A comprehensive guide on Number Systems for CBSE Class 11 Applied Mathematics, covering conversion methods between Decimal and Binary systems with step-by-step solved examples.
The decimal system is base-10, while the binary system is base-2. To convert a decimal number to binary, we repeatedly divide the decimal number by 2 and record the remainders. The binary representation is obtained by reading the remainders in reverse order (from the last remainder to the first).
• Divide 25 by 2: Quotient = 12, Remainder = 1
• Divide 12 by 2: Quotient = 6, Remainder = 0
• Divide 6 by 2: Quotient = 3, Remainder = 0
• Divide 3 by 2: Quotient = 1, Remainder = 1
• Divide 1 by 2: Quotient = 0, Remainder = 1
• Read remainders from bottom to top: 11001
Answer: The binary equivalent of 25 is 11001.
To convert a binary number to decimal, each digit is multiplied by its positional weight, which is a power of 2, starting from 2^0 for the rightmost bit. The sum of these values gives the equivalent decimal number.
• Write the positional values: 1*(2^3) + 0*(2^2) + 1*(2^1) + 1*(2^0)
• Calculate powers: 1*(8) + 0*(4) + 1*(2) + 1*(1)
• Sum the products: 8 + 0 + 2 + 1 = 11
Answer: The decimal equivalent of 1011 is 11.
The binary system is the foundation of digital technology. Computers use binary for data representation, storage, and processing because logic gates can easily distinguish between two states: ON (1) and OFF (0). It is essential for low-level programming, network addressing (IP addresses), and cryptography.
• Digital circuits consist of electronic components like transistors.
• Transistors can reliably exist in two states: high voltage (1) or low voltage (0).
• Binary allows for simple representation of complex data using only these two reliable states.
Answer: Binary is the fundamental language of computers due to its reliability in hardware representation.