# Number System - Grade 10 Notes

## Complete Chapter Notes with All Exercises - NEB Curriculum

## 🎯 Chapter Overview

Have you ever wondered how computers, which are just electronic circuits, can handle everything from photos and videos to complex calculations? The secret lies in the Number System. While we humans use a system with ten digits (0-9), computers speak a much simpler language with just two: 0 and 1. 

This chapter is your key to understanding the language of computers. We will explore the different number systems used in computing, with a focus on how to convert numbers between them and perform basic calculations in the binary system. This is a fundamental and high-scoring topic in computer science. Every concept is broken down into simple steps with plenty of examples to guide you.

**Weightage:** 6-8 marks in SEE examination  
**Time Required:** 12-15 periods

## 📚 Learning Objectives

After studying this chapter, you will be able to:

- Define what a number system is and identify its different types
- Understand the concept of base or radix
- Convert numbers from one number system (Decimal, Binary, Octal, Hexadecimal) to another
- Perform binary arithmetic: addition, subtraction, and multiplication
- Define fundamental data units like BIT and Byte

## 🔢 1. Introduction to Number Systems

A **Number System** is a systematic way of representing or expressing numbers. Every number system has a set of unique characters or digits and a specific **base (or radix)**, which is the total number of unique digits used in that system.

### Types of Number Systems

| System | Base | Digits Used | Example | Why it's important |
|--------|------|-------------|---------|-------------------|
| **Decimal** | **10** | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 | (257)₁₀ | The everyday system used by humans. |
| **Binary** | **2** | 0, 1 | (1101)₂ | The fundamental language of computers (0=Off, 1=On). |
| **Octal** | **8** | 0, 1, 2, 3, 4, 5, 6, 7 | (401)₈ | A shorthand for representing long binary numbers. |
| **Hexadecimal** | **16** | 0-9, A(10), B(11), C(12), D(13), E(14), F(15) | (1A5)₁₆ | A more compact shorthand for binary; used in memory addressing and color codes (e.g., #FFFFFF for white). |

### BITs and Bytes: The Units of Data

- **BIT (Binary Digit):** The smallest unit of data in a computer. It can have a value of either **0** or **1**.
- **Byte:** A group of **8 bits**. A byte is the basic unit used to represent a character, like the letter 'A' or the symbol '$'.
  - 1 Byte = 8 Bits
  - 1 Kilobyte (KB) = 1024 Bytes

## 🔄 2. Number System Conversions

This is the most important skill to learn in this chapter. Here are the step-by-step methods for all conversions.

### A. Conversion from Decimal to Other Bases (Division Method)

**Rule:** Continuously divide the decimal number by the new base until the quotient is 0. Then, write the remainders in **reverse order** (from bottom to top).

#### 1. Decimal to Binary

**Convert (25)₁₀ to Binary:**

```
2 | 25 | Remainder
2 | 12 |     1
2 |  6 |     0
2 |  3 |     0
2 |  1 |     1
  |  0 |     1   (Read upwards)
```

**Answer:** (25)₁₀ = (11001)₂

#### 2. Decimal to Octal

**Convert (150)₁₀ to Octal:**

```
8 | 150 | Remainder
8 |  18 |     6
8 |   2 |     2
  |   0 |     2   (Read upwards)
```

**Answer:** (150)₁₀ = (226)₈

#### 3. Decimal to Hexadecimal

**Convert (423)₁₀ to Hexadecimal:** (Remember: 10=A, 11=B, 12=C, 13=D, ...)

```
16 | 423 | Remainder
16 |  26 |     7
16 |   1 |    10 (which is A)
   |   0 |     1         (Read upwards)
```

**Answer:** (423)₁₀ = (1A7)₁₆

### B. Conversion to Decimal from Other Bases (Power Method)

**Rule:** Multiply each digit by its positional weight (base raised to the power of its position, starting from 0 on the right) and add the results.

#### 1. Binary to Decimal

Convert (1101)₂ to Decimal:
= (1×2³) + (1×2²) + (0×2¹) + (1×2⁰)
= (1×8) + (1×4) + (0×2) + (1×1)
= 8 + 4 + 0 + 1 = 13

**Answer:** (1101)₂ = (13)₁₀

#### 2. Octal to Decimal

Convert (345)₈ to Decimal:
= (3×8²) + (4×8¹) + (5×8⁰)
= (3×64) + (4×8) + (5×1)
= 192 + 32 + 5 = 229

**Answer:** (345)₈ = (229)₁₀

#### 3. Hexadecimal to Decimal

Convert (2A7)₁₆ to Decimal: (Remember: A=10)
= (2×16²) + (10×16¹) + (7×16⁰)
= (2×256) + (10×16) + (7×1)
= 512 + 160 + 7 = 679

**Answer:** (2A7)₁₆ = (679)₁₀

### C. Conversions Between Binary, Octal, and Hexadecimal (Shortcut Method)

**Rule:** Use grouping. This is much faster than converting to decimal first.

#### 1. Binary to Octal (Group in 3s)

**Convert (11010111)₂ to Octal:**
- Group digits in sets of three from the right: 11 010 111
- Add leading zeros to the first group to make it three digits: 011 010 111
- Convert each group: 011 = 3, 010 = 2, 111 = 7

