Chapter 1: Numbers - Binary & Decimal Systems

Exercise | Class 11 Applied Mathematics

Comprehensive coverage of number system conversions and binary arithmetic operations

📊Chapter Overview - Mind Map

Numbers - Binary & Decimal Systems

Introduction & Prerequisites

Binary vs Decimal Coordination

Prior Conversions Review

Binary Arithmetic

Decimal to Binary Conversion

Repeated Division by 2

Remainder Reading Trick

Binary to Decimal Conversion

Sum of Powers of 2

Powers Reference Table

Binary Operations

Addition

Subtraction

Multiplication

Division

Universal Tips & Verification

Powers Memorization

Decimal Verification

📚Key Topics

1. Introduction & Prerequisites Review

Before diving into conversions and arithmetic, review the fundamental coordination between binary and decimal systems.

ℹ️

Binary vs Decimal Coordination

Decimal (Base 10): Uses digits 0-9, powers of 10 Binary (Base 2): Uses digits 0-1, powers of 2 Every number has unique representation in both systems

Prior Conversions & Binary Arithmetic

Decimal → Binary:Repeated division by 2
Binary → Decimal:Sum of powers of 2
Binary Arithmetic:Add, subtract, multiply, divide without converting

2. Decimal to Binary Conversion

Converting a decimal number to binary involves repeatedly dividing by 2 and reading remainders from bottom to top.

Conversion Formula

Decimal_{10} = Remainder_1 \times 2^0 + Remainder_2 \times 2^1 + Remainder_3 \times 2^2 + ... (Read Remainders Bottom-to-Top)

ℹ️

Method: Repeated Division by 2

Step 1: Divide the number by 2\nStep 2: Record the remainder (0 or 1)\nStep 3: Divide the quotient by 2 again\nStep 4: Repeat until quotient becomes 0\nStep 5: Read remainders from BOTTOM to TOP to get binary

Example: Convert 13 to Binary

$13 div 2 = 6$ R $1$

$6 div 2 = 3$ R $0$

$3 div 2 = 1$ R $1$

$1 div 2 = 0$ R $1$

Read remainders bottom-to-top: $1101_2$

💡

Quick Trick

The binary representation has no leading zeros. The number of binary digits = ⌊log₂(n)⌋ + 1

More Examples:

47₁₀ = 101111₂

129₁₀ = 10000001₂

250₁₀ = 11111010₂

394₁₀ = 110001010₂

639₁₀ = 1001111111₂

1018₁₀ = 1111111010₂

3. Binary to Decimal Conversion

Converting binary to decimal involves summing the powers of 2 where 1s appear in the binary number.

Conversion Formula

Binary_2 = (b_n \times 2^n) + (b_{n-1} \times 2^{n-1}) + ... + (b_1 \times 2^1) + (b_0 \times 2^0), \text{ where } b_i \in {0,1}

ℹ️

Method: Sum of Powers of 2

Step 1: Write powers of 2 above each digit (rightmost =

2^0

)\nStep 2: Multiply each digit by its power of 2\nStep 3: Sum all the products\nStep 4: Result is the decimal number

Powers of 2 Reference Table

2⁰ = 1

2¹ = 2

2² = 4

2³ = 8

2⁴ = 16

2⁵ = 32

2⁶ = 64

2⁷ = 128

2⁸ = 256

2⁹ = 512

2¹⁰ = 1024

Example: Convert $11111001_2$ to Decimal

Powers: $2^7$ $2^6$ $2^5$ $2^4$ $2^3$ $2^2$ $2^1$ $2^0$

Binary: $1$ $1$ $1$ $1$ $1$ $0$ $0$ $1$

Sum: $128 + 64 + 32 + 16 + 8 + 0 + 0 + 1$

= $249_10$

4. Binary Addition

Binary addition follows simple rules similar to decimal addition, but with base 2.

