Binary to decimal conversion_

1. Binary to decimal conversion: Binary to decimal conversion: Binary to decimal conversion is the process of converting a number from the binary numeral system (base-2) to the decimal numeral system (base-10). This is an essential concept in computer science and digital systems, where binary is the primary language of machines. How It Works: Binary Representation A binary number consists of 0s and 1s, where each digit represents a power of 2, starting with 202^020 from the rightmost digit (least significant bit). Example: Binary number 101010101010 Assign Powers of 2 Each binary digit corresponds to a power of 2 based on its position: (1×23)+(0×22)+(1×21)+(0×20)(1 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (0 \times 2^0)(1×23)+(0×22)+(1×21)+(0×20) Perform the Conversion Multiply each binary digit by its respective power of 2 and sum the results: (1×8)+(0×4)+(1×2)+(0×1)=8+0+2+0=10(1 \times 8) + (0 \times 4) + (1 \times 2) + (0 \times 1) = 8 + 0 + 2 + 0 = 10(1×8)+(0×4)+(1×2)+(0×1)=8+0+2+0=10 So, 101010101010 in binary is 101010 in decimal. Formula for Conversion: The decimal equivalent of a binary number bnbn−1…b1b0b_n b_{n-1} \ldots b_1 b_0bnbn−1…b1b0 is calculated as: Decimal=∑i=0nbi⋅2i\text{Decimal} = \sum_{i=0}^{n} b_i \cdot 2^iDecimal=i=0∑nbi⋅2i Where bib_ibi is the binary digit at position iii, and iii starts at 0 for the rightmost digit. Example Conversion:Convert 111011110111101 to Decimal: Assign powers of 2: 24,23,22,21,202^4, 2^3, 2^2, 2^1, 2^024,23,22,21,20 Multiply and add: (1×16)+(1×8)+(1×4)+(0×2)+(1×1)(1 \times 16) + (1 \times 8) + (1 \times 4) + (0 \times 2) + (1 \times 1)(1×16)+(1×8)+(1×4)+(0×2)+(1×1)16+8+4+0+1=2916 + 8 + 4 + 0 + 1 = 2916+8+4+0+1=29 Thus, 111011110111101 in binary equals 292929 in decimal. Applications:Programming: Translating binary data for software development.Digital Systems: Interpreting binary outputs from circuits or microcontrollers.Networking: Converting binary IP addresses or data packets to decimal for human readability. Binary to decimal conversion is an essential tool for understanding and working with digital systems effectively.