**Answer:** (327)₈

#### 2. Binary to Hexadecimal (Group in 4s)

**Convert (11010111)₂ to Hexadecimal:**
- Group digits in sets of four from the right: 1101 0111
- Convert each group: 1101 = 13 = D, 0111 = 7

**Answer:** (D7)₁₆

#### 3. Octal to Binary (Expand to 3 bits)

**Convert (562)₈ to Binary:**
- Convert each octal digit to its 3-bit binary equivalent:
- 5 = 101, 6 = 110, 2 = 010
- Combine them: 101 110 010

**Answer:** (101110010)₂

#### 4. Hexadecimal to Binary (Expand to 4 bits)

**Convert (9F2)₁₆ to Binary:**
- Convert each hexadecimal digit to its 4-bit binary equivalent:
- 9 = 1001, F (15) = 1111, 2 = 0010
- Combine them: 1001 1111 0010

**Answer:** (100111110010)₂

## ➕ 3. Binary Arithmetic

Computers perform all calculations using binary arithmetic.

### A. Binary Addition

**Rules:**
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 0 (with a **carry** of 1 to the next column)
- 1 + 1 + 1 = 1 (with a **carry** of 1 to the next column)

**Example: Add (1101)₂ and (1011)₂**

```
  111   (Carries)
  1101
+ 1011
-------
 11000
```

**Answer:** (11000)₂

### B. Binary Subtraction

**Rules:**
- 0 − 0 = 0
- 1 − 0 = 1
- 1 − 1 = 0
- 0 − 1 = 1 (with a **borrow** of 1 from the next column)

**Example: Subtract (101)₂ from (1100)₂**

```
  1 1 0 0  (Original)
-   1 0 1
---------
    1 1 1
```

**Explanation:**
- Rightmost column: 0 - 1 requires a borrow. The next 0 also needs to borrow, so we borrow from the '1' two places over. The first '0' becomes '10' (2), then lends one, becoming '1'. The rightmost '0' becomes '10' (2).
- So, 10 - 1 = 1. Next column is 1 - 0 = 1. Next is 0 - 1, requiring a borrow. The leftmost '1' becomes '0', this column becomes 10 - 1 = 1.

**Answer:** (0111)₂ or (111)₂

### C. Binary Multiplication

**Rules:** (Same as decimal multiplication)
- 0 × 0 = 0
- 0 × 1 = 0
- 1 × 0 = 0
- 1 × 1 = 1

**Example: Multiply (101)₂ by (11)₂**

```
    101
  x  11
  ------
    101   (101 × 1)
+ 1010   (101 × 1, shifted one place left)
  --------
  1111
```

**Answer:** (1111)₂

## 📋 4. Complete Exercise Solutions

### 1. Answer the following questions:

**a. What is a number system? List the different types of number systems.**

**Answer:** A Number System is a method or system used to represent numbers. The four common types used in computing are:
- Decimal Number System (Base-10)
- Binary Number System (Base-2)
- Octal Number System (Base-8)
- Hexadecimal Number System (Base-16)

**b. What is a binary number system?**

**Answer:** The number system that uses only two digits, 0 and 1, is known as the binary number system. It is the fundamental language of computers.

**c. What is a hexadecimal number system? List its digits.**

**Answer:** The number system that uses 16 unique digits to represent numbers is known as the hexadecimal number system. Its digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.

**d. What is an octal number system? List its digits.**

**Answer:** The number system that uses 8 unique digits to represent numbers is known as the octal number system. Its digits are 0, 1, 2, 3, 4, 5, 6, and 7.

**e. What are BIT and Byte?**

**Answer:**
- **BIT (Binary Digit):** The smallest unit of data in a computer, representing either a 0 or a 1.
- **Byte:** A group of 8 bits, typically used to represent a single character.

### 2. Perform the following Binary Calculations:

**(More Solved Examples)**

**a. Addition:** (11101)₂ + (10110)₂

```
  111    (Carries)
  11101
+ 10110
---------
 110011
```

**Answer:** (110011)₂

**b. Subtraction:** (11011)₂ − (1110)₂

```
  11011
-  1110
---------
   1101
```

**Answer:** (1101)₂

**c. Multiplication:** (110)₂ × (101)₂

```
    110
  x 101
  ------
    110
   0000
+ 11000
---------
  11110
```

**Answer:** (11110)₂

## ✅ 5. Chapter Checklist & Exam Tips

**Exam Hotspots 🔥:**

- **Conversions:** This is the most frequently tested topic. Be perfect in converting from Decimal to Binary and Binary to Decimal. The shortcut methods (Binary to Octal/Hex) are also very important.
- **Binary Arithmetic:** Expect at least one question on binary addition or subtraction.
- **Definitions:** Be ready to define the four number systems, BIT, and Byte.

**Your Pre-Exam Checklist:**

- [ ] I can define the four main number systems and their bases.
- [ ] I can convert a decimal number to binary, octal, and hexadecimal.
- [ ] I can convert a binary, octal, or hexadecimal number to decimal.
- [ ] I can use the shortcut method to convert between binary and octal/hexadecimal.
- [ ] I can perform binary addition, subtraction, and multiplication.

## 🎊 Conclusion

Understanding number systems is like learning the alphabet of the digital world. While it seems abstract, it's the foundation upon which all modern technology is built. Mastering conversions and binary math will give you a deeper appreciation for how computers work and a solid base for further studies in science and technology.

**Key to Success:** Practice Conversions Daily + Memorize the Binary Math Rules = Top Marks!