Addition Formula

A_2 + B_2 = \text{Sum}_2 \mid \text{Rules: } 0+0=0, 0+1=1, 1+0=1, 1+1=10 \text{ (0 carry 1)}

ℹ️

Binary Addition Rules

0 + 0 = 0\n0 + 1 = 1\n1 + 0 = 1\n1 + 1 = 10 (Write 0, Carry 1)\n1 + 1 + 1 = 11 (Write 1, Carry 1)

Example: 10001₂ + 1101₂

10001

+ 1101

11110

Verification: 10001₂ = 17₁₀, 1101₂ = 13₁₀, 11110₂ = 30₁₀
17 + 13 = 30 ✓

More Examples:

10111₂ + 1111₂ = 100110₂ (23 + 15 = 38) ✓

1000011₂ + 110100₂ = 1110111₂ (67 + 52 = 119) ✓

10000000₂ + 1011010₂ = 11010110₂ (128 + 90 = 218) ✓

5. Binary Subtraction

Binary subtraction uses borrowing similar to decimal subtraction when needed.

Subtraction Formula

A_2 - B_2 = \text{Diff}_2 \mid \text{Rules: } 0-0=0, 1-0=1, 1-1=0, 0-1=\text{Borrow}

ℹ️

Binary Subtraction Rules

0 - 0 = 0\n1 - 0 = 1\n1 - 1 = 0\n0 - 1 = 1 (After Borrow)\nChain Borrowing: Borrow from the next available 1

Example: 10111₂ - 10000₂

10111

- 10000

111

Verification: 10111₂ = 23₁₀, 10000₂ = 16₁₀, 111₂ = 7₁₀
23 - 16 = 7 ✓

More Examples:

110001₂ - 11001₂ = 11000₂ (49 - 25 = 24) ✓

1101100₂ - 111001₂ = 10101₂ (108 - 57 = 51) ✓

1111110₂ - 1001010₂ = 110100₂ (126 - 74 = 52) ✓

💡

Chain Borrowing Tip

When you encounter 0 - 1, look left for the nearest 1. That 1 becomes 0, all 0s in between become 1s, and your current position becomes 10.

6. Binary Multiplication

Binary multiplication is simpler than decimal multiplication since we only multiply by 0 or 1.

Multiplication Formula

A_2 \times B_2 = \text{Product}_2 \mid \text{Rules: } 0 \times X = 0, 1 \times X = X

ℹ️

Binary Multiplication Rules

0 times X = 0\n1 times X = X\nProcess: Multiply digit by digit, then sum partial products (with shifts)

Example: 1100₂ × 111₂

1100

× 111

1100 (×1)

1100 (×1, shift 1)

1100 (×1, shift 2)

1010100

Verification: 1100₂ = 12₁₀, 111₂ = 7₁₀, 1010100₂ = 84₁₀
12 × 7 = 84 ✓

More Examples:

10111₂ × 1001₂ = 11001111₂ (23 × 9 = 207) ✓

101011₂ × 1101₂ = 100110111₂ (43 × 13 = 559) ✓

110110₂ × 10001₂ = 11100110₂ (54 × 17 = 918) ✓

7. Binary Division

Binary division uses long division similar to decimal division, but with base 2 logic.

Division Formula

A₂ \div B₂ = Quotient₂ with Remainder₂ \mid Dividend = (Divisor \times Quotient) + Remainder

ℹ️

Binary Division Method

Step 1: Use long division\nStep 2: Compare prefix with divisor\nStep 3: If Prefix >= Divisor, write 1 in quotient, subtract divisor\nStep 4: If Prefix < Divisor, write 0 in quotient\nStep 5: Bring down next digit and repeat

Example: 11100₂ ÷ 111₂

11100 ÷ 111 = 100 R 0

Verification: 11100₂ = 28₁₀, 111₂ = 7₁₀, 100₂ = 4₁₀
28 ÷ 7 = 4 remainder 0 ✓

More Examples:

10000001₂ ÷ 1111₂ = 1000 R 1001 (129 ÷ 15 = 8 R 9) ✓

100000100₂ ÷ 101₂ = 10100 R 0 (260 ÷ 5 = 52 R 0) ✓

8. Universal Tips & Verification

💡

Memorization: Powers of 2

Memorize powers from 2⁰ = 1 to 2¹⁰ = 1024. These are essential for quick conversions.

