Grade 9||Number System|| Notes

Number System

1. Evolution & Introduction

Early counting methods: fingers, sticks, stones, knots.

Development of symbols → systematic number systems.

Modern usage: Different systems for different fields (daily life, computing, engineering).

2. What is a Number System?

A set of symbols and rules used to represent numbers and perform arithmetic.

Base/Radix: Total number of distinct digits used.

Example: Decimal base = 10 → digits 0–9.

3. Types of Number Systems

🔢 Decimal System (Base 10)

Digits: 0–9.

Place values: Units (10⁰), Tens (10¹), Hundreds (10²), etc.

Example:

• (719)10=7×102+1×101+9×100(719)10​=7×102+1×101+9×100

🔢 Binary System (Base 2)

Digits: 0, 1.

Used by computers (on/off states, transistors).

Example:

= 32 + 16 + 0 + 4 + 2 + 1 = (55)₁₀

🔢 Octal System (Base 8)

Digits: 0–7.

Used in computing (shorter than binary).

Example:

🔢 Hexadecimal System (Base 16)

Digits: 0–9, A(10), B(11), C(12), D(13), E(14), F(15).

Widely used in programming, memory addressing, color codes.

Example:

4. Binary Arithmetic (Deep Explanation)

4.1 Binary Addition

Rules:

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 0 carry 1 (i.e., 10 in binary)

Example:

  1 1 1 0 1  (29 in decimal)
+   1 0 1 1  (11 in decimal)
------------
1 0 1 0 0 0  (40 in decimal)

Carry explained:

Column from right:
1 + 1 = 0, carry 1
0 + 1 + carry 1 = 0, carry 1
1 + 0 + carry 1 = 0, carry 1
1 + 1 + carry 1 = 1, carry 1
1 + carry 1 = 10

4.2 Binary Subtraction

Rules:

0 – 0 = 0

1 – 0 = 1

1 – 1 = 0

0 – 1 = 1 borrow 1 from left column

Example:

  1 0 1 1 0  (22 in decimal)
-   1 0 1 1  (11 in decimal)
------------
 0 1 1 0 1  (13 in decimal)

In column 2 (from right): 0 − 1 → borrow from column 3, becomes 10 − 1 = 1.

4.3 Binary Multiplication

Rules: Same as decimal, but simpler:

0 × 0 = 0
0 × 1 = 0
1 × 0 = 0
1 × 1 = 1

Example:

      1 0 1  (5)
   × 1 1 0  (6)
   --------
     0 0 0
   1 0 1
+ 1 0 1
-----------
 1 1 1 1 0  (30)

14.4 Binary Division

Process: Similar to long division in decimal.
Example: (1101)₂ ÷ (10)₂

      1 1 0  (quotient = 6 in decimal)
    --------
10 | 1 1 0 1
     - 1 0
     -----
       1 0
      - 1 0
      -----
         0 1  (remainder = 1)

Result: Quotient = 110₂ (6), Remainder = 1.

5. Number System Conversions

5.1 Decimal to Other Bases

General Rule: Repeated division by the target base.

Decimal → Binary

Divide by 2, note remainder.

Repeat until quotient is 0.

Write remainders bottom to top.

Example: (46)₁₀ → (?)₂

(46)₁₀ → (?)₂
46 ÷ 2 = 23 rem 0
23 ÷ 2 = 11 rem 1
11 ÷ 2 = 5 rem 1
5 ÷ 2 = 2 rem 1
2 ÷ 2 = 1 rem 0
1 ÷ 2 = 0 rem 1
Read ↑ = 101110₂

(46)₁₀ = (101110)₂

Decimal → Octal

Divide by 8, note remainder.

Example: (345)₁₀ → (?)₈

(345)₁₀ → (?)₈
345 ÷ 8 = 43 rem 1
43 ÷ 8 = 5 rem 3
5 ÷ 8 = 0 rem 5
Read ↑ = 531₈

Decimal → Hexadecimal: Divide by 16, note remainder.
(255)₁₀ → (?)₁₆
255 ÷ 16 = 15 rem F
15 ÷ 16 = 0 rem F
Read ↑ = FF₁₆

5.2 Binary/Octal/Hex → Decimal

Method: Expand using place values (powers of base).

Binary → Decimal:
(1011)₂ = 1×2³ + 0×2² + 1×2¹ + 1×2⁰
= 8 + 0 + 2 + 1 = 11₁₀

Octal → Decimal:
(157)₈ = 1×8² + 5×8¹ + 7×8⁰
= 64 + 40 + 7 = 111₁₀

Hex → Decimal:
(1A3)₁₆ = 1×16² + 10×16¹ + 3×16⁰
= 256 + 160 + 3 = 419₁₀

Binary ↔ Octal

Binary → Octal: Group binary digits into 3 from right, convert each to octal.

(101110)₂ → 101 | 110 → 5 | 6 → (56)₈

Octal → Binary: Convert each octal digit to 3 binary digits.

(F3)₁₆ → F=1111, 3=0011 → (11110011)₂

octal ↔ Hexadecimal (via Binary)

Convert octal → binary → hex (or reverse).

Example: (345)₈ → binary → hex:

3=011, 4=100, 5=101 → (011100101)₂
Group 4: 0001 1100 1010? (pad leading zeros)
Actually: 011100101 → 0|1110|0101 → 0 E 5 → (E5)₁₆

6. Why Conversions Matter

1. Computers use binary (0s and 1s).
2. Humans use decimal.
3. Octal and hexadecimal are shorter forms of binary used in programming, networking (IP addresses), digital circuits, cryptography, etc.
4. Conversions allow communication between humans and machines.

7. Memory Aid & Tips

Base Values: B2 O8 D10 H16 (Binary 2, Octal 8, Decimal 10, Hex 16).

Hex Letters: A=10, B=11, C=12, D=13, E=14, F=15.

Binary Groups: Octal → 3 bits, Hex → 4 bits.

Conversion Flowchart:

Decimal ←→ Binary ←→ Octal
         ↓
       Hex

8. Common Mistakes to Avoid

Forgetting to write base subscript: (101)₂ not just 101.

In binary addition: 1 + 1 = 0 carry 1, not 2.

In conversions: Group binary from right to left for octal/hex.

Hex letters A–F are case-insensitive but often written uppercase.