Decimal to Binary Converter

What is Decimal to Binary Conversion?

Decimal to binary conversion is the process of transforming a number from the decimal system (base-10) to the binary system (base-2). This conversion is fundamental in computer science and digital systems, as computers operate using binary code.

Formula and Representation

The formula for converting a decimal number to binary is:

\[D_{10} = \sum_{i=0}^{n} b_i \cdot 2^i\]

Where:

• \(D_{10}\) is the decimal number
• \(b_i\) represents the binary digits (0 or 1)
• \(i\) is the position of the binary digit (starting from 0)
• \(n\) is the number of binary digits minus 1

Calculation Steps

To convert a decimal number to binary:

1. Divide the decimal number by 2
2. Keep track of the remainder (0 or 1)
3. Continue dividing the quotient by 2 until the quotient becomes 0
4. Read the remainders from bottom to top to get the binary number

Example

Let's convert the decimal number 13 to binary:

\[ \begin{align*} 13 \div 2 &= 6 \text{ remainder } 1 \\ 6 \div 2 &= 3 \text{ remainder } 0 \\ 3 \div 2 &= 1 \text{ remainder } 1 \\ 1 \div 2 &= 0 \text{ remainder } 1 \end{align*} \]

Reading the remainders from bottom to top, we get: 1101

Therefore, \(13_{10} = 1101_2\)

Visual Representation

This visual representation illustrates the conversion of the decimal number 13 to its binary equivalent 1101. The arrow indicates the transformation from base-10 to base-2, demonstrating how a single decimal number can be represented using only two digits (0 and 1) in the binary system.