❗

Always Verify Your Work

For every binary operation, convert to decimal, perform the operation, and verify the result. This catches calculation errors.

ℹ️

Formatting Rules

• No leading zeros in binary\n• Use subscript 2 for binary (e.g.,

1101_2

)\n• Use subscript 10 for decimal (e.g.,

13_{10}

)

💡

Exam Strategy

✓ Show all working steps\n✓ Verify using decimal conversion\n✓ Double-check for small carry/borrow errors

✓Key Takeaways

Decimal to binary: Repeated division by 2, read remainders bottom-to-top
Binary to decimal: Sum of powers of 2 where 1s appear
Binary addition: Simple rules with carry (1+1=10)
Binary subtraction: Use borrowing from left when needed (0-1 requires borrow)
Binary multiplication: 0×X=0, 1×X=X, then add partial products
Binary division: Long division with base-2 logic
Always verify: Convert results to decimal to check correctness
Memorize powers of 2 up to 2¹⁰ = 1024 for efficiency
Practice is essential - these skills form the foundation for computer science and digital systems

Loading Knowledge...

Preparing your next chapter

Chapter 1: Numbers - Binary & Decimal Systems

Exercise | Class 11 Applied Mathematics

Comprehensive coverage of number system conversions and binary arithmetic operations

📊Chapter Overview - Mind Map

Numbers - Binary & Decimal Systems

Introduction & Prerequisites

Binary vs Decimal Coordination

Prior Conversions Review

Binary Arithmetic

Decimal to Binary Conversion

Repeated Division by 2

Remainder Reading Trick

Binary to Decimal Conversion

Sum of Powers of 2

Powers Reference Table

Binary Operations

Addition

Subtraction

Multiplication

Division

Universal Tips & Verification

Powers Memorization

Decimal Verification

📚Key Topics

1. Introduction & Prerequisites Review

Before diving into conversions and arithmetic, review the fundamental coordination between binary and decimal systems.

ℹ️

Binary vs Decimal Coordination

Decimal (Base 10): Uses digits 0-9, powers of 10 Binary (Base 2): Uses digits 0-1, powers of 2 Every number has unique representation in both systems

Prior Conversions & Binary Arithmetic

Decimal → Binary:Repeated division by 2
Binary → Decimal:Sum of powers of 2
Binary Arithmetic:Add, subtract, multiply, divide without converting

2. Decimal to Binary Conversion

Converting a decimal number to binary involves repeatedly dividing by 2 and reading remainders from bottom to top.

Conversion Formula

Decimal_{10} = Remainder_1 \times 2^0 + Remainder_2 \times 2^1 + Remainder_3 \times 2^2 + ... (Read Remainders Bottom-to-Top)

ℹ️

Method: Repeated Division by 2

Example: Convert 13 to Binary

$13 div 2 = 6$ R $1$

$6 div 2 = 3$ R $0$

$3 div 2 = 1$ R $1$

$1 div 2 = 0$ R $1$

Read remainders bottom-to-top: $1101_2$

💡

Quick Trick

The binary representation has no leading zeros. The number of binary digits = ⌊log₂(n)⌋ + 1

More Examples:

47₁₀ = 101111₂

129₁₀ = 10000001₂

250₁₀ = 11111010₂

394₁₀ = 110001010₂

639₁₀ = 1001111111₂

1018₁₀ = 1111111010₂

3. Binary to Decimal Conversion

Converting binary to decimal involves summing the powers of 2 where 1s appear in the binary number.

