Section 1: Mindmap

Section 2: Main Content with Inline Diagrams

Introduction to Number Systems

Before learning how to convert decimal numbers to binary, it is essential to understand the two number systems involved:

• Decimal Number System (Base 10): This is the standard number system used in daily life. It uses digits from 0 to 9. Each digit's place value is a power of 10. For example, the number 345 means \(3 \times 10^2 + 4 \times 10^1 + 5 \times 10^0\).
• Binary Number System (Base 2): This system is used internally by almost all modern computers and digital systems. It uses only two digits: 0 and 1. Each digit's place value is a power of 2. For example, the binary number 1011 means \(1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 11\) in decimal.

Why Convert Decimal to Binary?

Computers operate using binary numbers because digital circuits have two states: ON (1) and OFF (0). Hence, understanding how to convert decimal numbers (human-friendly) to binary numbers (machine-friendly) is fundamental in computer science and digital electronics.

Conversion Method: Division by 2

The most common and straightforward method to convert a decimal number to binary is the Division by 2 Method. This method involves repeatedly dividing the decimal number by 2 and recording the remainders.

Procedure:

1. Divide the decimal number by 2.
2. Write down the remainder (0 or 1).
3. Update the decimal number to the quotient obtained.
4. Repeat steps 1 to 3 until the quotient becomes 0.
5. The binary number is the sequence of remainders read from bottom to top (last remainder is the most significant bit).

Example Diagram:

Decimal Number: 13Step 1: 13 ÷ 2 = 6 remainder 1Step 2: 6 ÷ 2 = 3 remainder 0Step 3: 3 ÷ 2 = 1 remainder 1Step 4: 1 ÷ 2 = 0 remainder 1Reading remainders from bottom to top: 1101Therefore, 13 in decimal = 1101 in binary.

Mathematical Explanation

Any decimal number \(N\) can be expressed in binary as:

where each \(b_i\) is either 0 or 1.

The division by 2 method essentially finds the coefficients \(b_i\) by repeatedly dividing \(N\) by 2 and taking the remainder as \(b_0, b_1, \ldots\).

Example of Conversion for a Larger Number

Convert decimal 156 to binary:

156 ÷ 2 = 78 remainder 078 ÷ 2 = 39 remainder 039 ÷ 2 = 19 remainder 119 ÷ 2 = 9 remainder 19 ÷ 2 = 4 remainder 14 ÷ 2 = 2 remainder 02 ÷ 2 = 1 remainder 01 ÷ 2 = 0 remainder 1Binary number (read bottom to top): 10011100

So, \(156_{10} = 10011100_2\).

Other Conversion Methods (Brief Overview)

• Subtraction Method: Subtract the largest power of 2 less than or equal to the decimal number, mark a 1 in that position, and 0 in others. Repeat for the remainder.
• Using Place Values: Express the decimal number as a sum of powers of 2 and write 1 for those powers and 0 otherwise.

However, the division method is the easiest and most systematic for all decimal numbers.

Applications of Decimal to Binary Conversion

• Computers: All data in computers is stored and processed in binary form.
• Digital Electronics: Binary numbers are used in logic gates, flip-flops, and microcontrollers.
• Networking: IP addresses and subnet masks are often represented in binary.

Common Mistakes to Avoid

• Not reading the remainders in reverse order after division.
• Confusing the quotient and remainder during division steps.
• Forgetting that binary digits can only be 0 or 1.
• Misplacing the binary digits when writing the final answer.

Summary

To convert any decimal number to binary:

1. Divide the number by 2.
2. Record the remainder.
3. Repeat until quotient is zero.
4. Write the binary number by reading the remainders from last to first.

This method is reliable, easy to perform manually, and forms the basis for understanding binary representation in computers.

Section 3: Worked Examples

10 ÷ 2 = 5 remainder 05 ÷ 2 = 2 remainder 12 ÷ 2 = 1 remainder 01 ÷ 2 = 0 remainder 1Binary number (read bottom to top): 1010Answer: \(10_{10} = 1010_2\)  

45 ÷ 2 = 22 remainder 122 ÷ 2 = 11 remainder 011 ÷ 2 = 5 remainder 15 ÷ 2 = 2 remainder 12 ÷ 2 = 1 remainder 01 ÷ 2 = 0 remainder 1Binary number: 101101Answer: \(45_{10} = 101101_2\)  

100 ÷ 2 = 50 remainder 050 ÷ 2 = 25 remainder 025 ÷ 2 = 12 remainder 112 ÷ 2 = 6 remainder 06 ÷ 2 = 3 remainder 03 ÷ 2 = 1 remainder 11 ÷ 2 = 0 remainder 1Binary number: 1100100Answer: \(100_{10} = 1100100_2\)  

255 ÷ 2 = 127 remainder 1127 ÷ 2 = 63 remainder 163 ÷ 2 = 31 remainder 131 ÷ 2 = 15 remainder 115 ÷ 2 = 7 remainder 17 ÷ 2 = 3 remainder 13 ÷ 2 = 1 remainder 11 ÷ 2 = 0 remainder 1Binary number: 11111111Answer: \(255_{10} = 11111111_2\)  

156 ÷ 2 = 78 remainder 078 ÷ 2 = 39 remainder 039 ÷ 2 = 19 remainder 119 ÷ 2 = 9 remainder 19 ÷ 2 = 4 remainder 14 ÷ 2 = 2 remainder 02 ÷ 2 = 1 remainder 01 ÷ 2 = 0 remainder 1Binary number: 10011100Answer: \(156_{10} = 10011100_2\)  

Section 4: Formula Bank

• Decimal to Binary Conversion: Repeated division by 2, binary digits are remainders.
• Binary Number Representation: \[ N = \sum_{i=0}^{k} b_i \times 2^i, \quad b_i \in {0,1} \]
• Number of Binary Digits (Bits) Required: For decimal number \(N\), \[ \text{Number of bits} = \lfloor \log_2 N \rfloor + 1 \]
• Power Set Cardinality: For a set with \(n\) elements, \[ |\mathcal{P}(S)| = 2^n \]