21...) Example: (25)10 = (11001)2 (16 + 8 + 1 = 25) BASE 2 = TRANSISTOR SWITCHES (0/1) • 1 + 1 = 10 (CARRY 1) • HEX USES A TO F

Chapter Summary & 10 Key Takeaways

Takeaway 1
A number system is a mathematical code using a set of symbols whose values depend on positional weights.
Takeaway 2
The Base (Radix) is the total number of unique symbols used: Decimal (10), Binary (2), Octal (8), Hex (16).
Takeaway 3
Computers use the binary system (0 and 1) because electronic transistors operate as binary on/off switches.
Takeaway 4
Hexadecimal uses digits 0-9 and letters A-F (where A=10, B=11, C=12, D=13, E=14, F=15).
Takeaway 5
Decimal to Binary conversion uses successive division by 2, collecting remainders from bottom to top.
Takeaway 6
Binary to Decimal conversion expands digits by multiplying by ascending powers of 2 (2^0, 2^1, 2^2...).
Takeaway 7
Binary addition rules: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (0 with carry 1), 1+1+1=11 (1 with carry 1).
Takeaway 8
In binary subtraction, borrowing 1 from the next column gives 10 in binary (which equals 2 in decimal).
Takeaway 9
Octal groups 3 binary bits into a single digit (2^3 = 8); Hexadecimal groups 4 bits (2^4 = 16).
Takeaway 10
Hexadecimal numbers are widely used in computing for memory addresses and HTML web color codes.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
Convert the decimal number (53)10 into its equivalent Binary representation using the successive division method.
Reveal Answer & Explanation
Answer: Divide 53 successively by 2:
• 53 / 2 = 26, remainder 1 (LSB)
• 26 / 2 = 13, remainder 0
• 13 / 2 = 6, remainder 1
• 6 / 2 = 3, remainder 0
• 3 / 2 = 1, remainder 1
• 1 / 2 = 0, remainder 1 (MSB)
Reading remainders from bottom to top gives: (110101)2.
Divide by 2 repeatedly: remainders from bottom to top give 110101. Check: 32 + 16 + 4 + 1 = 53.
2
Convert the binary number (110110)2 into its equivalent Decimal value.
Reveal Answer & Explanation
Answer: Expand using powers of 2 from right to left:
(110110)2 = (1 * 32) + (1 * 16) + (0 * 8) + (1 * 4) + (1 * 2) + (0 * 1)
= 32 + 16 + 0 + 4 + 2 + 0 = (54)10.
Sum powers of 2: 32 + 16 + 4 + 2 = 54.
3
Perform the binary addition: (11011)2 + (10110)2.
Reveal Answer & Explanation
Answer: Adding column by column from right to left with carries:
1 + 0 = 1
1 + 1 = 10 (0, carry 1)
1 + 0 + 1 = 10 (0, carry 1)
1 + 1 + 0 = 10 (0, carry 1)
1 + 1 + 1 = 11 (1, carry 1)
Result: (110001)2 [27 + 22 = 49].
Result is 110001 in binary. In decimal: 27 + 22 = 49.
4
Why do computers utilize the Binary number system instead of the human Decimal number system?
Reveal Answer & Explanation
Answer: • Digital computers are built from billions of microscopic electronic switches (transistors).
• An electronic circuit can reliably and unambiguously detect only two distinct voltage states: ON / High Voltage (1) and OFF / Low Voltage (0).
• Designing circuits to distinguish 10 different voltage levels for decimal digits would result in extreme noise sensitivity and calculation errors.
Transistors are two-state electronic switches (ON/OFF). Distinguishing 2 states is simple and error-free.
5
Convert the decimal number (92)10 into its Hexadecimal equivalent.
Reveal Answer & Explanation
Answer: Divide 92 successively by 16:
• 92 / 16 = 5, remainder 12 (represented by the letter C in hexadecimal).
• 5 / 16 = 0, remainder 5.
Reading remainders from bottom to top gives: (5C)16.
Check: (5 * 16) + 12 = 80 + 12 = 92.
92 / 16 = 5 with remainder 12 (C). Result is (5C)16.
6
Convert the hexadecimal number (3B)16 into Decimal.
Reveal Answer & Explanation
Answer: In hexadecimal, B = 11.
(3B)16 = (3 * 16) + (11 * 1) = 48 + 11 = (59)10.
(3 * 16) + (11 * 1) = 48 + 11 = 59.
7
Perform the binary subtraction: (1101)2 - (1010)2.
Reveal Answer & Explanation
Answer: Subtracting bit by bit:
1 - 0 = 1
0 - 1 = 1 (with borrow 1 from next column)
0 - 0 = 0
1 - 1 = 0
Result: (0011)2 = (11)2 [13 - 10 = 3].
1101 (13) - 1010 (10) = 0011 (3).
8
What are the alphanumeric symbols used in the Hexadecimal number system? What do the letters A through F represent?
Reveal Answer & Explanation
Answer: • The Hexadecimal number system uses 16 unique alphanumeric symbols: digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and the six letters A, B, C, D, E, F.
• Numerical Equivalents: A = 10, B = 11, C = 12, D = 13, E = 14, F = 15.
Uses digits 0-9 and letters A through F representing numerical values 10 to 15.
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