Conversion Formula

Binary_2 = (b_n \times 2^n) + (b_{n-1} \times 2^{n-1}) + ... + (b_1 \times 2^1) + (b_0 \times 2^0), \text{ where } b_i \in {0,1}

ℹ️

Method: Sum of Powers of 2

Step 1: Write powers of 2 above each digit (rightmost =

2^0

)\nStep 2: Multiply each digit by its power of 2\nStep 3: Sum all the products\nStep 4: Result is the decimal number

Powers of 2 Reference Table

2⁰ = 1

2¹ = 2

2² = 4

2³ = 8

2⁴ = 16

2⁵ = 32

2⁶ = 64

2⁷ = 128

2⁸ = 256

2⁹ = 512

2¹⁰ = 1024

Example: Convert $11111001_2$ to Decimal

Powers: $2^7$ $2^6$ $2^5$ $2^4$ $2^3$ $2^2$ $2^1$ $2^0$

Binary: $1$ $1$ $1$ $1$ $1$ $0$ $0$ $1$

Sum: $128 + 64 + 32 + 16 + 8 + 0 + 0 + 1$

= $249_10$

4. Binary Addition

Binary addition follows simple rules similar to decimal addition, but with base 2.

Addition Formula

A_2 + B_2 = \text{Sum}_2 \mid \text{Rules: } 0+0=0, 0+1=1, 1+0=1, 1+1=10 \text{ (0 carry 1)}

ℹ️

Binary Addition Rules

0 + 0 = 0\n0 + 1 = 1\n1 + 0 = 1\n1 + 1 = 10 (Write 0, Carry 1)\n1 + 1 + 1 = 11 (Write 1, Carry 1)

Example: 10001₂ + 1101₂

10001

+ 1101

11110

Verification: 10001₂ = 17₁₀, 1101₂ = 13₁₀, 11110₂ = 30₁₀
17 + 13 = 30 ✓

More Examples:

10111₂ + 1111₂ = 100110₂ (23 + 15 = 38) ✓

1000011₂ + 110100₂ = 1110111₂ (67 + 52 = 119) ✓

10000000₂ + 1011010₂ = 11010110₂ (128 + 90 = 218) ✓

5. Binary Subtraction

Binary subtraction uses borrowing similar to decimal subtraction when needed.

Subtraction Formula

A_2 - B_2 = \text{Diff}_2 \mid \text{Rules: } 0-0=0, 1-0=1, 1-1=0, 0-1=\text{Borrow}

ℹ️

Binary Subtraction Rules

0 - 0 = 0\n1 - 0 = 1\n1 - 1 = 0\n0 - 1 = 1 (After Borrow)\nChain Borrowing: Borrow from the next available 1

Example: 10111₂ - 10000₂

10111

- 10000

111

Verification: 10111₂ = 23₁₀, 10000₂ = 16₁₀, 111₂ = 7₁₀
23 - 16 = 7 ✓

More Examples:

110001₂ - 11001₂ = 11000₂ (49 - 25 = 24) ✓

1101100₂ - 111001₂ = 10101₂ (108 - 57 = 51) ✓

1111110₂ - 1001010₂ = 110100₂ (126 - 74 = 52) ✓

💡

Chain Borrowing Tip

When you encounter 0 - 1, look left for the nearest 1. That 1 becomes 0, all 0s in between become 1s, and your current position becomes 10.

6. Binary Multiplication

Binary multiplication is simpler than decimal multiplication since we only multiply by 0 or 1.

Multiplication Formula

A_2 \times B_2 = \text{Product}_2 \mid \text{Rules: } 0 \times X = 0, 1 \times X = X

ℹ️

Binary Multiplication Rules

0 times X = 0\n1 times X = X\nProcess: Multiply digit by digit, then sum partial products (with shifts)

Example: 1100₂ × 111₂

1100

× 111

1100 (×1)

1100 (×1, shift 1)

1100 (×1, shift 2)

1010100

Verification: 1100₂ = 12₁₀, 111₂ = 7₁₀, 1010100₂ = 84₁₀
12 × 7 = 84 ✓

More Examples:

10111₂ × 1001₂ = 11001111₂ (23 × 9 = 207) ✓

101011₂ × 1101₂ = 100110111₂ (43 × 13 = 559) ✓

110110₂ × 10001₂ = 11100110₂ (54 × 17 = 918) ✓

7. Binary Division

Binary division uses long division similar to decimal division, but with base 2 logic.

Division Formula

A₂ \div B₂ = Quotient₂ with Remainder₂ \mid Dividend = (Divisor \times Quotient) + Remainder

ℹ️

Binary Division Method

Example: 11100₂ ÷ 111₂

11100 ÷ 111 = 100 R 0

Verification: 11100₂ = 28₁₀, 111₂ = 7₁₀, 100₂ = 4₁₀
28 ÷ 7 = 4 remainder 0 ✓

More Examples:

10000001₂ ÷ 1111₂ = 1000 R 1001 (129 ÷ 15 = 8 R 9) ✓

100000100₂ ÷ 101₂ = 10100 R 0 (260 ÷ 5 = 52 R 0) ✓

8. Universal Tips & Verification

💡

Memorization: Powers of 2

Memorize powers from 2⁰ = 1 to 2¹⁰ = 1024. These are essential for quick conversions.

❗

Always Verify Your Work

For every binary operation, convert to decimal, perform the operation, and verify the result. This catches calculation errors.

ℹ️

Formatting Rules

• No leading zeros in binary\n• Use subscript 2 for binary (e.g.,

1101_2

)\n• Use subscript 10 for decimal (e.g.,

13_{10}

)

💡

Exam Strategy

✓ Show all working steps\n✓ Verify using decimal conversion\n✓ Double-check for small carry/borrow errors

✓Key Takeaways

Decimal to binary: Repeated division by 2, read remainders bottom-to-top
Binary to decimal: Sum of powers of 2 where 1s appear
Binary addition: Simple rules with carry (1+1=10)
Binary subtraction: Use borrowing from left when needed (0-1 requires borrow)
Binary multiplication: 0×X=0, 1×X=X, then add partial products
Binary division: Long division with base-2 logic
Always verify: Convert results to decimal to check correctness
Memorize powers of 2 up to 2¹⁰ = 1024 for efficiency
Practice is essential - these skills form the foundation for computer science and digital systems

📊Chapter Overview - Mind Map

📚Key Topics

1. Introduction & Prerequisites Review

Binary vs Decimal Coordination

Prior Conversions & Binary Arithmetic

2. Decimal to Binary Conversion

Conversion Formula

Method: Repeated Division by 2

Example: Convert 13 to Binary

Quick Trick

More Examples:

3. Binary to Decimal Conversion

Conversion Formula

Method: Sum of Powers of 2

Powers of 2 Reference Table

Example: Convert $11111001_2$ to Decimal

4. Binary Addition

Addition Formula

Binary Addition Rules

Example: 10001₂ + 1101₂

More Examples:

5. Binary Subtraction

Subtraction Formula

Binary Subtraction Rules

Example: 10111₂ - 10000₂

More Examples:

Chain Borrowing Tip

6. Binary Multiplication

Multiplication Formula

Binary Multiplication Rules

Example: 1100₂ × 111₂

More Examples:

7. Binary Division

Division Formula

Binary Division Method

Example: 11100₂ ÷ 111₂

More Examples:

8. Universal Tips & Verification

Memorization: Powers of 2

Always Verify Your Work

Formatting Rules

Exam Strategy

❓Important Questions & Problems

Chapter 1 Practice Problems

✓Key Takeaways

Loading Knowledge...

📊Chapter Overview - Mind Map

📚Key Topics

1. Introduction & Prerequisites Review

Binary vs Decimal Coordination

Prior Conversions & Binary Arithmetic

2. Decimal to Binary Conversion

Conversion Formula

Method: Repeated Division by 2

Example: Convert 13 to Binary

Quick Trick

More Examples:

3. Binary to Decimal Conversion

Conversion Formula

Method: Sum of Powers of 2

Powers of 2 Reference Table

Example: Convert $11111001_2$ to Decimal

4. Binary Addition

Addition Formula

Binary Addition Rules

Example: 10001₂ + 1101₂

More Examples:

5. Binary Subtraction

Subtraction Formula

Binary Subtraction Rules

Example: 10111₂ - 10000₂

More Examples:

Chain Borrowing Tip

6. Binary Multiplication

Multiplication Formula

Binary Multiplication Rules

Example: 1100₂ × 111₂

More Examples:

7. Binary Division

Division Formula